
We present a general computational framework that enables one to generate realistic 3D microstructure models of heterogeneous materials from limited morphological information via stochastic optimization. In our framework, the 3D material microstructure is represented as a 3D array, whose entries indicate the local state of that voxel. The limited structural data obtained in various experiments correspond to different mathematical transformations of the 3D array. Reconstructing the 3D material structure from such limited data is formulated as an inverse problem, originally proposed by Yeong and Torquato [Phys. Rev. E 57, 495 (1998)], which is solved using the simulated annealing procedure. The utility, versatility and robustness of our general framework are illustrated by reconstructing a polycrystalline microstructure from 2D EBSD micrographs and a binary metallic alloy from limited angle projections. Our framework can be also applied in the reconstructions based on small-angle x-ray scattering (SAXS) data and has ramifications in 4D materials science (e.g., charactering structural evolution over time).
Citation: Jiao Yang. Three-Dimensional Heterogeneous Material Microstructure Reconstruction from Limited Morphological Information via Stochastic Optimization[J]. AIMS Materials Science, 2014, 1(1): 28-40. doi: 10.3934/matersci.2014.1.28
[1] | Said G. Nassr, Amal S. Hassan, Rehab Alsultan, Ahmed R. El-Saeed . Acceptance sampling plans for the three-parameter inverted Topp–Leone model. Mathematical Biosciences and Engineering, 2022, 19(12): 13628-13659. doi: 10.3934/mbe.2022636 |
[2] | Ghada Mohammed Mansour, Haroon Mohamed Barakat, Islam Abdullah Husseiny, Magdy Nagy, Ahmed Hamdi Mansi, Metwally Alsayed Alawady . Measures of cumulative residual Tsallis entropy for concomitants of generalized order statistics based on the Morgenstern family with application to medical data. Mathematical Biosciences and Engineering, 2025, 22(6): 1572-1597. doi: 10.3934/mbe.2025058 |
[3] | Walid Emam, Ghadah Alomani . Predictive modeling of reliability engineering data using a new version of the flexible Weibull model. Mathematical Biosciences and Engineering, 2023, 20(6): 9948-9964. doi: 10.3934/mbe.2023436 |
[4] | Manal M. Yousef, Rehab Alsultan, Said G. Nassr . Parametric inference on partially accelerated life testing for the inverted Kumaraswamy distribution based on Type-II progressive censoring data. Mathematical Biosciences and Engineering, 2023, 20(2): 1674-1694. doi: 10.3934/mbe.2023076 |
[5] | M. Nagy, Adel Fahad Alrasheedi . The lifetime analysis of the Weibull model based on Generalized Type-I progressive hybrid censoring schemes. Mathematical Biosciences and Engineering, 2022, 19(3): 2330-2354. doi: 10.3934/mbe.2022108 |
[6] | Walid Emam, Khalaf S. Sultan . Bayesian and maximum likelihood estimations of the Dagum parameters under combined-unified hybrid censoring. Mathematical Biosciences and Engineering, 2021, 18(3): 2930-2951. doi: 10.3934/mbe.2021148 |
[7] | Lernik Asserian, Susan E. Luczak, I. G. Rosen . Computation of nonparametric, mixed effects, maximum likelihood, biosensor data based-estimators for the distributions of random parameters in an abstract parabolic model for the transdermal transport of alcohol. Mathematical Biosciences and Engineering, 2023, 20(11): 20345-20377. doi: 10.3934/mbe.2023900 |
[8] | M. G. M. Ghazal, H. M. M. Radwan . A reduced distribution of the modified Weibull distribution and its applications to medical and engineering data. Mathematical Biosciences and Engineering, 2022, 19(12): 13193-13213. doi: 10.3934/mbe.2022617 |
[9] | Wael S. Abu El Azm, Ramy Aldallal, Hassan M. Aljohani, Said G. Nassr . Estimations of competing lifetime data from inverse Weibull distribution under adaptive progressively hybrid censored. Mathematical Biosciences and Engineering, 2022, 19(6): 6252-6275. doi: 10.3934/mbe.2022292 |
[10] | Raquel Caballero-Águila, María J. García-Ligero, Aurora Hermoso-Carazo, Josefa Linares-Pérez . Unreliable networks with random parameter matrices and time-correlated noises: distributed estimation under deception attacks. Mathematical Biosciences and Engineering, 2023, 20(8): 14550-14577. doi: 10.3934/mbe.2023651 |
We present a general computational framework that enables one to generate realistic 3D microstructure models of heterogeneous materials from limited morphological information via stochastic optimization. In our framework, the 3D material microstructure is represented as a 3D array, whose entries indicate the local state of that voxel. The limited structural data obtained in various experiments correspond to different mathematical transformations of the 3D array. Reconstructing the 3D material structure from such limited data is formulated as an inverse problem, originally proposed by Yeong and Torquato [Phys. Rev. E 57, 495 (1998)], which is solved using the simulated annealing procedure. The utility, versatility and robustness of our general framework are illustrated by reconstructing a polycrystalline microstructure from 2D EBSD micrographs and a binary metallic alloy from limited angle projections. Our framework can be also applied in the reconstructions based on small-angle x-ray scattering (SAXS) data and has ramifications in 4D materials science (e.g., charactering structural evolution over time).
Weighted distributions (WDs) provide an approach to deal with model specification and data interpretations problems. They adjust the probabilities of the actual occurrence of events to arrive at a specification of the probabilities when those events are recorded. Reference [1] extended the basic ideas of the methods of ascertainment upon the estimation of frequencies in [2]. The author defined a unifying concept of the WDs and described several sample conditions that the WDs can model. The usefulness and applications of the WDs in various areas, including medicine, ecology, reliability, and branching processes, can also be seen in [3,4,5]. Important findings on the WDs have been reported by several research. For examples, reference [6] suggested a weighted x-gamma distribution, reference [7] derived a new generalized weighted Weibull distribution, reference [8] introduced the weighted exponential-Gompertz distribution, reference [9] studied the new weighted inverse Rayleigh distribution, reference [10] introduced a weighted version of the generalized inverse Weibull distribution, reference [11] proposed a bounded weighted exponential distribution, reference [12] derived a new weighted exponential distribution, reference [13] proposed a weighted power Lomax distribution, reference [14] derived a new generalized weighted exponential distribution, reference [15] introduced a new version of the weighted Weibull distribution, reference [16] proposed the modified weighted exponential distribution, and reference [17] proposed a weighted Nwikpe distribution, reference [18] introduced a new version of the double weighted quasi Lindley distribution and reference [19] proposed the modified length-biased weighted Lomax distribution.
In contrast, statistical models have the capacity to depict and predict real-world phenomena. Over the past few decades, numerous extended distributions have been extensively utilized in data modeling. Recent progress has been centered on the development of novel distribution families that not only enhance existing distributions but also offer significant versatility in practical data modeling. Engineering, economics, biology, and environmental science are particular examples of this. Regarding this, a number of writers suggested some of the created families of continuous distributions, (see for example [20,21,22]). Our interest here is in the same scheme used for the beta-G (B-G) family prepared in [23]. The following is the cumulative distribution function (cdf) for the B-G family:
$ F(x)=∫G(x)0r(t)dt, $
|
(1.1) |
where $ G(x) $ is a cdf of a continuous distribution and $ r(t) $ is the probability density function (pdf) of the beta distribution. Naturally, any new family can be created by taking another pdf for $ r(t) $ with support $ [0, 1] $ (see reference [23]).
As a matter of fact, few works about the weighted-G family have been proposed in the literature. For example, reference [24] studied the weighted exponential-G family, reference [25] introduced the weighted exponentiated family, reference [26] proposed a weighted general family, and reference [27] developed a weighted Topp-Leone-G family.
The primary purpose of this study is to introduce the length-biased truncated Lomax-G (LBTLo-G) family. The following arguments give enough motivation to study it:
$ 1) $ The LBTLo-G family is very flexible and simple.
$ 2) $ The LBTLo-G family contains some new distributions.
$ 3) $ The shapes of the pdfs of the generated distributions can be unimodal, decreasing, bathtub, right-skewed, and symmetric. Also, the hazard rate function (hrf) shapes for these distributions can be increasing, decreasing, U-shaped, upside-down-shaped, or J- shaped.
After emphasizing these important aspects, some statistical and mathematical properties of the newly suggested family are discussed. The maximum likelihood (ML) method of estimation is used to estimate the LBTLo Weibull (LBTLoW) model parameters based on complete and type Ⅱ censoring (T2C).
The variability of the LBTLoW distribution is demonstrated through four authentic data sets. The first data set describes age data on rest times (in minutes) for analgesic patients. The second data set shows the percentage of natural gas reserves in 44 countries in 2020. The third authentic data set listed the top 20 countries by oil reserves. Proven reserves refer to the quantities of petroleum that can be predicted as commercially recoverable from known reservoirs, based on the analysis of geological and engineering data. These estimates are made considering existing economic conditions and are projected from a specific period onwards. The fourth data set displays the top 100 central banks in terms of gold reserves. This gold reserve data, collected from IMF IFS figures, tracks central banks' reported gold purchases and sales as a percentage of their international reserves. The application results show that the LBTLoW distribution can indeed match the data better than other competing distributions.
The following is the structure for this article: Section 2 defines the crucial functions of the LBTLo-G family and provides four special distributions of the family. In Section 3, some statistical properties of the LBTLo-G family are provided. Section 4 deals with the ML estimates (MLEs) of the unknown parameters. A simulation study to examine the theoretical performance of MLEs for the LBTLoW distribution is studied in Section 5. Section 6 presents the applicability and goodness of fit of the proposed models using four real data sets. The paper ends with a few last observations, as may be seen in Section 7.
Here, we suggest a new weighted family based on the weighted version of the truncated Lomax distribution, which is called the LBTLo distribution [28]. The cdf and pdf of the LBTLo distribution are, respectively, given by
$ G(t;α)=Λ(α)[(1+t)−α(1+αt)−1],0<t<1,α>0, $
|
(2.1) |
$ g(t;α)=α(1−α)Λ(α)t(1+t)−(α+1), $
|
(2.2) |
where $ \Lambda\left(\alpha \right) = \left[2^{-\alpha } \; \; \left(1+\alpha \right)-1\right]^{-1}. $ For these functions, it is assumed the standard complementary values for $ t\le 0 $ and $ t\ge 1 $.
As mentioned in [28], the following advantages of the LBTLo distribution are outlined: (ⅰ) It depends on only one parameter; (ⅱ) the pdf has only one maximum point with a relatively sharp peak and a heavy tail; (ⅲ) the hrf has increasing behavior or is N-shaped; and (ⅳ) it outperforms some other competing models in real-world applications to medical data and the percentage of household spending on education out of total household expenditure from the Household Income, Expenditure, and Consumption Survey data for North Sinai Governorate.
In light of these merits, the LBTLo distribution is a great choice to use in various fields. As a consequence, we present a novel generated family that is based on the LBTLo distribution. In order to define the LBTLo-G family, let $ G(x; \zeta) $ and $ g(x; \zeta) $ be the baseline cdf and pdf, respectively, of a continuous distribution, and $ \zeta $ is a vector of parameters. The generalized B-G generator specified in (1.1) and the LBTLo distribution (2.2) are combined to generate the cdf of the LBTLo-G family:
$ F(x;α,ζ)=α(1−α)Λ(α)∫G(x;ζ)0t(1+t)−α−1dt=Λ(α)[(1+G(x;ζ))−α(1+αG(x;ζ))−1],x∈R,α>0, $
|
(2.3) |
where $ \alpha $ is a shape parameter. Therefore, the pdf of the LBTLo-G family is given by
$ f(x;α,ζ)=α(1−α)Λ(α)g(x;ζ)G(x;ζ)(1+G(x;ζ))−α−1,x∈R,α>0. $
|
(2.4) |
A random variable X with the pdf (2.4) is designated as $ X\sim $ LBTLo-G from here on out. The complementary cdf (ccdf), and hrf, are, provided by
$ S(x;α,ζ)=1−Λ(α)[(1+G(x;ζ))−α(1+αG(x;ζ))−1], $
|
$ h(x;α,ζ)=α(1−α)Λ(α)g(x)G(x)(1+G(x))−α−11−Λ(α)[(1+G(x;ζ))−α(1+αG(x;ζ))−1]. $
|
We create four new LBTLo-G family sub-distributions in the subsections that follow: LBTLo-inverse exponential, LBTLo-uniform, LBTLo-Weibull, and LBTLo-Kumaraswamy distributions.
The cdf and pdf of the LBTLo-inverse exponential (LBTLoIE) distribution are obtained from (2.3) and (2.4) for $ G\left(x; \beta \right) = e^{-(\beta /x)}, \beta, x > 0, $ as follows:
$ F(x;α,β)=Λ(α)[(1+e−(β/x))−α(1+αe−(β/x))−1],x>0,α,β>0, $
|
$ f(x;α,β)=α(1−α)Λ(α)βx−2e−2(β/x)(1+e−(β/x))−α−1. $
|
Further, the hrf is as follows:
$ h(x;α,β)=α(1−α)Λ(α)βx−2e−2(β/x)(1+e−(β/x))−α−11−Λ(α)[(1+e−(β/x))−α(1+αe−(β/x))−1]. $
|
The cdf and pdf of the LBTLo-uniform (LBTLoU) distribution are derived from (2.3) and (2.4) by taking $ G\left(x; \beta \right) = \beta ^{-1} x, 0 < x < \beta, $ as follows:
$ F(x;α,β)=[(1+αβ−1x)(1+β−1x)−α−1]Λ(α),0<x<β,α,β>0, $
|
$ f(x;α,β)=αβ−2x(1+β−1x)−α−1(1−α)Λ(α). $
|
Further, the hrf is as follows:
$ h(x;α,β)=αβ−2x(1+β−1x)−α−1(1−α)Λ(α)1−Λ(α)[(1+αβ−1x)(1+β−1x)−α−1]. $
|
The cdf and pdf of the LBTLoW distribution are derived from (2.3) and (2.4) taking $ G\left(x; \beta, \gamma \right) = 1-e^{-\beta x^{\gamma } }, \, \, x, \beta, \gamma > 0, $ as follows:
$ F(x;α,β,γ)=[(2−e−βxγ)−α(1+α−αe−βxγ)−1]Λ(α),x>0,α,β,γ>0, $
|
(2.5) |
$ f(x;α,β,γ)=αβγ(1−α)xγ−1e−βxγ(1−e−βxγ)(2−e−βxγ)−α−1Λ(α). $
|
(2.6) |
Further, the hrf is:
$ h(x;α,β,γ)=αβγ(1−α)xγ−1e−βxγ(1−e−βxγ)(2−e−βxγ)−α−1Λ(α)1−Λ(α)[(2−e−βxγ)−α(1+α−αe−βxγ)−1]. $
|
The cdf and pdf of the LBTLo- Kumaraswamy (LBTLoKw) distribution are obtained from (2.3) and (2.4) by taking $ G(x; \mu, b) = 1-(1-x^{\mu })^{b}, \, \, 0 < x < 1, b, \mu > 0, $ as follows:
$ F(x;α,μ,b)=[(2−(1−xμ)b)−α(1+α−α(1−xμ)b)−1]Λ(α),0<x<1,α,μ,b>0, $
|
$ f(x;α,μ,b)=αμb(1−α)xμ−1(1−xμ)b−1(1−(1−xμ)b)×(2−(1−xμ)b)−α−1Λ(α). $
|
Further, the hrf is as follows:
$ h(x;α,μ,b)=αμb(1−α)xμ−1(1−xμ)b−1(1−(1−xμ)b)Λ(α)(2−(1−xμ)b)−α−11−Λ(α)[(2−(1−xμ)b)−α(1+α−α(1−xμ)b)−1]. $
|
The plots of pdf and hrf for the LBTLoIE, LBTLoU, LBTLoW and LBTLoKw distributions are given in Figures 1 and 2, respectively.
The pdfs of the investigated distributions can have a variety of forms, including right- and left-skewed, bathtub, uni-modal, declining, and symmetric shapes, as shown in Figure 1. The corresponding hrf can take any form, including U, J, reverse J, growing, or decreasing, as seen in Figure 1.
In this part, we give some statistical properties of the LBTLo-G family.
The LBTLo-G family representations in pdf and cdf format are displayed here. The generalized binomial theorem says that
$ (1+z)−β=∞∑i=0(−1)i(β+i−1i)zi, $
|
(3.1) |
for $ |z| < 1 $. Hence, by using (3.1) in (2.4), the pdf of the LBTLo-G family can be written as follows:
$ f(x;α,ζ)=∞∑i=0ϑig(x;ζ)G(x;ζ)i+1,x∈R, $
|
(3.2) |
where $ \vartheta _{i} = \left(-1\right)^{i} \alpha \left(1-\alpha \right)\Lambda\left(\alpha \right)\left(α+ii
$ f(x;α,β,γ)=βγ∞∑i=0ϑixγ−1e−βxγ(1−e−βxγ)i+1,x>0,α,β,γ>0. $
|
(3.3) |
But, in the special case where $ b $ is a positive integer, the standard generalized binomial theorem says that
$ (1−z)b=b∑ν=0(−1)ν(bν)zν. $
|
(3.4) |
Then using the binomial expansion (3.4) in (3.3), we get
$ f(x;α,β,γ)=∞∑i=0i+1∑ν=0ϑi,νxγ−1e−β(ν+1)xγ, $
|
(3.5) |
where $ \vartheta _{i, \nu } = \beta \gamma \vartheta _{i} \left(-1\right)^{\nu } \left(i+1ν
$ F(x;α,ζ)h=Λ(α)h[(1+G(x;ζ))−α(1+αG(x;ζ))−1]h. $
|
(3.6) |
Using the binomial expansion (3.4) in (3.6), we get
$ F(x;α,ζ)h=Λ(α)hh∑j=0(−1)h−j(hj)(1+G(x;ζ))−αj(1+αG(x;ζ))j. $
|
(3.7) |
Using the binomial expansion (3.1), we obtain
$ F(x;α,ζ)h=Λ(α)h∞∑d=0h∑j=0(−1)d+h−j(hj)(αj+d−1d)×G(x;ζ)d(1+αG(x;ζ))j. $
|
(3.8) |
By using (3.4) in (3.8), we obtain
$ F(x;α,ζ)h=∞∑d=0ϖd,j,mG(x;ζ)d+m, $
|
(3.9) |
where $ \varpi _{{d, j, m}} = \sum _{j = 0}^{h}\sum _{m = 0}^{j}\left(-1\right)^{d+m+h-j} \alpha ^{m} \left(hj
For example, the expansion of the cdf of the LBTLoW distribution is derived from (3.9), where $ G(x, \zeta) = 1-e^{-\beta x^{\gamma } } $, as follows:
$ F(x;α,β,γ)h=∞∑d=0ϖd,j,m(1−e−βxγ)d+m. $
|
By using the binomial expansion (3.4) in the last term of the previous equation, we get
$ F(x;α,β,γ)h=∞∑d=0d+m∑l=0(−1)l(d+ml)ϖd,j,me−βlxγ. $
|
(3.10) |
The above representations are of interest to express various important moment measures as series. By truncating the index of summation, we can have a precise approximation with a reasonable computation cost.
As a special class of moments, the probability weighted moments (PWMs) have been proposed in [29]. This class is used to derive estimates of the parameters and quantiles of distributions expressible in inverse form. Let $ X $ be a random variable with pdf and cdf $ f(x) $ and $ F(x) $, respectively, and $ r $ and $ q $ be non-negative integers. Then, the $ (r, q)^{th} $ PWM of $ X $, denoted by $ {\pi }_{r, q} $, can be calculated through the following relation:
$ πr,q=E[XrF(X)q]=∫∞−∞xrf(x)F(x)qdx. $
|
(3.11) |
On this basis, the $ (r, q)^{th} $ PMW of $ X $ with pdf and cdf of the LBTLo-G family is obtained by substituting (3.2) and (3.9) into (3.11), as follows:
$ πr,q=E[XrF(X;α,ζ)q]=∫∞−∞∞∑i,d=0ϑiϖd,j,mxrg(x;ζ)[G(x;ζ)]i+d+m+1dx. $
|
Then, provided that the interchange of the integral and sum is valid, depending on the definitions of $ g\left(x; \zeta \right) $ and $ G\left(x; \zeta \right) $, we have
$ πr,q=∞∑i,d=0ϑiϖd,j,mρr,i+d+m+1, $
|
where
$ ρr,i+d+m+1=∫∞−∞xrg(x;ζ)[G(x;ζ)]i+d+m+1dx. $
|
For example, the $ (r, q)^{th} $ of a random variable $ X $ that follows the LBTLoW distribution can be obtained by substituting (3.5) and (3.10) into (3.11), and replacing $ h $ with $ q $. We thus obtain
$ πr,q=∞∑i,d=0i+1∑v=0d+m∑l=0(−1)lϑi,vϖd,j,m(β(ν+l+1))rγ+1(d+ml)Γ(rγ+1), $
|
where $ \Gamma \left(.\right) $ stands for gamma function.
In this part, for any non-negative integer $ r $, the $ r^{th} $ moment associated with the LBTLo-G family is derived.
Let $ X $ be a random variable having the pdf of the LBTLo-G family. Then, the $ r^{th} $ moment of $ X $ is obtained as follows:
$ μ′r=E(Xr)=∫∞−∞∞∑i=0ϑixrg(x;ζ)[G(x;ζ)]i+1dx=∞∑i=0ϑiυr,i+1, $
|
where $ \upsilon _{r, i+1} $ is the $ (r, i+1)^{th} $ PWM of the baseline distribution. For example, after some developments, the $ r^{th} $ moment associated with LBTLoW distribution is given by
$ μ′r=∞∑i=0i+1∑ν=0ϑi,ν(β(ν+1))rγ+1Γ(rγ+1). $
|
Tables 1–3 show the numerical values of the first four moments $ \mu '_{1} $, $ \mu '_{2} $, $ \mu '_{3} $, $ \mu '_{4}, $ also the numerical values of variance ($ {\sigma }^{\mathrm{2}} $), coefficient of skewness (CS), coefficient of kurtosis (CK) and coefficient of variation (CV) associated with the LBTLoW and LBTLoIE distribution.
$ \gamma $ | $ \alpha $ | $ \mu'_1 $ | $ \mu'_2 $ | $ \mu'_3 $ | $ \mu'_4 $ | $ {\sigma }^{\mathrm{2}} $ | $ CS $ | $ CK $ | $ CV $ |
0.4 | 0.2 | 2.629 | 8.323 | 30.531 | 126.387 | 1.413 | 0.729 | 3.596 | 0.452 |
0.6 | 0.5 | 1.558 | 2.937 | 6.430 | 15.922 | 0.508 | 0.746 | 3.627 | 0.457 |
0.8 | 0.8 | 1.184 | 1.700 | 2.847 | 5.403 | 0.300 | 0.765 | 3.660 | 0.463 |
1.1 | 1.2 | 0.925 | 1.044 | 1.380 | 2.074 | 0.189 | 0.793 | 3.717 | 0.470 |
1.3 | 1.5 | 0.805 | 0.794 | 0.920 | 1.216 | 0.146 | 0.814 | 3.758 | 0.475 |
1.7 | 1.8 | 0.705 | 0.615 | 0.634 | 0.748 | 0.117 | 0.856 | 3.853 | 0.486 |
1.9 | 2.0 | 0.655 | 0.532 | 0.513 | 0.568 | 0.103 | 0.878 | 3.906 | 0.491 |
2.4 | 2.3 | 0.582 | 0.425 | 0.371 | 0.374 | 0.086 | 0.937 | 4.058 | 0.503 |
2.7 | 2.6 | 0.530 | 0.355 | 0.285 | 0.266 | 0.073 | 0.974 | 4.162 | 0.511 |
3.2 | 3.0 | 0.469 | 0.280 | 0.203 | 0.172 | 0.060 | 1.038 | 4.359 | 0.522 |
$ \gamma $ | $ \alpha $ | $ \mu'_1 $ | $ \mu'_2 $ | $ \mu'_3 $ | $ \mu'_4 $ | $ {\sigma }^{\mathrm{2}} $ | $ CS $ | $ CK $ | $ CV $ |
0.4 | 0.2 | 1.964 | 4.277 | 10.154 | 25.945 | 0.420 | 0.380 | 3.017 | 0.330 |
0.6 | 0.5 | 1.347 | 2.017 | 3.298 | 5.813 | 0.202 | 0.395 | 3.026 | 0.334 |
0.8 | 0.8 | 1.105 | 1.360 | 1.829 | 2.658 | 0.139 | 0.411 | 3.037 | 0.337 |
1.1 | 1.2 | 0.924 | 0.954 | 1.081 | 1.325 | 0.100 | 0.435 | 3.058 | 0.343 |
1.3 | 1.5 | 0.836 | 0.783 | 0.805 | 0.897 | 0.084 | 0.453 | 3.075 | 0.346 |
1.7 | 1.8 | 0.760 | 0.649 | 0.612 | 0.627 | 0.072 | 0.489 | 3.116 | 0.353 |
1.9 | 2.0 | 0.720 | 0.584 | 0.524 | 0.511 | 0.066 | 0.508 | 3.140 | 0.356 |
2.4 | 2.3 | 0.661 | 0.494 | 0.411 | 0.374 | 0.058 | 0.557 | 3.213 | 0.365 |
2.7 | 2.6 | 0.617 | 0.433 | 0.339 | 0.290 | 0.052 | 0.588 | 3.265 | 0.369 |
3.2 | 3.0 | 0.565 | 0.364 | 0.263 | 0.209 | 0.045 | 0.641 | 3.366 | 0.377 |
$ \beta $ | $ \alpha $ | $ \mu'_1 $ | $ \mu'_2 $ | $ \mu'_3 $ | $ \mu'_4 $ | $ {\sigma }^{\mathrm{2}} $ | $ CS $ | $ CK $ | $ CV $ |
1.5 | 0.2 | 0.029 | 0.026 | 0.023 | 0.021 | 0.025 | 5.273 | 29.151 | 5.386 |
0.5 | 0.032 | 0.028 | 0.025 | 0.023 | 0.027 | 5.014 | 26.463 | 5.138 | |
0.8 | 0.035 | 0.031 | 0.028 | 0.025 | 0.030 | 4.772 | 24.069 | 4.906 | |
1.2 | 0.039 | 0.035 | 0.031 | 0.028 | 0.033 | 4.472 | 21.275 | 4.622 | |
1.5 | 0.043 | 0.038 | 0.033 | 0.03 | 0.036 | 4.264 | 19.441 | 4.424 | |
1.8 | 0.046 | 0.041 | 0.036 | 0.033 | 0.038 | 4.069 | 17.801 | 4.241 | |
2 | 0.049 | 0.043 | 0.038 | 0.034 | 0.040 | 3.946 | 16.806 | 4.125 | |
2.3 | 0.053 | 0.046 | 0.041 | 0.037 | 0.043 | 3.771 | 15.445 | 3.961 | |
2.6 | 0.056 | 0.049 | 0.044 | 0.039 | 0.046 | 3.607 | 14.225 | 3.809 | |
3 | 0.062 | 0.054 | 0.048 | 0.043 | 0.050 | 3.405 | 12.794 | 3.621 | |
2.5 | 0.2 | 0.006 | 0.005 | 0.005 | 0.004 | 0.005 | 12.420 | 156.328 | 12.403 |
0.5 | 0.007 | 0.006 | 0.006 | 0.005 | 0.006 | 11.672 | 138.195 | 11.665 | |
0.8 | 0.007 | 0.007 | 0.006 | 0.006 | 0.007 | 10.981 | 122.434 | 10.984 | |
1.2 | 0.009 | 0.008 | 0.007 | 0.007 | 0.008 | 10.140 | 104.548 | 10.156 | |
1.5 | 0.01 | 0.009 | 0.008 | 0.007 | 0.009 | 9.563 | 93.121 | 9.589 | |
1.8 | 0.011 | 0.010 | 0.009 | 0.008 | 0.010 | 9.030 | 83.140 | 9.066 | |
2 | 0.012 | 0.011 | 0.010 | 0.009 | 0.010 | 8.697 | 77.190 | 8.739 | |
2.3 | 0.013 | 0.012 | 0.011 | 0.010 | 0.012 | 8.227 | 69.194 | 8.279 | |
2.6 | 0.014 | 0.013 | 0.012 | 0.011 | 0.013 | 7.792 | 62.179 | 7.853 | |
3 | 0.017 | 0.015 | 0.014 | 0.013 | 0.015 | 7.261 | 54.128 | 7.334 |
It can be seen from Tables 1–3 that, when the value of $ \alpha, \gamma $ increases for a fixed value of $ \beta $, the first four moments and $ {\sigma }^{\mathrm{2}} $ decrease, while the CS, CK, and CV measures increase. When the value of $ \beta $ increases for a fixed value of $ \alpha $ and $ \gamma $, we observe that the first four moments and $ \sigma $ decrease and then increase, while the CS, CK, and CV measures increase. The LBTLoW distribution is skewed to the right by leptokurtic curves.
Furthermore, if $ X $ is a random variable having the pdf of the LBTLo-G family, then the $ r^{th} $ incomplete moment of $ X $ is obtained as follows:
$ φr(t)=E(XrI{X≤t})=∫t−∞xrf(x;α,ζ)dx=∫t−∞∞∑i=0ϑixrg(x;ζ)G(x;ζ)i+1dx. $
|
For example, after some developments, the $ r^{th} $ incomplete moment associated with the LBTLoW distribution is given by
$ φr(t)=∞∑i=0i+1∑ν=0ϑi,ν[β(ν+1)]rγ+1Γ(rγ+1,β(ν+1)tγ), $
|
where $ \Gamma \left(., x\right) $ is the lower incomplete gamma function.
Here, some uncertainty measures of the LBTLo-G family are derived. Then, these measures are specialized to the LBTLoW distribution. To begin, the Rényi entropy (RE), presented in [30], associated with a distribution with pdf $ f(x) $, is defined by
$ IR(ε)=11−εlog[∫∞−∞f(x)εdx],ε≠1,ε>0. $
|
A numerical study with integral calculus is possible; here, we focus on a series expansion. In what follows, an expansion for $ f\left(x; \alpha, \zeta \right)^{\varepsilon } $ is derived, for $ \varepsilon $ is a non-integer (again, the generalized binomial expansion is worked out):
$ f(x;α,ζ)ε=∞∑i=0Δig(x;ζ)εG(x;ζ)i+ε, $
|
where
$ Δi=(−1)i[α(1−α)Λ(α)]ε(ε(α+1)+i−1i). $
|
Then, the RE associated with the LBTLo-G family is given by
$ IR(ε)=(1−ε)−1log{∫∞−∞∞∑i=0Δig(x;ζ)εG(x;ζ)i+εdx}. $
|
For example, the RE associated with the LBTLoW distribution can be obtained as follows:
$ IR(ε)=(1−ε)−1log{∞∑i,j=0Δi,jγ[β(ε+j)]−ε+(ε/γ)−(1/γ)Γ(ε−εγ+1γ)}. $
|
The Havrda and Charvát entropy (HaCE) (see [31]) associated with a distribution with pdf $ f(x) $ is defined by
$ HCR(ε)=121−ε−1({∫∞−∞f(x)εdx}1/ε−1),ε≠1,ε>0. $
|
Hence, the HaCE of the LBTLo-G family is given by
$ HC_{R} \left(\varepsilon \right) = \frac{1}{2^{1-\varepsilon } -1} \left(\left\{ \int _{-\infty }^{\infty }\sum\limits_{i = 0}^{\infty }\Delta _{i} g\left(x;\zeta \right)^{\varepsilon } G\left(x;\zeta \right)^{i+\varepsilon } dx\right\}^{{1/ \varepsilon } } -1\right). $ |
For example, the HaCE of the LBTLoW distribution can be obtained as follows:
$ HCR(ε)=121−ε−1({∞∑i,j=0Δi,jγ[β(ε+j)]−ε+(ε/γ)−(1/γ)Γ(ε−εγ+1γ)}1/ε−1). $
|
The Arimoto entropy (ArE) (see [32]) associated with a distribution with pdf $ f(x) $ is defined by
$ AR(ε)=ε1−ε({∫∞−∞f(x)εdx}1/ε−1),ε≠1,ε>0. $
|
Hence, the ArE of the LBTLo-G family is given by
$ AR(ε)=ε1−ε({∫∞−∞∞∑i=0Δig(x;ζ)εG(x;ζ)i+εdx}1/ε−1). $
|
For example, the ArE of the LBTLoW distribution can be obtained as follows:
$ AR(ε)=ε1−ε({∞∑i,j=0Δi,jγ[β(ε+j)]−ε+(ε/γ)−(1/γ)Γ(ε−εγ+1γ)}1/ε−1). $
|
The Tsallis entropy (TsE) (see [33]) associated with a distribution with pdf $ f(x) $, is defined by
$ TR(ε)=1ε−1{1−∫∞−∞f(x)εdx},ε≠1,ε>0. $
|
Hence, the TsE of the LBTLo-G family is obtained as follows:
$ TR(ε)=1ε−1{1−∫∞−∞∞∑i=0Δig(x;ζ)εG(x;ζ)i+εdx}. $
|
For example, the TsE of the LBTLoW distribution can be obtained as follows:
$ TR(ε)=1ε−1{1−∞∑i,j=0Δi,jγ[β(ε+j)]−ε+(ε/γ)−(1/γ)Γ(ε−εγ+1γ)}. $
|
Some numerical values for the proposed entropy measures are obtained for the LBTLoW and LBTLoIE distribution in Tables 4 and 5.
$ \varepsilon $ | $ \beta $ | $ \alpha $ | $ \gamma $ | RE | HaCE | ArE | TsE |
1.5 | 0.25 | 0.2 | 0.4 | 3.331 | 3.340 | 2.767 | 1.957 |
0.5 | 0.6 | 3.252 | 3.333 | 2.753 | 1.953 | ||
0.8 | 0.8 | 3.099 | 3.318 | 2.722 | 1.944 | ||
1.2 | 1.1 | 2.875 | 3.290 | 2.670 | 1.927 | ||
1.5 | 1.3 | 2.716 | 3.264 | 2.627 | 1.912 | ||
1.8 | 1.7 | 2.524 | 3.227 | 2.568 | 1.891 | ||
2.0 | 1.9 | 2.415 | 3.203 | 2.53 | 1.876 | ||
2.3 | 2.4 | 2.229 | 3.152 | 2.458 | 1.846 | ||
2.6 | 2.7 | 2.084 | 3.104 | 2.394 | 1.818 | ||
3.0 | 3.2 | 1.889 | 3.026 | 2.296 | 1.773 | ||
0.5 | 0.2 | 0.4 | 1.930 | 3.044 | 2.318 | 1.783 | |
0.5 | 0.6 | 1.924 | 3.042 | 2.315 | 1.782 | ||
0.8 | 0.8 | 1.824 | 2.996 | 2.260 | 1.755 | ||
1.2 | 1.1 | 1.660 | 2.909 | 2.161 | 1.704 | ||
1.5 | 1.3 | 1.541 | 2.835 | 2.081 | 1.661 | ||
1.8 | 1.7 | 1.392 | 2.726 | 1.969 | 1.597 | ||
2.0 | 1.9 | 1.308 | 2.657 | 1.901 | 1.556 | ||
2.3 | 2.4 | 1.161 | 2.517 | 1.770 | 1.475 | ||
2.6 | 2.7 | 1.051 | 2.397 | 1.661 | 1.404 | ||
3.0 | 3.2 | 0.903 | 2.207 | 1.500 | 1.293 | ||
2.0 | 0.25 | 0.2 | 0.4 | 2.180 | 1.987 | 1.837 | 0.993 |
0.5 | 0.6 | 2.167 | 1.986 | 1.835 | 0.993 | ||
0.8 | 0.8 | 2.053 | 1.982 | 1.812 | 0.991 | ||
1.2 | 1.1 | 1.876 | 1.973 | 1.769 | 0.987 | ||
1.5 | 1.3 | 1.753 | 1.965 | 1.734 | 0.982 | ||
1.8 | 1.7 | 1.595 | 1.949 | 1.681 | 0.975 | ||
2.0 | 1.9 | 1.510 | 1.938 | 1.648 | 0.969 | ||
2.3 | 2.4 | 1.358 | 1.912 | 1.581 | 0.956 | ||
2.6 | 2.7 | 1.247 | 1.887 | 1.524 | 0.943 | ||
3.0 | 3.2 | 1.098 | 1.840 | 1.435 | 0.920 | ||
0.5 | 0.2 | 0.4 | 1.210 | 1.877 | 1.503 | 0.938 | |
0.5 | 0.6 | 1.279 | 1.895 | 1.541 | 0.947 | ||
0.8 | 0.8 | 1.210 | 1.877 | 1.503 | 0.938 | ||
1.2 | 1.1 | 1.080 | 1.834 | 1.423 | 0.917 | ||
1.5 | 1.3 | 0.987 | 1.794 | 1.358 | 0.897 | ||
1.8 | 1.7 | 0.861 | 1.725 | 1.258 | 0.862 | ||
2.0 | 1.9 | 0.795 | 1.679 | 1.199 | 0.840 | ||
2.3 | 2.4 | 0.672 | 1.574 | 1.077 | 0.787 | ||
2.6 | 2.7 | 0.587 | 1.482 | 0.982 | 0.741 | ||
3.0 | 3.2 | 0.470 | 1.323 | 0.836 | 0.661 |
$ \varepsilon $ | $ \beta $ | $ \alpha $ | RE | HaCE | ArE | TsE |
1.5 | 0.25 | 0.2 | 7.001 | 3.311 | 2.709 | 1.94 |
0.5 | 7.075 | 3.315 | 2.716 | 1.942 | ||
0.8 | 7.156 | 3.319 | 2.724 | 1.944 | ||
1.2 | 7.274 | 3.324 | 2.735 | 1.947 | ||
1.5 | 7.372 | 3.329 | 2.743 | 1.95 | ||
1.8 | 7.476 | 3.333 | 2.752 | 1.952 | ||
2 | 7.55 | 3.336 | 2.758 | 1.954 | ||
2.3 | 7.667 | 3.34 | 2.767 | 1.957 | ||
2.6 | 7.792 | 3.345 | 2.777 | 1.959 | ||
3 | 7.971 | 3.351 | 2.79 | 1.963 | ||
0.4 | 0.2 | 6.441 | 3.278 | 2.65 | 1.92 | |
0.5 | 6.452 | 3.279 | 2.651 | 1.921 | ||
0.8 | 6.469 | 3.28 | 2.653 | 1.921 | ||
1.2 | 6.503 | 3.282 | 2.657 | 1.923 | ||
1.5 | 6.536 | 3.284 | 2.66 | 1.924 | ||
1.8 | 6.577 | 3.287 | 2.665 | 1.925 | ||
2 | 6.608 | 3.289 | 2.668 | 1.927 | ||
2.3 | 6.661 | 3.292 | 2.674 | 1.928 | ||
2.6 | 6.721 | 3.296 | 2.681 | 1.931 | ||
3 | 6.813 | 3.301 | 2.69 | 1.934 | ||
2.0 | 0.25 | 0.2 | 4.376 | 1.975 | 1.776 | 0.987 |
0.5 | 4.429 | 1.976 | 1.782 | 0.988 | ||
0.8 | 4.487 | 1.977 | 1.788 | 0.989 | ||
1.2 | 4.57 | 1.979 | 1.796 | 0.99 | ||
1.5 | 4.639 | 1.981 | 1.803 | 0.99 | ||
1.8 | 4.713 | 1.982 | 1.81 | 0.991 | ||
2 | 4.765 | 1.983 | 1.815 | 0.991 | ||
2.3 | 4.847 | 1.984 | 1.823 | 0.992 | ||
2.6 | 4.934 | 1.986 | 1.83 | 0.993 | ||
3 | 5.058 | 1.987 | 1.841 | 0.994 | ||
0.4 | 0.2 | 3.975 | 1.962 | 1.726 | 0.981 | |
0.5 | 3.987 | 1.963 | 1.728 | 0.981 | ||
0.8 | 4.003 | 1.963 | 1.73 | 0.982 | ||
1.2 | 4.031 | 1.965 | 1.734 | 0.982 | ||
1.5 | 4.058 | 1.965 | 1.737 | 0.983 | ||
1.8 | 4.09 | 1.967 | 1.741 | 0.983 | ||
2 | 4.114 | 1.967 | 1.744 | 0.984 | ||
2.3 | 4.154 | 1.969 | 1.749 | 0.984 | ||
2.6 | 4.199 | 1.97 | 1.755 | 0.985 | ||
3 | 4.267 | 1.972 | 1.763 | 0.986 |
We can see from these tables that, as the value of $ \varepsilon $ rises, all entropy values decrease, providing more information. For a fixed value of $ \beta $, as the values of $ \alpha $ and $ \gamma $ rise, we infer that all entropy metrics decrease, indicating that there is less fluctuation. Additionally, we deduce that all entropies have less variability as the values of $ \alpha $, $ \gamma $ and $ \beta $ increase. When compared to other measures, the TsE measure values typically have the smallest values.
Let $ x_{(1)} \le x_{(2)} \le \ldots\le x_{(n)} $ be a T2C of size $ r $ resulting from a life test on $ n $ items whose lifetimes are described by the LBTLo-G family with a given set of parameters $ \alpha $ and $ \zeta $, see [34,35,36,37]. The log-likelihood function of $ r $ failures and ($ n-r $) censored values, is given by
$ logL(α,ζ)=rlogα+rlog(1−α)+rlogΛ(α)+r∑i=1logg(xi;ζ)+r∑i=1logG(xi;ζ)−(α+1)r∑i=1log(1+G(xi;ζ))+(n−r)log[Ar(α,ζ)], $
|
where $ A_{r} (\alpha, \zeta) = 1-\Lambda\left(\alpha \right)\left[\left(1+G\left(x_{r}; \zeta \right)\right)^{-\alpha } \left(1+\alpha G\left(x_{r}; \zeta \right)\right)-1\right] $, and we write $ x_{(i)} = x_{i} $ for simplified form.
By maximizing the previous likelihood function, the MLEs of unknown parameters are determined. To achieve this, we can first compute the first derivative of the score function $ \left(U_{\alpha }, \, U_{\zeta _{k} } \right) $, given as follows:
$ Uα=rα−r1−α+rΛ(α)(∂∂αΛ(α))−r∑i=1log(1+G(xi;ζ))+(n−r)Ar(α,ζ)(∂∂αAr(α,ζ)), $
|
$ Uζk=−(α+1)r∑i=111+G(xi;ζ)∂∂ζk(G(xi;ζ))+(n−r)Ar(α,ζ)∂∂ζkAr(α,ζ), $
|
where
$ \frac{\partial }{\partial \alpha } \Lambda\left(\alpha \right) = \left[\Lambda\left(\alpha \right)\right]^{2} 2^{-\alpha } \left[ \left(1+\alpha \right)\log 2-1\right], $ |
$ ∂∂αAr(α,ζ)=−∂∂αΛ(α)[(1+G(xr;ζ))−α(1+αG(xr;ζ))−1]+Λ(α)[(1+G(xr;ζ))−α{(1+αG(xr;ζ))log(1+G(xr;ζ))−G(xr;ζ)}], $
|
and
$ \frac{\partial }{\partial _{\zeta _{k} } } \left(A_{r} (\alpha , \zeta )\right) = \alpha \left(\alpha -1\right)\Lambda\left(\alpha \right)G\left(x_{r} ;\zeta \right)\left(1+G\left(x_{r} ;\zeta \right)\right)^{-\alpha -1} \frac{\partial }{\partial _{\zeta _{k} } } G\left(x_{r} ;\zeta \right). $ |
By putting $ U_{\alpha } $ and $ U_{\zeta _{k} } $ equal to zero and solving these equations simultaneously, the MLEs of the LBTLo-G family are found. These equations are not amenable to analytical solution, however they are amenable to numerical solution by iterative techniques utilizing statistical software.
The confidence interval (CI) of the vector of the unknown parameters $ \xi = (\alpha, \, \zeta) $ could be obtained from the asymptotic distribution of the MLEs of the parameters as $ \left(\hat{\xi }_{MLE} -\xi \right)\to N_{2} \left(0, I^{-1} \left(\hat{\xi }_{MLE} \right)\right), $ where $ I\left(\xi \right) $ is the Fisher information matrix. Under particular regularity conditions, the two-sided $ 100\left(1-v\right)%, 0 < v < 1 $, asymptotic CI for the vector of unknown parameters $ \xi $ can be acquired in the following ways: $ \hat{\xi }_{MLE} \pm z_{{v/ 2} } {}_{} \sqrt{var(\hat{\xi })} $, where $ var(\hat{\xi }) $ is the element of the main diagonal of the asymptotic variance-covariance matrix $ I^{-1} \left(\hat{\xi }_{MLE} \right) $ and $ z_{{v/ 2} } $ is the upper $ {v^{th} / 2} $ percentile of the standard normal distribution.
This section includes a simulation study to evaluate the performance of the MLEs for the LBTLoW model $ (\alpha, \beta, \gamma), $ for complete and T2C. The Mathematica 9 package is used to get the mean squared error (MSE), lower bound (LB) of CI, upper bound (UB) of CI, average length (AL) of 95%, and coverage probability (CP) of 95% of the estimated values of $ \alpha $, $ \beta $ and $ \gamma $. The algorithm is developed in the way described below:
$ 1) $ From the LBTLoW distribution, 5000 random samples of sizes $ n $ = 50,100,150, and 200 are created.
$ 2) $ Values of the unknown parameters $ \left(\alpha, \beta, \gamma \right) $ are selected as Set 1 $ = (\alpha = 0.5, \beta = 0.5, \gamma = 0.5) $, Set 2 $ = \left(\alpha = 0.7, \beta = 0.5, \gamma = 0.25\right) $, Set 3 $ = \left(\alpha = 0.7, \beta = 0.7, \gamma = 0.5\right) $, and Set 4 $ = \left(\alpha = 0.6, \beta = 0.3, \gamma = 0.5\right) $.
$ 3) $ Three levels of censorship are chosen: $ r $ = 70%, 80% (T2C), and 100% (complete sample).
$ 4) $ The MLEs, Biases, and MSEs for all sample sizes and for all selected sets of parameters are computed. Furthermore, the LB, UB, AL, and CP with a confidence level of 0.95 for all sample sizes and for all selected sets of parameters are calculated.
$ 5) $ Numerical outcomes are reported in Table 6. Based on complete and T2C samples, we can detect the following about the performance of the estimated parameters.
n | r | Set1 ($ \alpha $ = 0.5, $ \beta $ = 0.5, $ \gamma $ = 0.5) | |||||||
MLE | Bias | MSE | LB | UB | AL | CP | |||
50 | 70% | $ \alpha $ | 0.4204 | 0.0796 | 0.0064 | 0.0019 | 0.839 | 0.8370 | 97.4% |
$ \beta $ | 0.7041 | 0.2041 | 0.0471 | 0.5036 | 0.9046 | 0.4010 | 96.9% | ||
$ \gamma $ | 0.4201 | 0.0799 | 0.0069 | 0.3023 | 0.5379 | 0.2356 | 96.0% | ||
80% | $ \alpha $ | 0.4218 | 0.0782 | 0.0061 | 0.0191 | 0.8245 | 0.8053 | 94.8% | |
$ \beta $ | 0.6382 | 0.1382 | 0.0242 | 0.4508 | 0.8256 | 0.3748 | 95.8% | ||
$ \gamma $ | 0.4386 | 0.0614 | 0.0053 | 0.3282 | 0.5490 | 0.2208 | 97.1% | ||
100% | $ \alpha $ | 0.4234 | 0.0766 | 0.0059 | 0.0357 | 0.8111 | 0.7754 | 95.4% | |
$ \beta $ | 0.5177 | 0.0177 | 0.0056 | 0.3661 | 0.6694 | 0.3033 | 95.5% | ||
$ \gamma $ | 0.5316 | 0.0316 | 0.0027 | 0.4303 | 0.6328 | 0.2025 | 96.0% | ||
100 | 70% | $ \alpha $ | 0.4213 | 0.0787 | 0.0062 | 0.0844 | 0.7583 | 0.6740 | 96.2% |
$ \beta $ | 0.6750 | 0.1750 | 0.0312 | 0.5375 | 0.8125 | 0.2750 | 95.9% | ||
$ \gamma $ | 0.4237 | 0.0763 | 0.0065 | 0.3389 | 0.5084 | 0.1694 | 96.0% | ||
80% | $ \alpha $ | 0.4230 | 0.0770 | 0.0061 | 0.2099 | 0.6360 | 0.4262 | 96.2% | |
$ \beta $ | 0.6099 | 0.1099 | 0.0127 | 0.4819 | 0.7379 | 0.2560 | 96.1% | ||
$ \gamma $ | 0.4487 | 0.0513 | 0.0033 | 0.3652 | 0.5321 | 0.1669 | 97.3% | ||
100% | $ \alpha $ | 0.4238 | 0.0762 | 0.0058 | 0.2501 | 0.5975 | 0.3473 | 95.6% | |
$ \beta $ | 0.4683 | 0.0317 | 0.0027 | 0.3558 | 0.5807 | 0.2249 | 95.8% | ||
$ \gamma $ | 0.4967 | 0.0033 | 0.0025 | 0.4199 | 0.5734 | 0.1535 | 96.0% | ||
150 | 70% | $ \alpha $ | 0.4217 | 0.0783 | 0.0061 | 0.2710 | 0.5725 | 0.3015 | 95.2% |
$ \beta $ | 0.6626 | 0.1626 | 0.0281 | 0.5577 | 0.7675 | 0.2097 | 95.6% | ||
$ \gamma $ | 0.4277 | 0.0723 | 0.0058 | 0.3571 | 0.4983 | 0.1412 | 97.3% | ||
80% | $ \alpha $ | 0.4236 | 0.0764 | 0.0059 | 0.3005 | 0.5466 | 0.2461 | 95.7% | |
$ \beta $ | 0.5977 | 0.0977 | 0.0113 | 0.4957 | 0.6997 | 0.2040 | 96.2% | ||
$ \gamma $ | 0.4649 | 0.0351 | 0.0022 | 0.3972 | 0.5325 | 0.1353 | 97.0% | ||
100% | $ \alpha $ | 0.4238 | 0.0762 | 0.0058 | 0.3010 | 0.5467 | 0.2457 | 95.6% | |
$ \beta $ | 0.4766 | 0.0234 | 0.0023 | 0.3784 | 0.5749 | 0.1965 | 96.4% | ||
$ \gamma $ | 0.5277 | 0.0277 | 0.0015 | 0.4659 | 0.5894 | 0.1236 | 96.9% | ||
200 | 70% | $ \alpha $ | 0.4219 | 0.0781 | 0.0061 | 0.3154 | 0.5285 | 0.2132 | 96.1% |
$ \beta $ | 0.6592 | 0.1592 | 0.0268 | 0.5675 | 0.7510 | 0.1835 | 96.3% | ||
$ \gamma $ | 0.4375 | 0.0625 | 0.0046 | 0.3789 | 0.4962 | 0.1173 | 96.7% | ||
80% | $ \alpha $ | 0.4239 | 0.0761 | 0.0058 | 0.3236 | 0.5242 | 0.2006 | 96.3% | |
$ \beta $ | 0.5912 | 0.0912 | 0.0099 | 0.5074 | 0.6750 | 0.1676 | 97.0% | ||
$ \gamma $ | 0.4667 | 0.0333 | 0.0020 | 0.4101 | 0.5233 | 0.1132 | 97.5% | ||
100% | $ \alpha $ | 0.4240 | 0.0760 | 0.0058 | 0.3372 | 0.5109 | 0.1737 | 96.5% | |
$ \beta $ | 0.4905 | 0.0095 | 0.0009 | 0.4167 | 0.5642 | 0.1475 | 96.7% | ||
$ \gamma $ | 0.5035 | 0.0035 | 0.0006 | 0.4524 | 0.5546 | 0.1022 | 97.1% | ||
n | r | Set2 ($ \alpha $ = 0.7, $ \beta $ = 0.5, $ \gamma $ = 0.25) | |||||||
MLE | Bias | MSE | LB | UB | AL | CP | |||
50 | 70% | $ \alpha $ | 0.4206 | 0.2794 | 0.0782 | 0.0055 | 0.8358 | 0.8304 | 97.7% |
$ \beta $ | 0.7016 | 0.2016 | 0.0462 | 0.5015 | 0.9016 | 0.4001 | 96.5% | ||
$ \gamma $ | 0.2172 | 0.0328 | 0.0017 | 0.1568 | 0.2776 | 0.1208 | 100% | ||
80% | $ \alpha $ | 0.4214 | 0.2786 | 0.0776 | 0.2084 | 0.6345 | 0.4261 | 97.9% | |
$ \beta $ | 0.6361 | 0.1361 | 0.0243 | 0.4490 | 0.8231 | 0.3741 | 98.5% | ||
$ \gamma $ | 0.2297 | 0.0203 | 0.0010 | 0.1719 | 0.2874 | 0.1155 | 100% | ||
100% | $ \alpha $ | 0.4234 | 0.2766 | 0.0765 | 0.2497 | 0.5970 | 0.3473 | 98.3% | |
$ \beta $ | 0.5161 | 0.0161 | 0.0062 | 0.3649 | 0.6673 | 0.3025 | 97.6% | ||
$ \gamma $ | 0.2565 | 0.0065 | 0.0007 | 0.2044 | 0.3086 | 0.1043 | 100% | ||
100 | 70% | $ \alpha $ | 0.4210 | 0.2790 | 0.0779 | 0.0208 | 0.8212 | 0.8004 | 96.4% |
$ \beta $ | 0.7006 | 0.2006 | 0.0431 | 0.5593 | 0.8419 | 0.2826 | 98.0% | ||
$ \gamma $ | 0.2141 | 0.0359 | 0.0016 | 0.1720 | 0.2562 | 0.0842 | 100% | ||
80% | $ \alpha $ | 0.4215 | 0.2785 | 0.0776 | 0.2708 | 0.5721 | 0.3013 | 97.2% | |
$ \beta $ | 0.6357 | 0.1357 | 0.0214 | 0.5035 | 0.7680 | 0.2645 | 97.7% | ||
$ \gamma $ | 0.2270 | 0.0230 | 0.0008 | 0.1866 | 0.2673 | 0.0807 | 100% | ||
100% | $ \alpha $ | 0.4234 | 0.2766 | 0.0765 | 0.3006 | 0.5462 | 0.2456 | 97.3% | |
$ \beta $ | 0.5158 | 0.0158 | 0.0033 | 0.4088 | 0.6227 | 0.2140 | 98.2% | ||
$ \gamma $ | 0.2540 | 0.0040 | 0.0003 | 0.2176 | 0.2905 | 0.0729 | 100% | ||
150 | 70% | $ \alpha $ | 0.4212 | 0.2788 | 0.0778 | 0.0330 | 0.8093 | 0.7763 | 97.7% |
$ \beta $ | 0.7000 | 0.2000 | 0.0419 | 0.5847 | 0.8153 | 0.2306 | 97.7% | ||
$ \gamma $ | 0.2122 | 0.0378 | 0.0016 | 0.1781 | 0.2463 | 0.0682 | 100% | ||
80% | $ \alpha $ | 0.4215 | 0.2785 | 0.0776 | 0.2985 | 0.5445 | 0.2460 | 98.8% | |
$ \beta $ | 0.6350 | 0.1350 | 0.0203 | 0.5271 | 0.7430 | 0.2159 | 98.1% | ||
$ \gamma $ | 0.2259 | 0.0241 | 0.0008 | 0.1931 | 0.2587 | 0.0656 | 96.0% | ||
100% | $ \alpha $ | 0.4234 | 0.2766 | 0.0765 | 0.3232 | 0.5237 | 0.2005 | 97.2% | |
$ \beta $ | 0.5151 | 0.0151 | 0.0023 | 0.4278 | 0.6024 | 0.1746 | 97.0% | ||
$ \gamma $ | 0.2529 | 0.0029 | 0.0002 | 0.2232 | 0.2825 | 0.0593 | 95.4% | ||
200 | 70% | $ \alpha $ | 0.4209 | 0.2791 | 0.0779 | 0.0849 | 0.7569 | 0.6720 | 100% |
$ \beta $ | 0.6981 | 0.1981 | 0.0405 | 0.5984 | 0.7978 | 0.1994 | 97.2% | ||
$ \gamma $ | 0.2118 | 0.0382 | 0.0016 | 0.1823 | 0.2412 | 0.0589 | 97.3% | ||
80% | $ \alpha $ | 0.4215 | 0.2785 | 0.0776 | 0.3150 | 0.5280 | 0.2131 | 100% | |
$ \beta $ | 0.6331 | 0.1331 | 0.0191 | 0.5398 | 0.7265 | 0.1867 | 98.2% | ||
$ \gamma $ | 0.2256 | 0.0244 | 0.0007 | 0.1973 | 0.2540 | 0.0567 | 98.0% | ||
100% | $ \alpha $ | 0.4234 | 0.2766 | 0.0765 | 0.3366 | 0.5103 | 0.1737 | 100% | |
$ \beta $ | 0.5136 | 0.0136 | 0.0016 | 0.4381 | 0.5891 | 0.1510 | 98.8% | ||
$ \gamma $ | 0.2523 | 0.0023 | 0.0002 | 0.2267 | 0.2779 | 0.0512 | 100% | ||
n | r | Set3 ($ \alpha $ = 0.7, $ \beta $ = 0.7, $ \gamma $ = 0.5) | |||||||
MLE | Bias | MSE | LB | UB | AL | CP | |||
50 | 70% | $ \alpha $ | 0.4178 | 0.2822 | 0.0797 | 0.1937 | 0.6419 | 0.4482 | 96.2% |
$ \beta $ | 0.8994 | 0.1994 | 0.0477 | 0.6064 | 1.1923 | 0.5859 | 95.9% | ||
$ \gamma $ | 0.6151 | 0.1151 | 0.0239 | 0.3763 | 0.8540 | 0.4776 | 95.0% | ||
80% | $ \alpha $ | 0.4193 | 0.2807 | 0.0788 | 0.2255 | 0.6131 | 0.3875 | 95.9% | |
$ \beta $ | 0.8238 | 0.1238 | 0.0218 | 0.5471 | 1.1006 | 0.5535 | 95.9% | ||
$ \gamma $ | 0.5695 | 0.0695 | 0.0182 | 0.3447 | 0.7943 | 0.4496 | 96.7% | ||
100% | $ \alpha $ | 0.4211 | 0.2789 | 0.0778 | 0.2480 | 0.5942 | 0.3461 | 96.8% | |
$ \beta $ | 0.7612 | 0.0612 | 0.0104 | 0.5012 | 1.0213 | 0.5201 | 97.0% | ||
$ \gamma $ | 0.5425 | 0.0425 | 0.0163 | 0.3395 | 0.7456 | 0.4061 | 95.0% | ||
100 | 70% | $ \alpha $ | 0.4174 | 0.2826 | 0.0798 | 0.2439 | 0.5910 | 0.3470 | 95.0% |
$ \beta $ | 0.8787 | 0.1787 | 0.0353 | 0.6426 | 1.1148 | 0.4722 | 96.3% | ||
$ \gamma $ | 0.5696 | 0.0696 | 0.0201 | 0.3697 | 0.7696 | 0.3998 | 96.0% | ||
80% | $ \alpha $ | 0.4191 | 0.2809 | 0.0789 | 0.2690 | 0.5691 | 0.3001 | 95.5% | |
$ \beta $ | 0.7802 | 0.0802 | 0.0134 | 0.5600 | 1.0004 | 0.4404 | 95.7% | ||
$ \gamma $ | 0.5479 | 0.0479 | 0.0091 | 0.3814 | 0.7144 | 0.3330 | 96.0% | ||
100% | $ \alpha $ | 0.4206 | 0.2794 | 0.0781 | 0.2866 | 0.5546 | 0.2680 | 95.6% | |
$ \beta $ | 0.7146 | 0.0146 | 0.0068 | 0.5069 | 0.9223 | 0.4154 | 95.7% | ||
$ \gamma $ | 0.5597 | 0.0597 | 0.0074 | 0.4063 | 0.7131 | 0.3067 | 96.0% | ||
150 | 70% | $ \alpha $ | 0.4176 | 0.2824 | 0.0798 | 0.2949 | 0.5403 | 0.2454 | 95.8% |
$ \beta $ | 0.8697 | 0.1697 | 0.0305 | 0.7056 | 1.0338 | 0.3282 | 96.2% | ||
$ \gamma $ | 0.5924 | 0.0924 | 0.0174 | 0.4449 | 0.7399 | 0.2950 | 97.1% | ||
80% | $ \alpha $ | 0.4193 | 0.2807 | 0.0788 | 0.3131 | 0.5254 | 0.2122 | 96.2% | |
$ \beta $ | 0.8023 | 0.1023 | 0.0124 | 0.6492 | 0.9555 | 0.3063 | 96.1% | ||
$ \gamma $ | 0.5524 | 0.0524 | 0.0058 | 0.4332 | 0.6716 | 0.2384 | 97.0% | ||
100% | $ \alpha $ | 0.4209 | 0.2791 | 0.0779 | 0.3261 | 0.5156 | 0.1895 | 95.8% | |
$ \beta $ | 0.7374 | 0.0374 | 0.0032 | 0.5932 | 0.8817 | 0.2885 | 96.3% | ||
$ \gamma $ | 0.5507 | 0.0507 | 0.0041 | 0.4447 | 0.6568 | 0.2121 | 96.9% | ||
200 | 70% | $ \alpha $ | 0.4175 | 0.2825 | 0.0798 | 0.3173 | 0.5177 | 0.2004 | 96.1% |
$ \beta $ | 0.8406 | 0.1406 | 0.0265 | 0.7075 | 0.9736 | 0.2661 | 97.2% | ||
$ \gamma $ | 0.5683 | 0.0683 | 0.0069 | 0.4569 | 0.6798 | 0.2230 | 96.9% | ||
80% | $ \alpha $ | 0.4193 | 0.2807 | 0.0788 | 0.3327 | 0.5059 | 0.1733 | 96.3% | |
$ \beta $ | 0.8043 | 0.1043 | 0.0123 | 0.6790 | 0.9295 | 0.2505 | 96.6% | ||
$ \gamma $ | 0.5502 | 0.0502 | 0.0041 | 0.4533 | 0.6471 | 0.1938 | 97.0% | ||
100% | $ \alpha $ | 0.4209 | 0.2791 | 0.0779 | 0.3435 | 0.4983 | 0.1548 | 96.1% | |
$ \beta $ | 0.7411 | 0.0411 | 0.0030 | 0.6229 | 0.8592 | 0.2362 | 97.0% | ||
$ \gamma $ | 0.5463 | 0.0463 | 0.0041 | 0.4589 | 0.6338 | 0.1749 | 96.2% | ||
n | r | Set4 ($ \alpha $ = 0.6, $ \beta $ = 0.3, $ \gamma $ = 0.5) | |||||||
MLE | Bias | MSE | LB | UB | AL | CP | |||
50 | 70% | $ \alpha $ | 0.4197 | 0.1803 | 0.0325 | 0.2456 | 0.5939 | 0.3482 | 98.1% |
$ \beta $ | 0.5719 | 0.2719 | 0.0744 | 0.3742 | 0.7696 | 0.3954 | 97.0% | ||
$ \gamma $ | 0.2990 | 0.2010 | 0.0406 | 0.1769 | 0.4211 | 0.2442 | 98.0% | ||
80% | $ \alpha $ | 0.4221 | 0.1779 | 0.0317 | 0.2716 | 0.5725 | 0.3009 | 98.4% | |
$ \beta $ | 0.4934 | 0.1934 | 0.0384 | 0.3126 | 0.6742 | 0.3616 | 97.4% | ||
$ \gamma $ | 0.3593 | 0.1407 | 0.0200 | 0.2383 | 0.4803 | 0.2421 | 98.2% | ||
100% | $ \alpha $ | 0.4246 | 0.1754 | 0.0308 | 0.2903 | 0.5589 | 0.2686 | 97.9% | |
$ \beta $ | 0.4198 | 0.1198 | 0.0150 | 0.2576 | 0.5820 | 0.3244 | 97.4% | ||
$ \gamma $ | 0.4172 | 0.0828 | 0.0078 | 0.2968 | 0.5376 | 0.2408 | 98.4% | ||
100 | 70% | $ \alpha $ | 0.4198 | 0.1802 | 0.0325 | 0.2966 | 0.5429 | 0.2463 | 97.2% |
$ \beta $ | 0.5674 | 0.2674 | 0.0717 | 0.4278 | 0.7069 | 0.2791 | 97.7% | ||
$ \gamma $ | 0.3137 | 0.1863 | 0.0351 | 0.2278 | 0.3995 | 0.1717 | 98.0% | ||
80% | $ \alpha $ | 0.4222 | 0.1778 | 0.0316 | 0.3157 | 0.5286 | 0.2128 | 97.9% | |
$ \beta $ | 0.4857 | 0.1857 | 0.0350 | 0.3584 | 0.6130 | 0.2546 | 97.9% | ||
$ \gamma $ | 0.3669 | 0.1331 | 0.0179 | 0.2818 | 0.4519 | 0.1701 | 98.7% | ||
100% | $ \alpha $ | 0.4248 | 0.1752 | 0.0307 | 0.3242 | 0.5253 | 0.2011 | 98.1% | |
$ \beta $ | 0.4082 | 0.1082 | 0.0125 | 0.2944 | 0.5221 | 0.2277 | 97.8% | ||
$ \gamma $ | 0.4171 | 0.0829 | 0.0074 | 0.3349 | 0.4993 | 0.1644 | 98.3% | ||
150 | 70% | $ \alpha $ | 0.4198 | 0.1802 | 0.0325 | 0.3248 | 0.5148 | 0.1899 | 97.2% |
$ \beta $ | 0.5653 | 0.2653 | 0.0706 | 0.4518 | 0.6788 | 0.2270 | 98.3% | ||
$ \gamma $ | 0.3140 | 0.1860 | 0.0350 | 0.2421 | 0.3860 | 0.1439 | 99.3% | ||
80% | $ \alpha $ | 0.4222 | 0.1778 | 0.0316 | 0.3353 | 0.5091 | 0.1738 | 97.7% | |
$ \beta $ | 0.4832 | 0.1832 | 0.0338 | 0.3797 | 0.5866 | 0.2069 | 98.2% | ||
$ \gamma $ | 0.3677 | 0.1323 | 0.0178 | 0.2996 | 0.4358 | 0.1363 | 99.0% | ||
100% | $ \alpha $ | 0.4248 | 0.1752 | 0.0307 | 0.3377 | 0.5119 | 0.1742 | 97.3% | |
$ \beta $ | 0.4018 | 0.1018 | 0.0107 | 0.3032 | 0.5004 | 0.1972 | 97.9% | ||
$ \gamma $ | 0.4248 | 0.0752 | 0.0061 | 0.3608 | 0.4888 | 0.1280 | 99.7% | ||
200 | 70% | $ \alpha $ | 0.4198 | 0.1802 | 0.0325 | 0.3423 | 0.4973 | 0.1551 | 99.0% |
$ \beta $ | 0.5650 | 0.2650 | 0.0705 | 0.4738 | 0.6561 | 0.1823 | 99.1% | ||
$ \gamma $ | 0.3256 | 0.1744 | 0.0309 | 0.2637 | 0.3875 | 0.1239 | 98.7% | ||
80% | $ \alpha $ | 0.4222 | 0.1778 | 0.0316 | 0.3470 | 0.4975 | 0.1505 | 99.6% | |
$ \beta $ | 0.4781 | 0.1781 | 0.0321 | 0.3882 | 0.5679 | 0.1798 | 99.3% | ||
$ \gamma $ | 0.3733 | 0.1267 | 0.0173 | 0.3129 | 0.4336 | 0.1206 | 99.5% | ||
100% | $ \alpha $ | 0.4250 | 0.1750 | 0.0306 | 0.3578 | 0.4921 | 0.1343 | 98.7% | |
$ \beta $ | 0.3904 | 0.0904 | 0.0088 | 0.3106 | 0.4703 | 0.1596 | 99.6% | ||
$ \gamma $ | 0.4262 | 0.0738 | 0.0060 | 0.3681 | 0.4843 | 0.1162 | 100% |
$ {\rm{A}}. $ For almost all the true values, the MSE of all the estimates decreases as the sample sizes and the censoring level r increase, demonstrating that the various estimates are consistent (see Table 6 and Figure 3).
$ {\rm{B}}. $ For all true parameter values, the ALs of all the estimates decrease as the sample sizes and the censoring level r increase (see Table 6 and Figure 4).
$ {\rm{C}}. $ For all true parameter values, the CP of all the estimates increases as the sample sizes and the censoring level r increase (see Table 6).
$ {\rm{D}}. $ The MSE of the estimate of $ \alpha $ at the true value of Set1 yields the lowest values in comparison to the other actual parameter values for all sample sizes (see Table 6 and Figure 5).
$ {\rm{E}}. $ At all actual values, the MSE of the estimate of $ \beta $ produces the largest results for all sample sizes (see Table 6 and Figure 6). Also, it is evident that except for $ n = 50 $ and 200, the MSE of $ \beta $ estimates obtains the smallest values for the actual value of Set1 compared to the other actual sets at the censoring level 70%. At the censoring level 80%, the MSE of $ \beta $ estimates gets the smallest values at all sets of parameters except at $ n = 50 $.
$ {\rm{F}}. $ The MSE of the estimate of $ \gamma $ at the true value of Set2 gets the smallest values in comparison to the other actual parameter values for all sample sizes (see Table 6 and Figure 7).
$ {\rm{G}}. $ The MSEs, biases, and ALs of $ \gamma $ are smaller than the other estimates of $ \alpha $ and $ \beta $ in almost all of the cases.
$ {\rm{H}}. $ As $ n $ rises, the CI's lengths get shorter.
$ {\rm{I}}. $ As $ n $ increases, parameter estimates grow increasingly accurate, suggesting that they are asymptotically unbiased.
$ {\rm{J}}. $ For the parameter values examined, the CI's overall performance is fairly strong.
Here, we provide applications to four real data sets to illustrate the importance and potentiality of the LBTLoW distribution. The goodness-of-fit statistics for these distributions and other competitive distributions are compared, and the MLEs of their parameters are provided.
The first real data set [38] on the relief times of twenty patients receiving an analgesic is 1.1, 1.4, 3, 1.7, 2.3, 1.4, 1.3, 1.7, 2.2, 1.7, 2.7, 4.1, 1.8, 1.5, 1.9, 1.8, 1.6, 1.2, 1.6, 2.
The second dataset illustrates the proportion of global reserves of natural gas in various countries as of the year 2020. In contrast to other nations, Russia possesses the largest natural gas reserves globally and maintains its position as the leading exporter of natural gas. Iran, on the other hand, ranks second in terms of natural gas reserves worldwide. Qatar, although holding slightly over 13% of the total global natural gas reserves, also plays a significant role in the natural gas market. Lastly, Saudi Arabia possesses the fifth-largest natural gas reserves globally. The electronic address from which it was taken is as follows: https://worldpopulationreview.com/. The data set is reported in Table 7.
Rank | Country | % Global Reserves | Rank | Country | % Global Reserves |
1 | Russia | 19.9 | 23 | Ukraine | 0.6 |
2 | Iran | 17.1 | 24 | Malaysia | 0.5 |
3 | Qatar | 13.1 | 25 | Uzbekistan | 0.4 |
4 | Turkmenistan | 7.2 | 26 | Oman | 0.4 |
5 | United States | 6.7 | 27 | Vietnam | 0.3 |
6 | China | 4.5 | 28 | Israel | 0.3 |
7 | Venezuela | 3.3 | 29 | Argentina | 0.2 |
8 | Saudi Arabia | 3.2 | 30 | Pakistan | 0.2 |
9 | United Arab Emirates | 3.2 | 31 | Trinidad | 0.2 |
10 | Nigeria | 2.9 | 32 | Brazil | 0.2 |
11 | Iraq | 1.9 | 33 | Myanmar | 0.2 |
12 | Canada | 1.3 | 34 | United Kingdom | 0.1 |
13 | Australia | 1.3 | 35 | Thailand | 0.1 |
14 | Azerbaijan | 1.3 | 36 | Mexico | 0.1 |
15 | Algeria | 1.2 | 37 | Bangladesh | 0.1 |
16 | Kazakhstan | 1.2 | 38 | Netherlands | 0.1 |
17 | Egypt | 1.1 | 39 | Bolivia | 0.1 |
18 | Kuwait | 0.9 | 40 | Brunei | 0.1 |
19 | Norway | 0.8 | 41 | Peru | 0.1 |
20 | Libya | 0.8 | 42 | Syria | 0.1 |
21 | Indonesia | 0.7 | 43 | Yemen | 0.1 |
22 | India | 0.7 | 44 | Papua New Guinea | 0.1 |
The third dataset pertains to the Top 20 Countries with the Largest Oil Reserves, measured in thousand million barrels. Crude oil serves as the predominant fuel source globally and is the primary source of energy on a wide scale. In the year 2020, global oil consumption reached around 88.6 million barrels per day, or 30.1% of the overall primary energy consumption. Venezuela possesses the largest oil reserves globally, over 300 billion barrels in total. Saudi Arabia holds the world's second-largest oil reserves, with 297.5 billion barrels. The United States is the world's leading producer of oil as well as the world's greatest user of oil, necessitating additional imports from dozens of other oil-producing countries. Despite having the world's highest oil production, the United States is only 9th in the world in terms of available oil reserves. It was obtained from the following electronic address: https://worldpopulationreview.com/. The data set is reported in Table 8.
Rank | Country | reserves2020 | Rank | Country | reserves2020 |
1 | Venezuela | 303.8 | 11 | Nigeria | 36.9 |
2 | Saudi Arabia | 297.5 | 12 | Kazakhstan | 30 |
3 | Canada | 168.1 | 13 | China | 26 |
4 | Iran | 157.8 | 14 | Qatar | 25.2 |
5 | Iraq | 145 | 15 | Algeria | 12.2 |
6 | Russia | 107.8 | 16 | Brazil | 11.9 |
7 | Kuwait | 101.5 | 17 | Norway | 7.9 |
8 | United Arab Emirates | 97.8 | 18 | Angola | 7.8 |
9 | United States | 68.8 | 19 | Azerbaijan | 7 |
10 | Libya | 48.4 | 20 | Mexico | 6.1 |
The fourth data set represents the Top 100 central banks that owned the largest gold Reserves (in thousand tons). Because of its safety, liquidity, and return qualities-the three major investment objectives for central banks-gold is an essential component of central bank reserves. As such, they are significant gold holders, accounting for around one-fifth of all gold extracted throughout history. They present gold reserve data derived using IMF IFS figures to help comprehend this sector of the gold market, which records central banks' (and other official institutions, when appropriate) reported purchases and sales of gold as a percentage of their international reserves. It was obtained from the following electronic address: https://www.gold.org/. The data set is reported in Table 9.
Rank | Country | Reserves of Gold | Rank | Country | Reserves of Gold | Rank | Country | Reserves of Gold |
1 | USA | 8.1335 | 35 | LBY | 0.1166 | 68 | CYP | 0.0139 |
2 | DEU | 3.3585 | 36 | GRC | 0.1141 | 69 | CUW | 0.0131 |
3 | IMF | 2.814 | 37 | ROK | 0.1045 | 70 | MUS | 0.0124 |
4 | ITA | 2.4518 | 38 | ROU | 0.1036 | 71 | IRL | 0.012 |
5 | FRA | 2.4365 | 39 | BIS | 0.102 | 72 | CZE | 0.0109 |
6 | RUS | 2.2985 | 40 | IRQ | 0.0964 | 73 | KGZ | 0.0102 |
7 | CHN | 1.9483 | 41 | HUN | 0.0945 | 74 | GHA | 0.0087 |
8 | CHE | 1.04 | 42 | AUS | 0.0798 | 75 | PRY | 0.0082 |
9 | JPN | 0.846 | 43 | KWT | 0.079 | 76 | NPL | 0.008 |
10 | IND | 0.7604 | 44 | IDN | 0.0786 | 77 | MNG | 0.0076 |
11 | NLD | 0.6125 | 45 | DNK | 0.0666 | 78 | MMR | 0.0073 |
12 | ECB | 0.5048 | 46 | PAK | 0.0647 | 79 | GTM | 0.0069 |
13 | TUR | 0.4311 | 47 | ARG | 0.0617 | 80 | MKD | 0.0069 |
14 | TAI | 0.4236 | 48 | ARE | 0.0553 | 81 | TUN | 0.0068 |
15 | PRT | 0.3826 | 49 | BLR | 0.0535 | 82 | LVA | 0.0067 |
16 | KAZ | 0.3681 | 50 | QAT | 0.0513 | 83 | LTU | 0.0058 |
17 | UZB | 0.3375 | 51 | KHM | 0.0504 | 84 | COL | 0.0047 |
18 | SAU | 0.3231 | 52 | FIN | 0.049 | 85 | BHR | 0.0047 |
19 | GBR | 0.3103 | 53 | JOR | 0.0435 | 86 | BRN | 0.0046 |
20 | LBN | 0.2868 | 54 | BOL | 0.0425 | 87 | GIN | 0.0042 |
21 | ESP | 0.2816 | 55 | BGR | 0.0408 | 88 | MOZ | 0.0039 |
22 | AUT | 0.28 | 56 | MYS | 0.0389 | 89 | SVN | 0.0032 |
23 | THA | 0.2442 | 57 | SRB | 0.0378 | 90 | ABW | 0.0031 |
24 | POL | 0.2287 | 58 | WAEMU | 0.0365 | 91 | BIH | 0.003 |
25 | BEL | 0.2274 | 59 | PER | 0.0347 | 92 | ALB | 0.0028 |
26 | DZA | 0.1736 | 60 | SVK | 0.0317 | 93 | LUX | 0.0022 |
27 | VEN | 0.1612 | 61 | UKR | 0.0271 | 94 | HKG | 0.0021 |
28 | PHL | 0.1563 | 62 | SYR | 0.0258 | 95 | ISL | 0.002 |
29 | SGP | 0.1537 | 63 | MAR | 0.0221 | 96 | TTO | 0.0019 |
30 | BRA | 0.1297 | 64 | ECU | 0.0219 | 97 | HTI | 0.0018 |
31 | SWE | 0.1257 | 65 | AFG | 0.0219 | 98 | YEM | 0.0016 |
32 | ZAF | 0.1254 | 66 | NGA | 0.0215 | 99 | SUR | 0.0015 |
33 | EGY | 0.125 | 67 | BGD | 0.014 | 100 | SLV | 0.0014 |
34 | MEX | 0.1199 |
The descriptive analysis of all the data sets is reported in Table 10.
n | Mean | Median | Skewness | Kurtosis | Range | Min | Max | Sum | |
Data1 | 20 | 1.900 | 1.700 | 1.860 | 4.185 | 3.000 | 1.100 | 4.100 | 38.000 |
Data2 | 44 | 2.248 | 0.650 | 2.990 | 8.864 | 19.800 | 0.100 | 19.900 | 98.900 |
Data3 | 20 | 83.375 | 42.650 | 1.430 | 1.420 | 297.700 | 6.100 | 303.800 | 1667.500 |
Data4 | 100 | 0.347 | 0.050 | 5.590 | 38.257 | 8.130 | 0.001 | 8.133 | 34.676 |
These real data sets are utilized to assess the goodness of fit of the LBTLoW distribution. The suggested model is compared with exponentiated transmuted generalized Rayleigh (ETGR) [39], beta Weibull (BW) [40], transmuted Lindley (T-Li) [41], McDonald log-logistic (McLL) [42], new modified Weibull (NMW) [43], weighted exponentiated inverted Weibull (WEIW) [44], transmuted complementary Weibull geometric (TCWG) [45], transmuted modified Weibull (TMW) [46], exponentiated Kumaraswamy Weibull (EKW) [47] and Weibull (W) models.
The maximum likelihood estimators (MLEs) and standard errors (SEs) of the model parameters are computed. In order to assess the distribution models, various criteria are taken into account, including the Akaike information criterion ($ A_{IC} $), correct $ A_{IC} $ ($ C_{AIC} $), Bayesian IC ($ B_{IC} $), Hannan-Quinn IC ($ H_{QIC} $), Kolmogorov-Smirnov ($ K_S $) test, and p-value ($ P_V $) test. In contrast, the broader dissemination is associated with reduced values of $ A_{IC} $, $ C_{AIC} $, $ B_{IC} $, $ H_{QIC} $, $ K_S $, and the highest magnitude of $ P_V $. The maximum likelihood estimators (MLEs) of the competitive models, along with their standard errors (SEs) and values of $ A_{IC} $, $ C_{AIC} $, $ B_{IC} $, $ H_{QIC} $, $ P_V $, and $ K_S $ for the suggested data sets, are displayed in Tables 11-18. It has been observed that the LBTLoW distribution, characterized by three parameters, exhibits superior goodness of fit compared to alternative models. This distribution exhibits the lowest values of $ A_{IC} $, $ C_{AIC} $, $ B_{IC} $, $ H_{QIC} $, and $ K_S $, and the highest value of $ P_V $ among the distributions under consideration in this analysis. Furthermore, Figures 8-15 exhibit the graphical representations of the estimated pdf, cdf, ccdf, and probability-probability (PP) plots for the competitive model applied to the given data sets.
Distributions | MLE and SE | ||||
$ \alpha $ | $ \beta $ | $ \gamma $ | $ \lambda $ | $ \theta $ | |
LBTLoW | 8.648 | 3.074 | 0.042 | ||
(3.545) | (0.474) | (0.025) | |||
ETGR | 0.103 | 0.692 | 23.539 | -0.342 | |
(0.436) | (0.086) | (105.137) | (1.971) | ||
BW | 0.831 | 0.613 | 29.947 | 11.632 | |
(0.954) | (0.340) | (40.414) | (21.900) | ||
T-Li | 0.665 | 0.359 | |||
(0.332) | (0.048) | ||||
McLL | 0.881 | 2.070 | 1.926 | 19.225 | 32.033 |
(0.109) | (3.693) | (5.165) | (22.341) | (43.081) | |
NMW | 0.121 | 2.784 | 2.787 | 0.003 | 0.008 |
(0.056) | (20.370) | (0.428) | (0.025) | (0.002) | |
W | 0.122 | 2.787 | |||
(0.056) | (0.427) |
Distributions | $ A_{IC} $ | $ C_{AIC} $ | $ B_{IC} $ | $ H_{QIC} $ | $ K_S $ | $ P_V $ |
LBTLoW | 40.140 | 41.640 | 38.040 | 40.720 | 0.146 | 0.790 |
ETGR | 44.860 | 47.520 | 42.060 | 45.630 | 0.190 | 0.465 |
BW | 42.400 | 45.060 | 39.600 | 43.170 | 0.160 | 0.683 |
T-Li | 65.730 | 66.440 | 64.330 | 66.120 | 0.380 | 0.006 |
McLL | 43.850 | 48.140 | 40.360 | 44.830 | 0.147 | 0.734 |
NMW | 51.170 | 55.460 | 47.680 | 52.150 | 0.190 | 0.501 |
W | 45.170 | 45.880 | 43.780 | 45.560 | 0.180 | 0.509 |
Distributions | MLE and SE | ||||
$ \alpha $ | $ \beta $ | $ \gamma $ | $ \lambda $ | $ \theta $ | |
LBTLoW | 6.268 | 0.623 | 0.484 | ||
(2.631) | (0.066) | (0.210) | |||
ETGR | 0.055 | 0.071 | 8.773 | 0.947 | |
(0.027) | (0.029) | (7.043) | (0.081) | ||
TCWG | 34.076 | 0.802 | 0.005 | 1.12 | |
(81.023) | (0.021) | (0.013) | (0.285) | ||
EKW | 0.221 | 400.298 | 5.215 | 1 | 3.823 |
(0.038) | (718.99) | (0.649) | (0.004) | (3.036) | |
TMW | 0.851 | 1.159 | -0.554 | 0.519 | |
(0.163) | (1.026) | (0.985) | (0.379) | ||
BW | 2.861 | 0.075 | 78.550 | 42.576 | |
(69.095) | (0.090) | (167.320) | (187.300) | ||
T-Li | 0.604 | 0.671 | |||
(0.155) | (0.074) | ||||
McLL | 0.181 | 1.565 | 1.286 | 21.234 | 28.124 |
(0.193) | (9.254) | (5.432) | (34.701) | (45.757) | |
NMW | 6.8 x $ 10^{-8} $ | 0.680 | 0.223 | 0.015 | 0.806 |
(0.623) | (0.110) | (617.48) | (0.015) | (0.418) | |
W | 0.799 | 0.621 | |||
(0.136) | (0.068) |
Distributions | $ A_{IC} $ | $ C_{AIC} $ | $ B_{IC} $ | $ H_{QIC} $ | $ K_S $ | $ P_V $ |
LBTLoW | 132.210 | 132.810 | 131.140 | 134.200 | 0.130 | 0.425 |
ETGR | 143.470 | 144.490 | 142.040 | 146.110 | 0.180 | 0.118 |
TCWG | 137.690 | 138.710 | 136.260 | 140.330 | 0.150 | 0.251 |
EKW | 133.890 | 135.470 | 132.110 | 140.330 | 0.140 | 0.355 |
TMW | 140.900 | 142.480 | 139.120 | 144.210 | 0.150 | 0.276 |
BW | 133.180 | 134.200 | 131.750 | 135.820 | 0.130 | 0.408 |
T-Li | 174.360 | 174.660 | 173.650 | 175.690 | 0.200 | 0.057 |
McLL | 134.830 | 136.410 | 133.040 | 138.130 | 0.130 | 0.419 |
NMW | 143.780 | 145.360 | 142.000 | 147.090 | 0.160 | 0.243 |
W | 138.650 | 138.940 | 137.940 | 139.970 | 0.170 | 0.139 |
Distributions | MLE and SE | ||||
$ \alpha $ | $ \beta $ | $ \gamma $ | $ \lambda $ | $ \theta $ | |
LBTLoW | 2.515 | 0.756 | 0.040 | ||
(3.877) | (0.166) | (0.048) | |||
WEIW | 0.909 | 0.871 | 7.225 | ||
(106700) | (0.152) | (384700) | |||
TMW | 0.998 | 0.459 | -0.443 | 0.202 | |
(0.081) | (18.537) | (18.537) | (0.769) | ||
T-Li | 0.021 | 0.384 | |||
(0.345) | (0.004) | ||||
McLL | 0.208 | 93.978 | 1.279 | 24.759 | 32.815 |
(0.499) | (1721) | (19.272) | (142.806) | (161.611) | |
NMW | 10.7 x $ 10^{-8} $ | 0.930 | 0.859 | 7.46 x $ 10^{-8} $ | 0.017 |
(0.001) | (0.250) | (1.216) | (0.002) | (0.017) | |
EKW | 0.167 | 261.64 | 45.725 | 1.201 | 2.138 |
(0.079) | (1709) | (219.725) | (0.741) | (7.209) |
Distributions | $ A_{IC} $ | $ C_{AIC} $ | $ B_{IC} $ | $ H_{QIC} $ | $ K_S $ | $ P_V $ |
LBTLoW | 221.690 | 223.190 | 219.600 | 222.280 | 0.135 | 0.857 |
WEIW | 223.400 | 224.900 | 221.300 | 223.980 | 0.157 | 0.708 |
TMW | 226.410 | 230.690 | 222.910 | 227.380 | 0.153 | 0.734 |
T-Li | 230.480 | 231.180 | 229.080 | 230.870 | 0.265 | 0.120 |
McLL | 225.990 | 230.280 | 222.500 | 222.500 | 0.146 | 0.789 |
NMW | 226.570 | 230.860 | 223.080 | 227.540 | 0.140 | 0.826 |
EKW | 226.290 | 230.570 | 222.790 | 229.300 | 0.148 | 0.776 |
Distributions | MLE and SE | ||||
$ \alpha $ | $ \beta $ | $ \gamma $ | $ \lambda $ | $ \theta $ | |
LBTLoW | 6.498 | 0.482 | 1.490 | ||
(2.301) | (0.034) | (0.573) | |||
EKW | 0.221 | 1096 | 4.424 | 1 | 1.717 |
(0.030) | (1376) | (1.817) | (0.001) | (0.901) | |
TMW | 0.596 | 2.612 | 0.588 | -0.523 | |
(0.057) | (0.689) | (0.256) | (0.346) | ||
BW | 134.832 | 0.073 | 49.149 | 22.930 | |
(956.622) | (0.060) | (74.497) | (46.500) | ||
WEIW | 27.512 | 0.549 | 0.094 | ||
(3272000) | (0.042) | (856.967) | |||
W | 2.648 | 0.489 | |||
(0.281) | (0.035) |
Distributions | $ A_{IC} $ | $ C_{AIC} $ | $ B_{IC} $ | $ H_{QIC} $ | $ K_S $ | $ P_V $ |
LBTLoW | –170.510 | –170.260 | –170.510 | –167.350 | 0.070 | 0.704 |
ETGR | –167.710 | –167.070 | –167.710 | –157.710 | 0.078 | 0.584 |
BW | –158.180 | –157.540 | –158.180 | –152.910 | 0.077 | 0.593 |
T-Li | –170.220 | –169.800 | –170.220 | –166.010 | 0.071 | 0.703 |
McLL | –170.200 | –169.950 | –170.200 | –167.040 | 0.091 | 0.383 |
W | –157.390 | –157.260 | –157.390 | –155.280 | 0.100 | 0.269 |
From the previous figures, we conclude that the LBTLoW model clearly gives the best overall fit and so may be picked as the most appropriate model for explaining data.
The LBTLo-G family of distributions is explored in this article. The LBTLo-G family of probability distributions has a number of desirable characteristics, including being very flexible and simple, containing a number of new distributions, the ability for the generated distributions' pdfs to be unimodal, decreasing, bathtub-shaped, right-skewed, and symmetric, and the ability for their hrf shapes to be increasing, decreasing, U-shaped, upside-down-shaped, or J-shaped. These include discussion of the characteristics of the LBTLo-G family, including expansion for the density function, moments, incomplete moments, and certain entropy metrics. Estimating the model parameters is done using the ML technique. A simulation study demonstrated that the estimates of the model parameters are not far from their true values. Also, the biases and mean squared errors of estimates based on censored samples are larger than those based on complete samples. As the censoring levels and sample sizes increase, the coverage probability of estimates increases in approximately most cases.
As one distribution of the LBTLo-G family, the real datasets for global reserves of oil, gold, and natural gas were chosen to fit the LBTLoW distribution. The first data set proposed was the lifetime data relating to relief times (in minutes) of patients receiving an analgesic. The second data set provides the percent of global reserves of natural gas for 44 countries. We have considered the third real data analysis of the countries with the largest oil reserves in 20 countries. We consider another real-data analysis of the central bank owning the largest gold reserves in 100 countries. This gold reserve data, compiled using international monetary funds and international financial statistics, tracks central banks' reported purchases and sales of gold as a percentage of their international reserves. The LBTLoW model typically provides superior fits in comparison to certain other alternative models, as shown by real-world data applications.
The authors declare that they have not used artificial intelligence tools in the creation of this article.
Researchers Supporting Project number (RSPD2023R548), King Saud University, Riyadh, Saudi Arabia.
The authors declare that there are no conflicts of interest.
[1] | Christensen RM (1979) Mechanics of Composite Materials.New York: Wiley . |
[2] | Nemat-Nasser S, Hori M (1999) Micromechanics: Overall Properties of Heterogeneous Solids.Elsevier Science . |
[3] | Torquato S (2002) Random Heterogeneous Materials: Microstructure and Macroscopic Properties.New York: Springer . |
[4] | Sahimi M (2003) Heterogeneous Materials I: Linear Transport and Optical Properties, and II: Nonlinear and Breakdown Properties and Atomistic Modeling.New York: Springer . |
[5] | Millar DIA (2012) Energetic Materials at Extreme Conditions.Springer Ph.D. Thesis . |
[6] | Haymes RC (1971) Introduction to Space Science.New York: John Wiley and Sons Inc . |
[7] | Thornton K, Poulsen HF (2008) Three-dimensional materials science: An intersection of three-dimensional reconstructions and simulations.MRS Bull 33: 587. |
[8] | Brandon D, Kaplan WD (1999) Microstructural Characterization of Materials.New York: John Wiley and Sons . |
[9] | Baruchel J, Bleuet P, Bravin A, et al. (2008) Advances in synchrotron hard x-ray based imaging.C R Physique 9: 624. |
[10] | Kinney JH, Nichols MC (1992) X-ray tomographic microscopy (XTM) using synchrotron radiation.Annu Rev Mater Sci 22: 121. |
[11] | Kak A, Slaney M (1988) Principles of Computerized Tomographic Imaging.IEEE Press . |
[12] | Babout L, Maire E, Buffière JY (2001) Characterisationby X-ray computed tomography of decohesion, porosity growth and coalescence in model metal matrix composites.Acta Mater 49: 2055. |
[13] | Borbély A, Csikor FF, Zabler S, et al. (2004) Three-dimensional characterization of the microstructure of a metal-matrix composite by holotomography.Mater Sci Engng A 367: 40. |
[14] | Kenesei P, Biermann H, Borbély A (2005) Structure-property relationship in particle reinforced metal-matrix composites based on holotomography.Scr Mater 53: 787. |
[15] | Weck A, Wilkinson DS, Maire E (2008) Visualization by x-ray tomography of void growth and coalescence leading to fracture in model materials.Acta Mater 56: 2919. |
[16] | Toda H, Yamamoto S, Kobayashi M (2008) Direct measurement procedure for three-dimensional local crack driving force using synchrotron X-ray microtomography.Acta Mater 56: 6027. |
[17] | Williams JJ, Flom Z, Amell AA (2010) Damage evolution in SiC particle reinforced Al alloy matrix composites by X-ray synchrotron tomography.Acta Mater 58: 6194. |
[18] | Williams JJ, Yazzie KE, Phillips NC (2011) On the correlation between fatigue striation spacing and crack crowth rate: A three-dimensional (3-D) X-ray synchrotron tomography study.Metal Mater Trans 42: 3845. |
[19] | Williams JJ, Yazzie KE, Phillips NC, et al.Understanding fatigue crack growth in aluminum alloys by in situ x-ray synchrotron tomography.Int J Fatigue in press . |
[20] | Groeber M, Ghosh S, Uchic MD (2008) A framework for automated analysis and simulation of 3D polycrystalline microstructures. I: Statistical characterization.Acta Mater 56: 1257. |
[21] | Groeber M, Ghosh S, Uchic MD (2008) A framework for automated analysis and simulation of 3D polycrystalline microstructures. II: Synthetic structure generation.Acta Mater 56: 1274. |
[22] | Niezgoda SR, Fullwood DT, Kalidindi SR (2008) Delineation of the space of 2-point correlations in a composite material system.Acta Mater 56: 5286-5292. |
[23] | Debye P, Anderson HR, Brumberger H (1957) Scattering by an inhomogeneous solid. II. The correlation function and its applications.J. Appl Phys 28: 679-683. |
[24] | Yeong CLY, Torquato S (1998) Reconstructing random media.Phys Rev E 57: 495. |
[25] | Yeong CLY, Torquato S (1998) Reconstructing random media: II. Three-dimensional reconstruction from two-dimensional cuts.Phys Rev E 58: 224. |
[26] | Jiao Y, Stillinger FH, Torquato S (2007) Modeling heterogeneous materials via two-point correlation functions: Basic principles.Phys Rev E 76: 031110. |
[27] | Jiao Y, Stillinger FH, Torquato S (2008) Modeling heterogeneous materials via two-point correlation functions: II. Algorithmic details and applications.Phys Rev E 77: 031135. |
[28] | Jiao Y, Stillinger FH, Torquato S (2008) A superior descriptor of random textures and its predictive capacity.Proc Natl Acad Sci USA 106: 17634. |
[29] | Roberts AP (1997) Statistical reconstruction of three-dimensional porous media from twodimensional images.Phys Rev E 56: 3203-3212. |
[30] | Fullwood DT, Niezgoda SR, Kalidindi SR (2008) Microstructure reconstructions from 2-point statistics using phase-recovery algorithms.Acta Mater 56: 942-948. |
[31] | Alireza H, Aliakbar S, Farhad A. (2011) Farhadpour A multiple-point statistics algorithm for 3D pore space reconstruction from 2D images.Advances in Water Resources 34: 1256-1267. |
[32] | Tahmasebi P, Sahimi M (2013) Cross-correlation function for accurate reconstruction of heterogeneous media.Phys Rev Lett 110: 078002. |
[33] | Torquato S (1986) Interfacial surface statistics arising in diffusion and flow problems in porous media.J Chem Phys 85: 4622. |
[34] | Lu B, Torquato S (1992) Lineal path function for random heterogeneous materials.Phys Rev A 45: 922. |
[35] | Torquato S, Avellaneda M (1991) Diffusion and reaction in heterogeneous media: Pore-size distribution, relaxation rimes, and mean survival time.J Chem Phys 95: 6477. |
[36] | Prager S (1963) Interphase transfer in stationary two-phase media.Chem Eng Sci 18: 228. |
[37] | Torquato S, Beasley JD, Chiew YC (1988) Two-point cluster function for continuum percolation.J Chem Phys 88: 6540. |
[38] | Singh SS, Williams JJ, Jiao Y, et al. (2012) Modeling anisotropic multiphase heterogeneous materials via directional correlation functions: Simulations and experimental verification.Metall Mater Trans 43 A: 4470-4474. |
[39] | Jiao Y, Pallia E, Chawla N (2013) Modeling and predicting microstructure evolution in lead/tin alloy via correlation functions and stochastic material reconstruction.Acta Mater 61: 3370. |
[40] | Liu Y, Greene MS, Chen W, et al. (2013) Computational microstructure characterization and reconstruction for stochastic multiscale material design.Computer-Aided Design 45: 65. |
[41] | Mikdam A, Belouettar R, Fiorelli D, et al. (2013) A tool for design of heterogeneous materials with desired physical properties using statistical continuum theory.Mater Sci Eng A 564: 493. |
[42] | Kirkpatrick S, Gelatt CD, Vecchi MP (1983) Optimization by simulated annealing.Science 220: 671-680. |
[43] | Sheehan N, Torquato S (2001) Generating Microstructures with Specified Correlation Functions.J Appl Phys 89: 53. |
[44] | Rozman MG, Utz M (2002) Uniqueness of reconstruction of multiphase morphologies from two-point correlation functions.Physl Rev Lett 89: 135501. |
[45] | Manko HH (2001) Solders and Soldering: Materials, Design, Production, and Analysis for Reliable Bonding.New York: McGraw-Hill . |
[46] | Gommes CJ (2013) Three-dimensional reconstruction of liquid phases in disordered mesopores using in situ small-angle scattering.J Appl Cryst 46: 493-504. |
[47] | Zhang H, Srolovitz DJ, Douglas JF (2009) Grain boundaries exhibit the dynamics of glass-forming liquids.Proc Natl Acad Sci USA 106: 7729-7734. |
[48] | Chen Z, Chu KT, Srolovitz DJ (2010) Dislocation climb strengthening in systems with immobile obstacles: Three-dimensional level-set simulation study.Phys Rev B 81: 054104. |
1. | Najwan Alsadat, Amal S Hassan, Mohammed Elgarhy, Vasili B V Nagarjuna, Sid Ahmed Benchiha, Ahmed M Gemeay, A novel asymmetric extension of power XLindley distribution: properties, inference and applications to engineering data, 2024, 99, 0031-8949, 105262, 10.1088/1402-4896/ad77fa | |
2. | Mohammed Elgarhy, Arne Johannssen, Mohamed Kayid, An extended Rayleigh Weibull model with actuarial measures and applications, 2024, 10, 24058440, e32143, 10.1016/j.heliyon.2024.e32143 | |
3. | Naif Alotaibi, A.S. Al-Moisheer, Amal S. Hassan, Ibrahim Elbatal, Salem A. Alyami, Ehab M. Almetwally, Epidemiological modeling of COVID-19 data with Advanced statistical inference based on Type-II progressive censoring, 2024, 10, 24058440, e36774, 10.1016/j.heliyon.2024.e36774 | |
4. | Sara Almakhareez, Loai Alzoubi, Extension to Benrabia distribution with applications and parameter estimation, 2024, 2, 2768-4520, 10.1080/27684520.2024.2393597 | |
5. | Abdullah M. Alomair, Ayesha Babar, Muhammad Ahsan-ul-Haq, Saadia Tariq, An improved extension of Xgamma distribution: Its properties, estimation and application on failure time data, 2025, 24058440, e41976, 10.1016/j.heliyon.2025.e41976 | |
6. | Ahmed W. Shawki, Mohamed Kayid, Oluwafemi Samson Balogun, Tamer S. Helal, Modeling to radiotherapy, environmental and engineering data: Using a new approach to generating family of distributions, 2025, 18, 16878507, 101317, 10.1016/j.jrras.2025.101317 | |
7. | Mohammed Elgarhy, Amal S. Hassan, Najwan Alsadat, Oluwafemi Samson Balogun, Ahmed W. Shawki, Ibrahim E. Ragab, A Heavy Tailed Model Based on Power XLindley Distribution with Actuarial Data Applications, 2025, 142, 1526-1506, 2547, 10.32604/cmes.2025.058362 | |
8. | Ehab M. Almetwally, Amal S. Hassan, Mohamed Kayid, Arne Johannssen, Mohammed Elgarhy, A flexible statistical distribution for capturing complex patterns in industrial data, 2025, 126, 11100168, 651, 10.1016/j.aej.2025.05.004 | |
9. | Mohammed Elgarhy, Diaa S. Metwally, Amal S. Hassan, Ahmed W. Shawki, A new generalization of power Chris-Jerry distribution with different estimation methods, simulation and applications, 2025, 29, 24682276, e02769, 10.1016/j.sciaf.2025.e02769 |
$ \gamma $ | $ \alpha $ | $ \mu'_1 $ | $ \mu'_2 $ | $ \mu'_3 $ | $ \mu'_4 $ | $ {\sigma }^{\mathrm{2}} $ | $ CS $ | $ CK $ | $ CV $ |
0.4 | 0.2 | 2.629 | 8.323 | 30.531 | 126.387 | 1.413 | 0.729 | 3.596 | 0.452 |
0.6 | 0.5 | 1.558 | 2.937 | 6.430 | 15.922 | 0.508 | 0.746 | 3.627 | 0.457 |
0.8 | 0.8 | 1.184 | 1.700 | 2.847 | 5.403 | 0.300 | 0.765 | 3.660 | 0.463 |
1.1 | 1.2 | 0.925 | 1.044 | 1.380 | 2.074 | 0.189 | 0.793 | 3.717 | 0.470 |
1.3 | 1.5 | 0.805 | 0.794 | 0.920 | 1.216 | 0.146 | 0.814 | 3.758 | 0.475 |
1.7 | 1.8 | 0.705 | 0.615 | 0.634 | 0.748 | 0.117 | 0.856 | 3.853 | 0.486 |
1.9 | 2.0 | 0.655 | 0.532 | 0.513 | 0.568 | 0.103 | 0.878 | 3.906 | 0.491 |
2.4 | 2.3 | 0.582 | 0.425 | 0.371 | 0.374 | 0.086 | 0.937 | 4.058 | 0.503 |
2.7 | 2.6 | 0.530 | 0.355 | 0.285 | 0.266 | 0.073 | 0.974 | 4.162 | 0.511 |
3.2 | 3.0 | 0.469 | 0.280 | 0.203 | 0.172 | 0.060 | 1.038 | 4.359 | 0.522 |
$ \gamma $ | $ \alpha $ | $ \mu'_1 $ | $ \mu'_2 $ | $ \mu'_3 $ | $ \mu'_4 $ | $ {\sigma }^{\mathrm{2}} $ | $ CS $ | $ CK $ | $ CV $ |
0.4 | 0.2 | 1.964 | 4.277 | 10.154 | 25.945 | 0.420 | 0.380 | 3.017 | 0.330 |
0.6 | 0.5 | 1.347 | 2.017 | 3.298 | 5.813 | 0.202 | 0.395 | 3.026 | 0.334 |
0.8 | 0.8 | 1.105 | 1.360 | 1.829 | 2.658 | 0.139 | 0.411 | 3.037 | 0.337 |
1.1 | 1.2 | 0.924 | 0.954 | 1.081 | 1.325 | 0.100 | 0.435 | 3.058 | 0.343 |
1.3 | 1.5 | 0.836 | 0.783 | 0.805 | 0.897 | 0.084 | 0.453 | 3.075 | 0.346 |
1.7 | 1.8 | 0.760 | 0.649 | 0.612 | 0.627 | 0.072 | 0.489 | 3.116 | 0.353 |
1.9 | 2.0 | 0.720 | 0.584 | 0.524 | 0.511 | 0.066 | 0.508 | 3.140 | 0.356 |
2.4 | 2.3 | 0.661 | 0.494 | 0.411 | 0.374 | 0.058 | 0.557 | 3.213 | 0.365 |
2.7 | 2.6 | 0.617 | 0.433 | 0.339 | 0.290 | 0.052 | 0.588 | 3.265 | 0.369 |
3.2 | 3.0 | 0.565 | 0.364 | 0.263 | 0.209 | 0.045 | 0.641 | 3.366 | 0.377 |
$ \beta $ | $ \alpha $ | $ \mu'_1 $ | $ \mu'_2 $ | $ \mu'_3 $ | $ \mu'_4 $ | $ {\sigma }^{\mathrm{2}} $ | $ CS $ | $ CK $ | $ CV $ |
1.5 | 0.2 | 0.029 | 0.026 | 0.023 | 0.021 | 0.025 | 5.273 | 29.151 | 5.386 |
0.5 | 0.032 | 0.028 | 0.025 | 0.023 | 0.027 | 5.014 | 26.463 | 5.138 | |
0.8 | 0.035 | 0.031 | 0.028 | 0.025 | 0.030 | 4.772 | 24.069 | 4.906 | |
1.2 | 0.039 | 0.035 | 0.031 | 0.028 | 0.033 | 4.472 | 21.275 | 4.622 | |
1.5 | 0.043 | 0.038 | 0.033 | 0.03 | 0.036 | 4.264 | 19.441 | 4.424 | |
1.8 | 0.046 | 0.041 | 0.036 | 0.033 | 0.038 | 4.069 | 17.801 | 4.241 | |
2 | 0.049 | 0.043 | 0.038 | 0.034 | 0.040 | 3.946 | 16.806 | 4.125 | |
2.3 | 0.053 | 0.046 | 0.041 | 0.037 | 0.043 | 3.771 | 15.445 | 3.961 | |
2.6 | 0.056 | 0.049 | 0.044 | 0.039 | 0.046 | 3.607 | 14.225 | 3.809 | |
3 | 0.062 | 0.054 | 0.048 | 0.043 | 0.050 | 3.405 | 12.794 | 3.621 | |
2.5 | 0.2 | 0.006 | 0.005 | 0.005 | 0.004 | 0.005 | 12.420 | 156.328 | 12.403 |
0.5 | 0.007 | 0.006 | 0.006 | 0.005 | 0.006 | 11.672 | 138.195 | 11.665 | |
0.8 | 0.007 | 0.007 | 0.006 | 0.006 | 0.007 | 10.981 | 122.434 | 10.984 | |
1.2 | 0.009 | 0.008 | 0.007 | 0.007 | 0.008 | 10.140 | 104.548 | 10.156 | |
1.5 | 0.01 | 0.009 | 0.008 | 0.007 | 0.009 | 9.563 | 93.121 | 9.589 | |
1.8 | 0.011 | 0.010 | 0.009 | 0.008 | 0.010 | 9.030 | 83.140 | 9.066 | |
2 | 0.012 | 0.011 | 0.010 | 0.009 | 0.010 | 8.697 | 77.190 | 8.739 | |
2.3 | 0.013 | 0.012 | 0.011 | 0.010 | 0.012 | 8.227 | 69.194 | 8.279 | |
2.6 | 0.014 | 0.013 | 0.012 | 0.011 | 0.013 | 7.792 | 62.179 | 7.853 | |
3 | 0.017 | 0.015 | 0.014 | 0.013 | 0.015 | 7.261 | 54.128 | 7.334 |
$ \varepsilon $ | $ \beta $ | $ \alpha $ | $ \gamma $ | RE | HaCE | ArE | TsE |
1.5 | 0.25 | 0.2 | 0.4 | 3.331 | 3.340 | 2.767 | 1.957 |
0.5 | 0.6 | 3.252 | 3.333 | 2.753 | 1.953 | ||
0.8 | 0.8 | 3.099 | 3.318 | 2.722 | 1.944 | ||
1.2 | 1.1 | 2.875 | 3.290 | 2.670 | 1.927 | ||
1.5 | 1.3 | 2.716 | 3.264 | 2.627 | 1.912 | ||
1.8 | 1.7 | 2.524 | 3.227 | 2.568 | 1.891 | ||
2.0 | 1.9 | 2.415 | 3.203 | 2.53 | 1.876 | ||
2.3 | 2.4 | 2.229 | 3.152 | 2.458 | 1.846 | ||
2.6 | 2.7 | 2.084 | 3.104 | 2.394 | 1.818 | ||
3.0 | 3.2 | 1.889 | 3.026 | 2.296 | 1.773 | ||
0.5 | 0.2 | 0.4 | 1.930 | 3.044 | 2.318 | 1.783 | |
0.5 | 0.6 | 1.924 | 3.042 | 2.315 | 1.782 | ||
0.8 | 0.8 | 1.824 | 2.996 | 2.260 | 1.755 | ||
1.2 | 1.1 | 1.660 | 2.909 | 2.161 | 1.704 | ||
1.5 | 1.3 | 1.541 | 2.835 | 2.081 | 1.661 | ||
1.8 | 1.7 | 1.392 | 2.726 | 1.969 | 1.597 | ||
2.0 | 1.9 | 1.308 | 2.657 | 1.901 | 1.556 | ||
2.3 | 2.4 | 1.161 | 2.517 | 1.770 | 1.475 | ||
2.6 | 2.7 | 1.051 | 2.397 | 1.661 | 1.404 | ||
3.0 | 3.2 | 0.903 | 2.207 | 1.500 | 1.293 | ||
2.0 | 0.25 | 0.2 | 0.4 | 2.180 | 1.987 | 1.837 | 0.993 |
0.5 | 0.6 | 2.167 | 1.986 | 1.835 | 0.993 | ||
0.8 | 0.8 | 2.053 | 1.982 | 1.812 | 0.991 | ||
1.2 | 1.1 | 1.876 | 1.973 | 1.769 | 0.987 | ||
1.5 | 1.3 | 1.753 | 1.965 | 1.734 | 0.982 | ||
1.8 | 1.7 | 1.595 | 1.949 | 1.681 | 0.975 | ||
2.0 | 1.9 | 1.510 | 1.938 | 1.648 | 0.969 | ||
2.3 | 2.4 | 1.358 | 1.912 | 1.581 | 0.956 | ||
2.6 | 2.7 | 1.247 | 1.887 | 1.524 | 0.943 | ||
3.0 | 3.2 | 1.098 | 1.840 | 1.435 | 0.920 | ||
0.5 | 0.2 | 0.4 | 1.210 | 1.877 | 1.503 | 0.938 | |
0.5 | 0.6 | 1.279 | 1.895 | 1.541 | 0.947 | ||
0.8 | 0.8 | 1.210 | 1.877 | 1.503 | 0.938 | ||
1.2 | 1.1 | 1.080 | 1.834 | 1.423 | 0.917 | ||
1.5 | 1.3 | 0.987 | 1.794 | 1.358 | 0.897 | ||
1.8 | 1.7 | 0.861 | 1.725 | 1.258 | 0.862 | ||
2.0 | 1.9 | 0.795 | 1.679 | 1.199 | 0.840 | ||
2.3 | 2.4 | 0.672 | 1.574 | 1.077 | 0.787 | ||
2.6 | 2.7 | 0.587 | 1.482 | 0.982 | 0.741 | ||
3.0 | 3.2 | 0.470 | 1.323 | 0.836 | 0.661 |
$ \varepsilon $ | $ \beta $ | $ \alpha $ | RE | HaCE | ArE | TsE |
1.5 | 0.25 | 0.2 | 7.001 | 3.311 | 2.709 | 1.94 |
0.5 | 7.075 | 3.315 | 2.716 | 1.942 | ||
0.8 | 7.156 | 3.319 | 2.724 | 1.944 | ||
1.2 | 7.274 | 3.324 | 2.735 | 1.947 | ||
1.5 | 7.372 | 3.329 | 2.743 | 1.95 | ||
1.8 | 7.476 | 3.333 | 2.752 | 1.952 | ||
2 | 7.55 | 3.336 | 2.758 | 1.954 | ||
2.3 | 7.667 | 3.34 | 2.767 | 1.957 | ||
2.6 | 7.792 | 3.345 | 2.777 | 1.959 | ||
3 | 7.971 | 3.351 | 2.79 | 1.963 | ||
0.4 | 0.2 | 6.441 | 3.278 | 2.65 | 1.92 | |
0.5 | 6.452 | 3.279 | 2.651 | 1.921 | ||
0.8 | 6.469 | 3.28 | 2.653 | 1.921 | ||
1.2 | 6.503 | 3.282 | 2.657 | 1.923 | ||
1.5 | 6.536 | 3.284 | 2.66 | 1.924 | ||
1.8 | 6.577 | 3.287 | 2.665 | 1.925 | ||
2 | 6.608 | 3.289 | 2.668 | 1.927 | ||
2.3 | 6.661 | 3.292 | 2.674 | 1.928 | ||
2.6 | 6.721 | 3.296 | 2.681 | 1.931 | ||
3 | 6.813 | 3.301 | 2.69 | 1.934 | ||
2.0 | 0.25 | 0.2 | 4.376 | 1.975 | 1.776 | 0.987 |
0.5 | 4.429 | 1.976 | 1.782 | 0.988 | ||
0.8 | 4.487 | 1.977 | 1.788 | 0.989 | ||
1.2 | 4.57 | 1.979 | 1.796 | 0.99 | ||
1.5 | 4.639 | 1.981 | 1.803 | 0.99 | ||
1.8 | 4.713 | 1.982 | 1.81 | 0.991 | ||
2 | 4.765 | 1.983 | 1.815 | 0.991 | ||
2.3 | 4.847 | 1.984 | 1.823 | 0.992 | ||
2.6 | 4.934 | 1.986 | 1.83 | 0.993 | ||
3 | 5.058 | 1.987 | 1.841 | 0.994 | ||
0.4 | 0.2 | 3.975 | 1.962 | 1.726 | 0.981 | |
0.5 | 3.987 | 1.963 | 1.728 | 0.981 | ||
0.8 | 4.003 | 1.963 | 1.73 | 0.982 | ||
1.2 | 4.031 | 1.965 | 1.734 | 0.982 | ||
1.5 | 4.058 | 1.965 | 1.737 | 0.983 | ||
1.8 | 4.09 | 1.967 | 1.741 | 0.983 | ||
2 | 4.114 | 1.967 | 1.744 | 0.984 | ||
2.3 | 4.154 | 1.969 | 1.749 | 0.984 | ||
2.6 | 4.199 | 1.97 | 1.755 | 0.985 | ||
3 | 4.267 | 1.972 | 1.763 | 0.986 |
n | r | Set1 ($ \alpha $ = 0.5, $ \beta $ = 0.5, $ \gamma $ = 0.5) | |||||||
MLE | Bias | MSE | LB | UB | AL | CP | |||
50 | 70% | $ \alpha $ | 0.4204 | 0.0796 | 0.0064 | 0.0019 | 0.839 | 0.8370 | 97.4% |
$ \beta $ | 0.7041 | 0.2041 | 0.0471 | 0.5036 | 0.9046 | 0.4010 | 96.9% | ||
$ \gamma $ | 0.4201 | 0.0799 | 0.0069 | 0.3023 | 0.5379 | 0.2356 | 96.0% | ||
80% | $ \alpha $ | 0.4218 | 0.0782 | 0.0061 | 0.0191 | 0.8245 | 0.8053 | 94.8% | |
$ \beta $ | 0.6382 | 0.1382 | 0.0242 | 0.4508 | 0.8256 | 0.3748 | 95.8% | ||
$ \gamma $ | 0.4386 | 0.0614 | 0.0053 | 0.3282 | 0.5490 | 0.2208 | 97.1% | ||
100% | $ \alpha $ | 0.4234 | 0.0766 | 0.0059 | 0.0357 | 0.8111 | 0.7754 | 95.4% | |
$ \beta $ | 0.5177 | 0.0177 | 0.0056 | 0.3661 | 0.6694 | 0.3033 | 95.5% | ||
$ \gamma $ | 0.5316 | 0.0316 | 0.0027 | 0.4303 | 0.6328 | 0.2025 | 96.0% | ||
100 | 70% | $ \alpha $ | 0.4213 | 0.0787 | 0.0062 | 0.0844 | 0.7583 | 0.6740 | 96.2% |
$ \beta $ | 0.6750 | 0.1750 | 0.0312 | 0.5375 | 0.8125 | 0.2750 | 95.9% | ||
$ \gamma $ | 0.4237 | 0.0763 | 0.0065 | 0.3389 | 0.5084 | 0.1694 | 96.0% | ||
80% | $ \alpha $ | 0.4230 | 0.0770 | 0.0061 | 0.2099 | 0.6360 | 0.4262 | 96.2% | |
$ \beta $ | 0.6099 | 0.1099 | 0.0127 | 0.4819 | 0.7379 | 0.2560 | 96.1% | ||
$ \gamma $ | 0.4487 | 0.0513 | 0.0033 | 0.3652 | 0.5321 | 0.1669 | 97.3% | ||
100% | $ \alpha $ | 0.4238 | 0.0762 | 0.0058 | 0.2501 | 0.5975 | 0.3473 | 95.6% | |
$ \beta $ | 0.4683 | 0.0317 | 0.0027 | 0.3558 | 0.5807 | 0.2249 | 95.8% | ||
$ \gamma $ | 0.4967 | 0.0033 | 0.0025 | 0.4199 | 0.5734 | 0.1535 | 96.0% | ||
150 | 70% | $ \alpha $ | 0.4217 | 0.0783 | 0.0061 | 0.2710 | 0.5725 | 0.3015 | 95.2% |
$ \beta $ | 0.6626 | 0.1626 | 0.0281 | 0.5577 | 0.7675 | 0.2097 | 95.6% | ||
$ \gamma $ | 0.4277 | 0.0723 | 0.0058 | 0.3571 | 0.4983 | 0.1412 | 97.3% | ||
80% | $ \alpha $ | 0.4236 | 0.0764 | 0.0059 | 0.3005 | 0.5466 | 0.2461 | 95.7% | |
$ \beta $ | 0.5977 | 0.0977 | 0.0113 | 0.4957 | 0.6997 | 0.2040 | 96.2% | ||
$ \gamma $ | 0.4649 | 0.0351 | 0.0022 | 0.3972 | 0.5325 | 0.1353 | 97.0% | ||
100% | $ \alpha $ | 0.4238 | 0.0762 | 0.0058 | 0.3010 | 0.5467 | 0.2457 | 95.6% | |
$ \beta $ | 0.4766 | 0.0234 | 0.0023 | 0.3784 | 0.5749 | 0.1965 | 96.4% | ||
$ \gamma $ | 0.5277 | 0.0277 | 0.0015 | 0.4659 | 0.5894 | 0.1236 | 96.9% | ||
200 | 70% | $ \alpha $ | 0.4219 | 0.0781 | 0.0061 | 0.3154 | 0.5285 | 0.2132 | 96.1% |
$ \beta $ | 0.6592 | 0.1592 | 0.0268 | 0.5675 | 0.7510 | 0.1835 | 96.3% | ||
$ \gamma $ | 0.4375 | 0.0625 | 0.0046 | 0.3789 | 0.4962 | 0.1173 | 96.7% | ||
80% | $ \alpha $ | 0.4239 | 0.0761 | 0.0058 | 0.3236 | 0.5242 | 0.2006 | 96.3% | |
$ \beta $ | 0.5912 | 0.0912 | 0.0099 | 0.5074 | 0.6750 | 0.1676 | 97.0% | ||
$ \gamma $ | 0.4667 | 0.0333 | 0.0020 | 0.4101 | 0.5233 | 0.1132 | 97.5% | ||
100% | $ \alpha $ | 0.4240 | 0.0760 | 0.0058 | 0.3372 | 0.5109 | 0.1737 | 96.5% | |
$ \beta $ | 0.4905 | 0.0095 | 0.0009 | 0.4167 | 0.5642 | 0.1475 | 96.7% | ||
$ \gamma $ | 0.5035 | 0.0035 | 0.0006 | 0.4524 | 0.5546 | 0.1022 | 97.1% | ||
n | r | Set2 ($ \alpha $ = 0.7, $ \beta $ = 0.5, $ \gamma $ = 0.25) | |||||||
MLE | Bias | MSE | LB | UB | AL | CP | |||
50 | 70% | $ \alpha $ | 0.4206 | 0.2794 | 0.0782 | 0.0055 | 0.8358 | 0.8304 | 97.7% |
$ \beta $ | 0.7016 | 0.2016 | 0.0462 | 0.5015 | 0.9016 | 0.4001 | 96.5% | ||
$ \gamma $ | 0.2172 | 0.0328 | 0.0017 | 0.1568 | 0.2776 | 0.1208 | 100% | ||
80% | $ \alpha $ | 0.4214 | 0.2786 | 0.0776 | 0.2084 | 0.6345 | 0.4261 | 97.9% | |
$ \beta $ | 0.6361 | 0.1361 | 0.0243 | 0.4490 | 0.8231 | 0.3741 | 98.5% | ||
$ \gamma $ | 0.2297 | 0.0203 | 0.0010 | 0.1719 | 0.2874 | 0.1155 | 100% | ||
100% | $ \alpha $ | 0.4234 | 0.2766 | 0.0765 | 0.2497 | 0.5970 | 0.3473 | 98.3% | |
$ \beta $ | 0.5161 | 0.0161 | 0.0062 | 0.3649 | 0.6673 | 0.3025 | 97.6% | ||
$ \gamma $ | 0.2565 | 0.0065 | 0.0007 | 0.2044 | 0.3086 | 0.1043 | 100% | ||
100 | 70% | $ \alpha $ | 0.4210 | 0.2790 | 0.0779 | 0.0208 | 0.8212 | 0.8004 | 96.4% |
$ \beta $ | 0.7006 | 0.2006 | 0.0431 | 0.5593 | 0.8419 | 0.2826 | 98.0% | ||
$ \gamma $ | 0.2141 | 0.0359 | 0.0016 | 0.1720 | 0.2562 | 0.0842 | 100% | ||
80% | $ \alpha $ | 0.4215 | 0.2785 | 0.0776 | 0.2708 | 0.5721 | 0.3013 | 97.2% | |
$ \beta $ | 0.6357 | 0.1357 | 0.0214 | 0.5035 | 0.7680 | 0.2645 | 97.7% | ||
$ \gamma $ | 0.2270 | 0.0230 | 0.0008 | 0.1866 | 0.2673 | 0.0807 | 100% | ||
100% | $ \alpha $ | 0.4234 | 0.2766 | 0.0765 | 0.3006 | 0.5462 | 0.2456 | 97.3% | |
$ \beta $ | 0.5158 | 0.0158 | 0.0033 | 0.4088 | 0.6227 | 0.2140 | 98.2% | ||
$ \gamma $ | 0.2540 | 0.0040 | 0.0003 | 0.2176 | 0.2905 | 0.0729 | 100% | ||
150 | 70% | $ \alpha $ | 0.4212 | 0.2788 | 0.0778 | 0.0330 | 0.8093 | 0.7763 | 97.7% |
$ \beta $ | 0.7000 | 0.2000 | 0.0419 | 0.5847 | 0.8153 | 0.2306 | 97.7% | ||
$ \gamma $ | 0.2122 | 0.0378 | 0.0016 | 0.1781 | 0.2463 | 0.0682 | 100% | ||
80% | $ \alpha $ | 0.4215 | 0.2785 | 0.0776 | 0.2985 | 0.5445 | 0.2460 | 98.8% | |
$ \beta $ | 0.6350 | 0.1350 | 0.0203 | 0.5271 | 0.7430 | 0.2159 | 98.1% | ||
$ \gamma $ | 0.2259 | 0.0241 | 0.0008 | 0.1931 | 0.2587 | 0.0656 | 96.0% | ||
100% | $ \alpha $ | 0.4234 | 0.2766 | 0.0765 | 0.3232 | 0.5237 | 0.2005 | 97.2% | |
$ \beta $ | 0.5151 | 0.0151 | 0.0023 | 0.4278 | 0.6024 | 0.1746 | 97.0% | ||
$ \gamma $ | 0.2529 | 0.0029 | 0.0002 | 0.2232 | 0.2825 | 0.0593 | 95.4% | ||
200 | 70% | $ \alpha $ | 0.4209 | 0.2791 | 0.0779 | 0.0849 | 0.7569 | 0.6720 | 100% |
$ \beta $ | 0.6981 | 0.1981 | 0.0405 | 0.5984 | 0.7978 | 0.1994 | 97.2% | ||
$ \gamma $ | 0.2118 | 0.0382 | 0.0016 | 0.1823 | 0.2412 | 0.0589 | 97.3% | ||
80% | $ \alpha $ | 0.4215 | 0.2785 | 0.0776 | 0.3150 | 0.5280 | 0.2131 | 100% | |
$ \beta $ | 0.6331 | 0.1331 | 0.0191 | 0.5398 | 0.7265 | 0.1867 | 98.2% | ||
$ \gamma $ | 0.2256 | 0.0244 | 0.0007 | 0.1973 | 0.2540 | 0.0567 | 98.0% | ||
100% | $ \alpha $ | 0.4234 | 0.2766 | 0.0765 | 0.3366 | 0.5103 | 0.1737 | 100% | |
$ \beta $ | 0.5136 | 0.0136 | 0.0016 | 0.4381 | 0.5891 | 0.1510 | 98.8% | ||
$ \gamma $ | 0.2523 | 0.0023 | 0.0002 | 0.2267 | 0.2779 | 0.0512 | 100% | ||
n | r | Set3 ($ \alpha $ = 0.7, $ \beta $ = 0.7, $ \gamma $ = 0.5) | |||||||
MLE | Bias | MSE | LB | UB | AL | CP | |||
50 | 70% | $ \alpha $ | 0.4178 | 0.2822 | 0.0797 | 0.1937 | 0.6419 | 0.4482 | 96.2% |
$ \beta $ | 0.8994 | 0.1994 | 0.0477 | 0.6064 | 1.1923 | 0.5859 | 95.9% | ||
$ \gamma $ | 0.6151 | 0.1151 | 0.0239 | 0.3763 | 0.8540 | 0.4776 | 95.0% | ||
80% | $ \alpha $ | 0.4193 | 0.2807 | 0.0788 | 0.2255 | 0.6131 | 0.3875 | 95.9% | |
$ \beta $ | 0.8238 | 0.1238 | 0.0218 | 0.5471 | 1.1006 | 0.5535 | 95.9% | ||
$ \gamma $ | 0.5695 | 0.0695 | 0.0182 | 0.3447 | 0.7943 | 0.4496 | 96.7% | ||
100% | $ \alpha $ | 0.4211 | 0.2789 | 0.0778 | 0.2480 | 0.5942 | 0.3461 | 96.8% | |
$ \beta $ | 0.7612 | 0.0612 | 0.0104 | 0.5012 | 1.0213 | 0.5201 | 97.0% | ||
$ \gamma $ | 0.5425 | 0.0425 | 0.0163 | 0.3395 | 0.7456 | 0.4061 | 95.0% | ||
100 | 70% | $ \alpha $ | 0.4174 | 0.2826 | 0.0798 | 0.2439 | 0.5910 | 0.3470 | 95.0% |
$ \beta $ | 0.8787 | 0.1787 | 0.0353 | 0.6426 | 1.1148 | 0.4722 | 96.3% | ||
$ \gamma $ | 0.5696 | 0.0696 | 0.0201 | 0.3697 | 0.7696 | 0.3998 | 96.0% | ||
80% | $ \alpha $ | 0.4191 | 0.2809 | 0.0789 | 0.2690 | 0.5691 | 0.3001 | 95.5% | |
$ \beta $ | 0.7802 | 0.0802 | 0.0134 | 0.5600 | 1.0004 | 0.4404 | 95.7% | ||
$ \gamma $ | 0.5479 | 0.0479 | 0.0091 | 0.3814 | 0.7144 | 0.3330 | 96.0% | ||
100% | $ \alpha $ | 0.4206 | 0.2794 | 0.0781 | 0.2866 | 0.5546 | 0.2680 | 95.6% | |
$ \beta $ | 0.7146 | 0.0146 | 0.0068 | 0.5069 | 0.9223 | 0.4154 | 95.7% | ||
$ \gamma $ | 0.5597 | 0.0597 | 0.0074 | 0.4063 | 0.7131 | 0.3067 | 96.0% | ||
150 | 70% | $ \alpha $ | 0.4176 | 0.2824 | 0.0798 | 0.2949 | 0.5403 | 0.2454 | 95.8% |
$ \beta $ | 0.8697 | 0.1697 | 0.0305 | 0.7056 | 1.0338 | 0.3282 | 96.2% | ||
$ \gamma $ | 0.5924 | 0.0924 | 0.0174 | 0.4449 | 0.7399 | 0.2950 | 97.1% | ||
80% | $ \alpha $ | 0.4193 | 0.2807 | 0.0788 | 0.3131 | 0.5254 | 0.2122 | 96.2% | |
$ \beta $ | 0.8023 | 0.1023 | 0.0124 | 0.6492 | 0.9555 | 0.3063 | 96.1% | ||
$ \gamma $ | 0.5524 | 0.0524 | 0.0058 | 0.4332 | 0.6716 | 0.2384 | 97.0% | ||
100% | $ \alpha $ | 0.4209 | 0.2791 | 0.0779 | 0.3261 | 0.5156 | 0.1895 | 95.8% | |
$ \beta $ | 0.7374 | 0.0374 | 0.0032 | 0.5932 | 0.8817 | 0.2885 | 96.3% | ||
$ \gamma $ | 0.5507 | 0.0507 | 0.0041 | 0.4447 | 0.6568 | 0.2121 | 96.9% | ||
200 | 70% | $ \alpha $ | 0.4175 | 0.2825 | 0.0798 | 0.3173 | 0.5177 | 0.2004 | 96.1% |
$ \beta $ | 0.8406 | 0.1406 | 0.0265 | 0.7075 | 0.9736 | 0.2661 | 97.2% | ||
$ \gamma $ | 0.5683 | 0.0683 | 0.0069 | 0.4569 | 0.6798 | 0.2230 | 96.9% | ||
80% | $ \alpha $ | 0.4193 | 0.2807 | 0.0788 | 0.3327 | 0.5059 | 0.1733 | 96.3% | |
$ \beta $ | 0.8043 | 0.1043 | 0.0123 | 0.6790 | 0.9295 | 0.2505 | 96.6% | ||
$ \gamma $ | 0.5502 | 0.0502 | 0.0041 | 0.4533 | 0.6471 | 0.1938 | 97.0% | ||
100% | $ \alpha $ | 0.4209 | 0.2791 | 0.0779 | 0.3435 | 0.4983 | 0.1548 | 96.1% | |
$ \beta $ | 0.7411 | 0.0411 | 0.0030 | 0.6229 | 0.8592 | 0.2362 | 97.0% | ||
$ \gamma $ | 0.5463 | 0.0463 | 0.0041 | 0.4589 | 0.6338 | 0.1749 | 96.2% | ||
n | r | Set4 ($ \alpha $ = 0.6, $ \beta $ = 0.3, $ \gamma $ = 0.5) | |||||||
MLE | Bias | MSE | LB | UB | AL | CP | |||
50 | 70% | $ \alpha $ | 0.4197 | 0.1803 | 0.0325 | 0.2456 | 0.5939 | 0.3482 | 98.1% |
$ \beta $ | 0.5719 | 0.2719 | 0.0744 | 0.3742 | 0.7696 | 0.3954 | 97.0% | ||
$ \gamma $ | 0.2990 | 0.2010 | 0.0406 | 0.1769 | 0.4211 | 0.2442 | 98.0% | ||
80% | $ \alpha $ | 0.4221 | 0.1779 | 0.0317 | 0.2716 | 0.5725 | 0.3009 | 98.4% | |
$ \beta $ | 0.4934 | 0.1934 | 0.0384 | 0.3126 | 0.6742 | 0.3616 | 97.4% | ||
$ \gamma $ | 0.3593 | 0.1407 | 0.0200 | 0.2383 | 0.4803 | 0.2421 | 98.2% | ||
100% | $ \alpha $ | 0.4246 | 0.1754 | 0.0308 | 0.2903 | 0.5589 | 0.2686 | 97.9% | |
$ \beta $ | 0.4198 | 0.1198 | 0.0150 | 0.2576 | 0.5820 | 0.3244 | 97.4% | ||
$ \gamma $ | 0.4172 | 0.0828 | 0.0078 | 0.2968 | 0.5376 | 0.2408 | 98.4% | ||
100 | 70% | $ \alpha $ | 0.4198 | 0.1802 | 0.0325 | 0.2966 | 0.5429 | 0.2463 | 97.2% |
$ \beta $ | 0.5674 | 0.2674 | 0.0717 | 0.4278 | 0.7069 | 0.2791 | 97.7% | ||
$ \gamma $ | 0.3137 | 0.1863 | 0.0351 | 0.2278 | 0.3995 | 0.1717 | 98.0% | ||
80% | $ \alpha $ | 0.4222 | 0.1778 | 0.0316 | 0.3157 | 0.5286 | 0.2128 | 97.9% | |
$ \beta $ | 0.4857 | 0.1857 | 0.0350 | 0.3584 | 0.6130 | 0.2546 | 97.9% | ||
$ \gamma $ | 0.3669 | 0.1331 | 0.0179 | 0.2818 | 0.4519 | 0.1701 | 98.7% | ||
100% | $ \alpha $ | 0.4248 | 0.1752 | 0.0307 | 0.3242 | 0.5253 | 0.2011 | 98.1% | |
$ \beta $ | 0.4082 | 0.1082 | 0.0125 | 0.2944 | 0.5221 | 0.2277 | 97.8% | ||
$ \gamma $ | 0.4171 | 0.0829 | 0.0074 | 0.3349 | 0.4993 | 0.1644 | 98.3% | ||
150 | 70% | $ \alpha $ | 0.4198 | 0.1802 | 0.0325 | 0.3248 | 0.5148 | 0.1899 | 97.2% |
$ \beta $ | 0.5653 | 0.2653 | 0.0706 | 0.4518 | 0.6788 | 0.2270 | 98.3% | ||
$ \gamma $ | 0.3140 | 0.1860 | 0.0350 | 0.2421 | 0.3860 | 0.1439 | 99.3% | ||
80% | $ \alpha $ | 0.4222 | 0.1778 | 0.0316 | 0.3353 | 0.5091 | 0.1738 | 97.7% | |
$ \beta $ | 0.4832 | 0.1832 | 0.0338 | 0.3797 | 0.5866 | 0.2069 | 98.2% | ||
$ \gamma $ | 0.3677 | 0.1323 | 0.0178 | 0.2996 | 0.4358 | 0.1363 | 99.0% | ||
100% | $ \alpha $ | 0.4248 | 0.1752 | 0.0307 | 0.3377 | 0.5119 | 0.1742 | 97.3% | |
$ \beta $ | 0.4018 | 0.1018 | 0.0107 | 0.3032 | 0.5004 | 0.1972 | 97.9% | ||
$ \gamma $ | 0.4248 | 0.0752 | 0.0061 | 0.3608 | 0.4888 | 0.1280 | 99.7% | ||
200 | 70% | $ \alpha $ | 0.4198 | 0.1802 | 0.0325 | 0.3423 | 0.4973 | 0.1551 | 99.0% |
$ \beta $ | 0.5650 | 0.2650 | 0.0705 | 0.4738 | 0.6561 | 0.1823 | 99.1% | ||
$ \gamma $ | 0.3256 | 0.1744 | 0.0309 | 0.2637 | 0.3875 | 0.1239 | 98.7% | ||
80% | $ \alpha $ | 0.4222 | 0.1778 | 0.0316 | 0.3470 | 0.4975 | 0.1505 | 99.6% | |
$ \beta $ | 0.4781 | 0.1781 | 0.0321 | 0.3882 | 0.5679 | 0.1798 | 99.3% | ||
$ \gamma $ | 0.3733 | 0.1267 | 0.0173 | 0.3129 | 0.4336 | 0.1206 | 99.5% | ||
100% | $ \alpha $ | 0.4250 | 0.1750 | 0.0306 | 0.3578 | 0.4921 | 0.1343 | 98.7% | |
$ \beta $ | 0.3904 | 0.0904 | 0.0088 | 0.3106 | 0.4703 | 0.1596 | 99.6% | ||
$ \gamma $ | 0.4262 | 0.0738 | 0.0060 | 0.3681 | 0.4843 | 0.1162 | 100% |
Rank | Country | % Global Reserves | Rank | Country | % Global Reserves |
1 | Russia | 19.9 | 23 | Ukraine | 0.6 |
2 | Iran | 17.1 | 24 | Malaysia | 0.5 |
3 | Qatar | 13.1 | 25 | Uzbekistan | 0.4 |
4 | Turkmenistan | 7.2 | 26 | Oman | 0.4 |
5 | United States | 6.7 | 27 | Vietnam | 0.3 |
6 | China | 4.5 | 28 | Israel | 0.3 |
7 | Venezuela | 3.3 | 29 | Argentina | 0.2 |
8 | Saudi Arabia | 3.2 | 30 | Pakistan | 0.2 |
9 | United Arab Emirates | 3.2 | 31 | Trinidad | 0.2 |
10 | Nigeria | 2.9 | 32 | Brazil | 0.2 |
11 | Iraq | 1.9 | 33 | Myanmar | 0.2 |
12 | Canada | 1.3 | 34 | United Kingdom | 0.1 |
13 | Australia | 1.3 | 35 | Thailand | 0.1 |
14 | Azerbaijan | 1.3 | 36 | Mexico | 0.1 |
15 | Algeria | 1.2 | 37 | Bangladesh | 0.1 |
16 | Kazakhstan | 1.2 | 38 | Netherlands | 0.1 |
17 | Egypt | 1.1 | 39 | Bolivia | 0.1 |
18 | Kuwait | 0.9 | 40 | Brunei | 0.1 |
19 | Norway | 0.8 | 41 | Peru | 0.1 |
20 | Libya | 0.8 | 42 | Syria | 0.1 |
21 | Indonesia | 0.7 | 43 | Yemen | 0.1 |
22 | India | 0.7 | 44 | Papua New Guinea | 0.1 |
Rank | Country | reserves2020 | Rank | Country | reserves2020 |
1 | Venezuela | 303.8 | 11 | Nigeria | 36.9 |
2 | Saudi Arabia | 297.5 | 12 | Kazakhstan | 30 |
3 | Canada | 168.1 | 13 | China | 26 |
4 | Iran | 157.8 | 14 | Qatar | 25.2 |
5 | Iraq | 145 | 15 | Algeria | 12.2 |
6 | Russia | 107.8 | 16 | Brazil | 11.9 |
7 | Kuwait | 101.5 | 17 | Norway | 7.9 |
8 | United Arab Emirates | 97.8 | 18 | Angola | 7.8 |
9 | United States | 68.8 | 19 | Azerbaijan | 7 |
10 | Libya | 48.4 | 20 | Mexico | 6.1 |
Rank | Country | Reserves of Gold | Rank | Country | Reserves of Gold | Rank | Country | Reserves of Gold |
1 | USA | 8.1335 | 35 | LBY | 0.1166 | 68 | CYP | 0.0139 |
2 | DEU | 3.3585 | 36 | GRC | 0.1141 | 69 | CUW | 0.0131 |
3 | IMF | 2.814 | 37 | ROK | 0.1045 | 70 | MUS | 0.0124 |
4 | ITA | 2.4518 | 38 | ROU | 0.1036 | 71 | IRL | 0.012 |
5 | FRA | 2.4365 | 39 | BIS | 0.102 | 72 | CZE | 0.0109 |
6 | RUS | 2.2985 | 40 | IRQ | 0.0964 | 73 | KGZ | 0.0102 |
7 | CHN | 1.9483 | 41 | HUN | 0.0945 | 74 | GHA | 0.0087 |
8 | CHE | 1.04 | 42 | AUS | 0.0798 | 75 | PRY | 0.0082 |
9 | JPN | 0.846 | 43 | KWT | 0.079 | 76 | NPL | 0.008 |
10 | IND | 0.7604 | 44 | IDN | 0.0786 | 77 | MNG | 0.0076 |
11 | NLD | 0.6125 | 45 | DNK | 0.0666 | 78 | MMR | 0.0073 |
12 | ECB | 0.5048 | 46 | PAK | 0.0647 | 79 | GTM | 0.0069 |
13 | TUR | 0.4311 | 47 | ARG | 0.0617 | 80 | MKD | 0.0069 |
14 | TAI | 0.4236 | 48 | ARE | 0.0553 | 81 | TUN | 0.0068 |
15 | PRT | 0.3826 | 49 | BLR | 0.0535 | 82 | LVA | 0.0067 |
16 | KAZ | 0.3681 | 50 | QAT | 0.0513 | 83 | LTU | 0.0058 |
17 | UZB | 0.3375 | 51 | KHM | 0.0504 | 84 | COL | 0.0047 |
18 | SAU | 0.3231 | 52 | FIN | 0.049 | 85 | BHR | 0.0047 |
19 | GBR | 0.3103 | 53 | JOR | 0.0435 | 86 | BRN | 0.0046 |
20 | LBN | 0.2868 | 54 | BOL | 0.0425 | 87 | GIN | 0.0042 |
21 | ESP | 0.2816 | 55 | BGR | 0.0408 | 88 | MOZ | 0.0039 |
22 | AUT | 0.28 | 56 | MYS | 0.0389 | 89 | SVN | 0.0032 |
23 | THA | 0.2442 | 57 | SRB | 0.0378 | 90 | ABW | 0.0031 |
24 | POL | 0.2287 | 58 | WAEMU | 0.0365 | 91 | BIH | 0.003 |
25 | BEL | 0.2274 | 59 | PER | 0.0347 | 92 | ALB | 0.0028 |
26 | DZA | 0.1736 | 60 | SVK | 0.0317 | 93 | LUX | 0.0022 |
27 | VEN | 0.1612 | 61 | UKR | 0.0271 | 94 | HKG | 0.0021 |
28 | PHL | 0.1563 | 62 | SYR | 0.0258 | 95 | ISL | 0.002 |
29 | SGP | 0.1537 | 63 | MAR | 0.0221 | 96 | TTO | 0.0019 |
30 | BRA | 0.1297 | 64 | ECU | 0.0219 | 97 | HTI | 0.0018 |
31 | SWE | 0.1257 | 65 | AFG | 0.0219 | 98 | YEM | 0.0016 |
32 | ZAF | 0.1254 | 66 | NGA | 0.0215 | 99 | SUR | 0.0015 |
33 | EGY | 0.125 | 67 | BGD | 0.014 | 100 | SLV | 0.0014 |
34 | MEX | 0.1199 |
n | Mean | Median | Skewness | Kurtosis | Range | Min | Max | Sum | |
Data1 | 20 | 1.900 | 1.700 | 1.860 | 4.185 | 3.000 | 1.100 | 4.100 | 38.000 |
Data2 | 44 | 2.248 | 0.650 | 2.990 | 8.864 | 19.800 | 0.100 | 19.900 | 98.900 |
Data3 | 20 | 83.375 | 42.650 | 1.430 | 1.420 | 297.700 | 6.100 | 303.800 | 1667.500 |
Data4 | 100 | 0.347 | 0.050 | 5.590 | 38.257 | 8.130 | 0.001 | 8.133 | 34.676 |
Distributions | MLE and SE | ||||
$ \alpha $ | $ \beta $ | $ \gamma $ | $ \lambda $ | $ \theta $ | |
LBTLoW | 8.648 | 3.074 | 0.042 | ||
(3.545) | (0.474) | (0.025) | |||
ETGR | 0.103 | 0.692 | 23.539 | -0.342 | |
(0.436) | (0.086) | (105.137) | (1.971) | ||
BW | 0.831 | 0.613 | 29.947 | 11.632 | |
(0.954) | (0.340) | (40.414) | (21.900) | ||
T-Li | 0.665 | 0.359 | |||
(0.332) | (0.048) | ||||
McLL | 0.881 | 2.070 | 1.926 | 19.225 | 32.033 |
(0.109) | (3.693) | (5.165) | (22.341) | (43.081) | |
NMW | 0.121 | 2.784 | 2.787 | 0.003 | 0.008 |
(0.056) | (20.370) | (0.428) | (0.025) | (0.002) | |
W | 0.122 | 2.787 | |||
(0.056) | (0.427) |
Distributions | $ A_{IC} $ | $ C_{AIC} $ | $ B_{IC} $ | $ H_{QIC} $ | $ K_S $ | $ P_V $ |
LBTLoW | 40.140 | 41.640 | 38.040 | 40.720 | 0.146 | 0.790 |
ETGR | 44.860 | 47.520 | 42.060 | 45.630 | 0.190 | 0.465 |
BW | 42.400 | 45.060 | 39.600 | 43.170 | 0.160 | 0.683 |
T-Li | 65.730 | 66.440 | 64.330 | 66.120 | 0.380 | 0.006 |
McLL | 43.850 | 48.140 | 40.360 | 44.830 | 0.147 | 0.734 |
NMW | 51.170 | 55.460 | 47.680 | 52.150 | 0.190 | 0.501 |
W | 45.170 | 45.880 | 43.780 | 45.560 | 0.180 | 0.509 |
Distributions | MLE and SE | ||||
$ \alpha $ | $ \beta $ | $ \gamma $ | $ \lambda $ | $ \theta $ | |
LBTLoW | 6.268 | 0.623 | 0.484 | ||
(2.631) | (0.066) | (0.210) | |||
ETGR | 0.055 | 0.071 | 8.773 | 0.947 | |
(0.027) | (0.029) | (7.043) | (0.081) | ||
TCWG | 34.076 | 0.802 | 0.005 | 1.12 | |
(81.023) | (0.021) | (0.013) | (0.285) | ||
EKW | 0.221 | 400.298 | 5.215 | 1 | 3.823 |
(0.038) | (718.99) | (0.649) | (0.004) | (3.036) | |
TMW | 0.851 | 1.159 | -0.554 | 0.519 | |
(0.163) | (1.026) | (0.985) | (0.379) | ||
BW | 2.861 | 0.075 | 78.550 | 42.576 | |
(69.095) | (0.090) | (167.320) | (187.300) | ||
T-Li | 0.604 | 0.671 | |||
(0.155) | (0.074) | ||||
McLL | 0.181 | 1.565 | 1.286 | 21.234 | 28.124 |
(0.193) | (9.254) | (5.432) | (34.701) | (45.757) | |
NMW | 6.8 x $ 10^{-8} $ | 0.680 | 0.223 | 0.015 | 0.806 |
(0.623) | (0.110) | (617.48) | (0.015) | (0.418) | |
W | 0.799 | 0.621 | |||
(0.136) | (0.068) |
Distributions | $ A_{IC} $ | $ C_{AIC} $ | $ B_{IC} $ | $ H_{QIC} $ | $ K_S $ | $ P_V $ |
LBTLoW | 132.210 | 132.810 | 131.140 | 134.200 | 0.130 | 0.425 |
ETGR | 143.470 | 144.490 | 142.040 | 146.110 | 0.180 | 0.118 |
TCWG | 137.690 | 138.710 | 136.260 | 140.330 | 0.150 | 0.251 |
EKW | 133.890 | 135.470 | 132.110 | 140.330 | 0.140 | 0.355 |
TMW | 140.900 | 142.480 | 139.120 | 144.210 | 0.150 | 0.276 |
BW | 133.180 | 134.200 | 131.750 | 135.820 | 0.130 | 0.408 |
T-Li | 174.360 | 174.660 | 173.650 | 175.690 | 0.200 | 0.057 |
McLL | 134.830 | 136.410 | 133.040 | 138.130 | 0.130 | 0.419 |
NMW | 143.780 | 145.360 | 142.000 | 147.090 | 0.160 | 0.243 |
W | 138.650 | 138.940 | 137.940 | 139.970 | 0.170 | 0.139 |
Distributions | MLE and SE | ||||
$ \alpha $ | $ \beta $ | $ \gamma $ | $ \lambda $ | $ \theta $ | |
LBTLoW | 2.515 | 0.756 | 0.040 | ||
(3.877) | (0.166) | (0.048) | |||
WEIW | 0.909 | 0.871 | 7.225 | ||
(106700) | (0.152) | (384700) | |||
TMW | 0.998 | 0.459 | -0.443 | 0.202 | |
(0.081) | (18.537) | (18.537) | (0.769) | ||
T-Li | 0.021 | 0.384 | |||
(0.345) | (0.004) | ||||
McLL | 0.208 | 93.978 | 1.279 | 24.759 | 32.815 |
(0.499) | (1721) | (19.272) | (142.806) | (161.611) | |
NMW | 10.7 x $ 10^{-8} $ | 0.930 | 0.859 | 7.46 x $ 10^{-8} $ | 0.017 |
(0.001) | (0.250) | (1.216) | (0.002) | (0.017) | |
EKW | 0.167 | 261.64 | 45.725 | 1.201 | 2.138 |
(0.079) | (1709) | (219.725) | (0.741) | (7.209) |
Distributions | $ A_{IC} $ | $ C_{AIC} $ | $ B_{IC} $ | $ H_{QIC} $ | $ K_S $ | $ P_V $ |
LBTLoW | 221.690 | 223.190 | 219.600 | 222.280 | 0.135 | 0.857 |
WEIW | 223.400 | 224.900 | 221.300 | 223.980 | 0.157 | 0.708 |
TMW | 226.410 | 230.690 | 222.910 | 227.380 | 0.153 | 0.734 |
T-Li | 230.480 | 231.180 | 229.080 | 230.870 | 0.265 | 0.120 |
McLL | 225.990 | 230.280 | 222.500 | 222.500 | 0.146 | 0.789 |
NMW | 226.570 | 230.860 | 223.080 | 227.540 | 0.140 | 0.826 |
EKW | 226.290 | 230.570 | 222.790 | 229.300 | 0.148 | 0.776 |
Distributions | MLE and SE | ||||
$ \alpha $ | $ \beta $ | $ \gamma $ | $ \lambda $ | $ \theta $ | |
LBTLoW | 6.498 | 0.482 | 1.490 | ||
(2.301) | (0.034) | (0.573) | |||
EKW | 0.221 | 1096 | 4.424 | 1 | 1.717 |
(0.030) | (1376) | (1.817) | (0.001) | (0.901) | |
TMW | 0.596 | 2.612 | 0.588 | -0.523 | |
(0.057) | (0.689) | (0.256) | (0.346) | ||
BW | 134.832 | 0.073 | 49.149 | 22.930 | |
(956.622) | (0.060) | (74.497) | (46.500) | ||
WEIW | 27.512 | 0.549 | 0.094 | ||
(3272000) | (0.042) | (856.967) | |||
W | 2.648 | 0.489 | |||
(0.281) | (0.035) |
Distributions | $ A_{IC} $ | $ C_{AIC} $ | $ B_{IC} $ | $ H_{QIC} $ | $ K_S $ | $ P_V $ |
LBTLoW | –170.510 | –170.260 | –170.510 | –167.350 | 0.070 | 0.704 |
ETGR | –167.710 | –167.070 | –167.710 | –157.710 | 0.078 | 0.584 |
BW | –158.180 | –157.540 | –158.180 | –152.910 | 0.077 | 0.593 |
T-Li | –170.220 | –169.800 | –170.220 | –166.010 | 0.071 | 0.703 |
McLL | –170.200 | –169.950 | –170.200 | –167.040 | 0.091 | 0.383 |
W | –157.390 | –157.260 | –157.390 | –155.280 | 0.100 | 0.269 |
$ \gamma $ | $ \alpha $ | $ \mu'_1 $ | $ \mu'_2 $ | $ \mu'_3 $ | $ \mu'_4 $ | $ {\sigma }^{\mathrm{2}} $ | $ CS $ | $ CK $ | $ CV $ |
0.4 | 0.2 | 2.629 | 8.323 | 30.531 | 126.387 | 1.413 | 0.729 | 3.596 | 0.452 |
0.6 | 0.5 | 1.558 | 2.937 | 6.430 | 15.922 | 0.508 | 0.746 | 3.627 | 0.457 |
0.8 | 0.8 | 1.184 | 1.700 | 2.847 | 5.403 | 0.300 | 0.765 | 3.660 | 0.463 |
1.1 | 1.2 | 0.925 | 1.044 | 1.380 | 2.074 | 0.189 | 0.793 | 3.717 | 0.470 |
1.3 | 1.5 | 0.805 | 0.794 | 0.920 | 1.216 | 0.146 | 0.814 | 3.758 | 0.475 |
1.7 | 1.8 | 0.705 | 0.615 | 0.634 | 0.748 | 0.117 | 0.856 | 3.853 | 0.486 |
1.9 | 2.0 | 0.655 | 0.532 | 0.513 | 0.568 | 0.103 | 0.878 | 3.906 | 0.491 |
2.4 | 2.3 | 0.582 | 0.425 | 0.371 | 0.374 | 0.086 | 0.937 | 4.058 | 0.503 |
2.7 | 2.6 | 0.530 | 0.355 | 0.285 | 0.266 | 0.073 | 0.974 | 4.162 | 0.511 |
3.2 | 3.0 | 0.469 | 0.280 | 0.203 | 0.172 | 0.060 | 1.038 | 4.359 | 0.522 |
$ \gamma $ | $ \alpha $ | $ \mu'_1 $ | $ \mu'_2 $ | $ \mu'_3 $ | $ \mu'_4 $ | $ {\sigma }^{\mathrm{2}} $ | $ CS $ | $ CK $ | $ CV $ |
0.4 | 0.2 | 1.964 | 4.277 | 10.154 | 25.945 | 0.420 | 0.380 | 3.017 | 0.330 |
0.6 | 0.5 | 1.347 | 2.017 | 3.298 | 5.813 | 0.202 | 0.395 | 3.026 | 0.334 |
0.8 | 0.8 | 1.105 | 1.360 | 1.829 | 2.658 | 0.139 | 0.411 | 3.037 | 0.337 |
1.1 | 1.2 | 0.924 | 0.954 | 1.081 | 1.325 | 0.100 | 0.435 | 3.058 | 0.343 |
1.3 | 1.5 | 0.836 | 0.783 | 0.805 | 0.897 | 0.084 | 0.453 | 3.075 | 0.346 |
1.7 | 1.8 | 0.760 | 0.649 | 0.612 | 0.627 | 0.072 | 0.489 | 3.116 | 0.353 |
1.9 | 2.0 | 0.720 | 0.584 | 0.524 | 0.511 | 0.066 | 0.508 | 3.140 | 0.356 |
2.4 | 2.3 | 0.661 | 0.494 | 0.411 | 0.374 | 0.058 | 0.557 | 3.213 | 0.365 |
2.7 | 2.6 | 0.617 | 0.433 | 0.339 | 0.290 | 0.052 | 0.588 | 3.265 | 0.369 |
3.2 | 3.0 | 0.565 | 0.364 | 0.263 | 0.209 | 0.045 | 0.641 | 3.366 | 0.377 |
$ \beta $ | $ \alpha $ | $ \mu'_1 $ | $ \mu'_2 $ | $ \mu'_3 $ | $ \mu'_4 $ | $ {\sigma }^{\mathrm{2}} $ | $ CS $ | $ CK $ | $ CV $ |
1.5 | 0.2 | 0.029 | 0.026 | 0.023 | 0.021 | 0.025 | 5.273 | 29.151 | 5.386 |
0.5 | 0.032 | 0.028 | 0.025 | 0.023 | 0.027 | 5.014 | 26.463 | 5.138 | |
0.8 | 0.035 | 0.031 | 0.028 | 0.025 | 0.030 | 4.772 | 24.069 | 4.906 | |
1.2 | 0.039 | 0.035 | 0.031 | 0.028 | 0.033 | 4.472 | 21.275 | 4.622 | |
1.5 | 0.043 | 0.038 | 0.033 | 0.03 | 0.036 | 4.264 | 19.441 | 4.424 | |
1.8 | 0.046 | 0.041 | 0.036 | 0.033 | 0.038 | 4.069 | 17.801 | 4.241 | |
2 | 0.049 | 0.043 | 0.038 | 0.034 | 0.040 | 3.946 | 16.806 | 4.125 | |
2.3 | 0.053 | 0.046 | 0.041 | 0.037 | 0.043 | 3.771 | 15.445 | 3.961 | |
2.6 | 0.056 | 0.049 | 0.044 | 0.039 | 0.046 | 3.607 | 14.225 | 3.809 | |
3 | 0.062 | 0.054 | 0.048 | 0.043 | 0.050 | 3.405 | 12.794 | 3.621 | |
2.5 | 0.2 | 0.006 | 0.005 | 0.005 | 0.004 | 0.005 | 12.420 | 156.328 | 12.403 |
0.5 | 0.007 | 0.006 | 0.006 | 0.005 | 0.006 | 11.672 | 138.195 | 11.665 | |
0.8 | 0.007 | 0.007 | 0.006 | 0.006 | 0.007 | 10.981 | 122.434 | 10.984 | |
1.2 | 0.009 | 0.008 | 0.007 | 0.007 | 0.008 | 10.140 | 104.548 | 10.156 | |
1.5 | 0.01 | 0.009 | 0.008 | 0.007 | 0.009 | 9.563 | 93.121 | 9.589 | |
1.8 | 0.011 | 0.010 | 0.009 | 0.008 | 0.010 | 9.030 | 83.140 | 9.066 | |
2 | 0.012 | 0.011 | 0.010 | 0.009 | 0.010 | 8.697 | 77.190 | 8.739 | |
2.3 | 0.013 | 0.012 | 0.011 | 0.010 | 0.012 | 8.227 | 69.194 | 8.279 | |
2.6 | 0.014 | 0.013 | 0.012 | 0.011 | 0.013 | 7.792 | 62.179 | 7.853 | |
3 | 0.017 | 0.015 | 0.014 | 0.013 | 0.015 | 7.261 | 54.128 | 7.334 |
$ \varepsilon $ | $ \beta $ | $ \alpha $ | $ \gamma $ | RE | HaCE | ArE | TsE |
1.5 | 0.25 | 0.2 | 0.4 | 3.331 | 3.340 | 2.767 | 1.957 |
0.5 | 0.6 | 3.252 | 3.333 | 2.753 | 1.953 | ||
0.8 | 0.8 | 3.099 | 3.318 | 2.722 | 1.944 | ||
1.2 | 1.1 | 2.875 | 3.290 | 2.670 | 1.927 | ||
1.5 | 1.3 | 2.716 | 3.264 | 2.627 | 1.912 | ||
1.8 | 1.7 | 2.524 | 3.227 | 2.568 | 1.891 | ||
2.0 | 1.9 | 2.415 | 3.203 | 2.53 | 1.876 | ||
2.3 | 2.4 | 2.229 | 3.152 | 2.458 | 1.846 | ||
2.6 | 2.7 | 2.084 | 3.104 | 2.394 | 1.818 | ||
3.0 | 3.2 | 1.889 | 3.026 | 2.296 | 1.773 | ||
0.5 | 0.2 | 0.4 | 1.930 | 3.044 | 2.318 | 1.783 | |
0.5 | 0.6 | 1.924 | 3.042 | 2.315 | 1.782 | ||
0.8 | 0.8 | 1.824 | 2.996 | 2.260 | 1.755 | ||
1.2 | 1.1 | 1.660 | 2.909 | 2.161 | 1.704 | ||
1.5 | 1.3 | 1.541 | 2.835 | 2.081 | 1.661 | ||
1.8 | 1.7 | 1.392 | 2.726 | 1.969 | 1.597 | ||
2.0 | 1.9 | 1.308 | 2.657 | 1.901 | 1.556 | ||
2.3 | 2.4 | 1.161 | 2.517 | 1.770 | 1.475 | ||
2.6 | 2.7 | 1.051 | 2.397 | 1.661 | 1.404 | ||
3.0 | 3.2 | 0.903 | 2.207 | 1.500 | 1.293 | ||
2.0 | 0.25 | 0.2 | 0.4 | 2.180 | 1.987 | 1.837 | 0.993 |
0.5 | 0.6 | 2.167 | 1.986 | 1.835 | 0.993 | ||
0.8 | 0.8 | 2.053 | 1.982 | 1.812 | 0.991 | ||
1.2 | 1.1 | 1.876 | 1.973 | 1.769 | 0.987 | ||
1.5 | 1.3 | 1.753 | 1.965 | 1.734 | 0.982 | ||
1.8 | 1.7 | 1.595 | 1.949 | 1.681 | 0.975 | ||
2.0 | 1.9 | 1.510 | 1.938 | 1.648 | 0.969 | ||
2.3 | 2.4 | 1.358 | 1.912 | 1.581 | 0.956 | ||
2.6 | 2.7 | 1.247 | 1.887 | 1.524 | 0.943 | ||
3.0 | 3.2 | 1.098 | 1.840 | 1.435 | 0.920 | ||
0.5 | 0.2 | 0.4 | 1.210 | 1.877 | 1.503 | 0.938 | |
0.5 | 0.6 | 1.279 | 1.895 | 1.541 | 0.947 | ||
0.8 | 0.8 | 1.210 | 1.877 | 1.503 | 0.938 | ||
1.2 | 1.1 | 1.080 | 1.834 | 1.423 | 0.917 | ||
1.5 | 1.3 | 0.987 | 1.794 | 1.358 | 0.897 | ||
1.8 | 1.7 | 0.861 | 1.725 | 1.258 | 0.862 | ||
2.0 | 1.9 | 0.795 | 1.679 | 1.199 | 0.840 | ||
2.3 | 2.4 | 0.672 | 1.574 | 1.077 | 0.787 | ||
2.6 | 2.7 | 0.587 | 1.482 | 0.982 | 0.741 | ||
3.0 | 3.2 | 0.470 | 1.323 | 0.836 | 0.661 |
$ \varepsilon $ | $ \beta $ | $ \alpha $ | RE | HaCE | ArE | TsE |
1.5 | 0.25 | 0.2 | 7.001 | 3.311 | 2.709 | 1.94 |
0.5 | 7.075 | 3.315 | 2.716 | 1.942 | ||
0.8 | 7.156 | 3.319 | 2.724 | 1.944 | ||
1.2 | 7.274 | 3.324 | 2.735 | 1.947 | ||
1.5 | 7.372 | 3.329 | 2.743 | 1.95 | ||
1.8 | 7.476 | 3.333 | 2.752 | 1.952 | ||
2 | 7.55 | 3.336 | 2.758 | 1.954 | ||
2.3 | 7.667 | 3.34 | 2.767 | 1.957 | ||
2.6 | 7.792 | 3.345 | 2.777 | 1.959 | ||
3 | 7.971 | 3.351 | 2.79 | 1.963 | ||
0.4 | 0.2 | 6.441 | 3.278 | 2.65 | 1.92 | |
0.5 | 6.452 | 3.279 | 2.651 | 1.921 | ||
0.8 | 6.469 | 3.28 | 2.653 | 1.921 | ||
1.2 | 6.503 | 3.282 | 2.657 | 1.923 | ||
1.5 | 6.536 | 3.284 | 2.66 | 1.924 | ||
1.8 | 6.577 | 3.287 | 2.665 | 1.925 | ||
2 | 6.608 | 3.289 | 2.668 | 1.927 | ||
2.3 | 6.661 | 3.292 | 2.674 | 1.928 | ||
2.6 | 6.721 | 3.296 | 2.681 | 1.931 | ||
3 | 6.813 | 3.301 | 2.69 | 1.934 | ||
2.0 | 0.25 | 0.2 | 4.376 | 1.975 | 1.776 | 0.987 |
0.5 | 4.429 | 1.976 | 1.782 | 0.988 | ||
0.8 | 4.487 | 1.977 | 1.788 | 0.989 | ||
1.2 | 4.57 | 1.979 | 1.796 | 0.99 | ||
1.5 | 4.639 | 1.981 | 1.803 | 0.99 | ||
1.8 | 4.713 | 1.982 | 1.81 | 0.991 | ||
2 | 4.765 | 1.983 | 1.815 | 0.991 | ||
2.3 | 4.847 | 1.984 | 1.823 | 0.992 | ||
2.6 | 4.934 | 1.986 | 1.83 | 0.993 | ||
3 | 5.058 | 1.987 | 1.841 | 0.994 | ||
0.4 | 0.2 | 3.975 | 1.962 | 1.726 | 0.981 | |
0.5 | 3.987 | 1.963 | 1.728 | 0.981 | ||
0.8 | 4.003 | 1.963 | 1.73 | 0.982 | ||
1.2 | 4.031 | 1.965 | 1.734 | 0.982 | ||
1.5 | 4.058 | 1.965 | 1.737 | 0.983 | ||
1.8 | 4.09 | 1.967 | 1.741 | 0.983 | ||
2 | 4.114 | 1.967 | 1.744 | 0.984 | ||
2.3 | 4.154 | 1.969 | 1.749 | 0.984 | ||
2.6 | 4.199 | 1.97 | 1.755 | 0.985 | ||
3 | 4.267 | 1.972 | 1.763 | 0.986 |
n | r | Set1 ($ \alpha $ = 0.5, $ \beta $ = 0.5, $ \gamma $ = 0.5) | |||||||
MLE | Bias | MSE | LB | UB | AL | CP | |||
50 | 70% | $ \alpha $ | 0.4204 | 0.0796 | 0.0064 | 0.0019 | 0.839 | 0.8370 | 97.4% |
$ \beta $ | 0.7041 | 0.2041 | 0.0471 | 0.5036 | 0.9046 | 0.4010 | 96.9% | ||
$ \gamma $ | 0.4201 | 0.0799 | 0.0069 | 0.3023 | 0.5379 | 0.2356 | 96.0% | ||
80% | $ \alpha $ | 0.4218 | 0.0782 | 0.0061 | 0.0191 | 0.8245 | 0.8053 | 94.8% | |
$ \beta $ | 0.6382 | 0.1382 | 0.0242 | 0.4508 | 0.8256 | 0.3748 | 95.8% | ||
$ \gamma $ | 0.4386 | 0.0614 | 0.0053 | 0.3282 | 0.5490 | 0.2208 | 97.1% | ||
100% | $ \alpha $ | 0.4234 | 0.0766 | 0.0059 | 0.0357 | 0.8111 | 0.7754 | 95.4% | |
$ \beta $ | 0.5177 | 0.0177 | 0.0056 | 0.3661 | 0.6694 | 0.3033 | 95.5% | ||
$ \gamma $ | 0.5316 | 0.0316 | 0.0027 | 0.4303 | 0.6328 | 0.2025 | 96.0% | ||
100 | 70% | $ \alpha $ | 0.4213 | 0.0787 | 0.0062 | 0.0844 | 0.7583 | 0.6740 | 96.2% |
$ \beta $ | 0.6750 | 0.1750 | 0.0312 | 0.5375 | 0.8125 | 0.2750 | 95.9% | ||
$ \gamma $ | 0.4237 | 0.0763 | 0.0065 | 0.3389 | 0.5084 | 0.1694 | 96.0% | ||
80% | $ \alpha $ | 0.4230 | 0.0770 | 0.0061 | 0.2099 | 0.6360 | 0.4262 | 96.2% | |
$ \beta $ | 0.6099 | 0.1099 | 0.0127 | 0.4819 | 0.7379 | 0.2560 | 96.1% | ||
$ \gamma $ | 0.4487 | 0.0513 | 0.0033 | 0.3652 | 0.5321 | 0.1669 | 97.3% | ||
100% | $ \alpha $ | 0.4238 | 0.0762 | 0.0058 | 0.2501 | 0.5975 | 0.3473 | 95.6% | |
$ \beta $ | 0.4683 | 0.0317 | 0.0027 | 0.3558 | 0.5807 | 0.2249 | 95.8% | ||
$ \gamma $ | 0.4967 | 0.0033 | 0.0025 | 0.4199 | 0.5734 | 0.1535 | 96.0% | ||
150 | 70% | $ \alpha $ | 0.4217 | 0.0783 | 0.0061 | 0.2710 | 0.5725 | 0.3015 | 95.2% |
$ \beta $ | 0.6626 | 0.1626 | 0.0281 | 0.5577 | 0.7675 | 0.2097 | 95.6% | ||
$ \gamma $ | 0.4277 | 0.0723 | 0.0058 | 0.3571 | 0.4983 | 0.1412 | 97.3% | ||
80% | $ \alpha $ | 0.4236 | 0.0764 | 0.0059 | 0.3005 | 0.5466 | 0.2461 | 95.7% | |
$ \beta $ | 0.5977 | 0.0977 | 0.0113 | 0.4957 | 0.6997 | 0.2040 | 96.2% | ||
$ \gamma $ | 0.4649 | 0.0351 | 0.0022 | 0.3972 | 0.5325 | 0.1353 | 97.0% | ||
100% | $ \alpha $ | 0.4238 | 0.0762 | 0.0058 | 0.3010 | 0.5467 | 0.2457 | 95.6% | |
$ \beta $ | 0.4766 | 0.0234 | 0.0023 | 0.3784 | 0.5749 | 0.1965 | 96.4% | ||
$ \gamma $ | 0.5277 | 0.0277 | 0.0015 | 0.4659 | 0.5894 | 0.1236 | 96.9% | ||
200 | 70% | $ \alpha $ | 0.4219 | 0.0781 | 0.0061 | 0.3154 | 0.5285 | 0.2132 | 96.1% |
$ \beta $ | 0.6592 | 0.1592 | 0.0268 | 0.5675 | 0.7510 | 0.1835 | 96.3% | ||
$ \gamma $ | 0.4375 | 0.0625 | 0.0046 | 0.3789 | 0.4962 | 0.1173 | 96.7% | ||
80% | $ \alpha $ | 0.4239 | 0.0761 | 0.0058 | 0.3236 | 0.5242 | 0.2006 | 96.3% | |
$ \beta $ | 0.5912 | 0.0912 | 0.0099 | 0.5074 | 0.6750 | 0.1676 | 97.0% | ||
$ \gamma $ | 0.4667 | 0.0333 | 0.0020 | 0.4101 | 0.5233 | 0.1132 | 97.5% | ||
100% | $ \alpha $ | 0.4240 | 0.0760 | 0.0058 | 0.3372 | 0.5109 | 0.1737 | 96.5% | |
$ \beta $ | 0.4905 | 0.0095 | 0.0009 | 0.4167 | 0.5642 | 0.1475 | 96.7% | ||
$ \gamma $ | 0.5035 | 0.0035 | 0.0006 | 0.4524 | 0.5546 | 0.1022 | 97.1% | ||
n | r | Set2 ($ \alpha $ = 0.7, $ \beta $ = 0.5, $ \gamma $ = 0.25) | |||||||
MLE | Bias | MSE | LB | UB | AL | CP | |||
50 | 70% | $ \alpha $ | 0.4206 | 0.2794 | 0.0782 | 0.0055 | 0.8358 | 0.8304 | 97.7% |
$ \beta $ | 0.7016 | 0.2016 | 0.0462 | 0.5015 | 0.9016 | 0.4001 | 96.5% | ||
$ \gamma $ | 0.2172 | 0.0328 | 0.0017 | 0.1568 | 0.2776 | 0.1208 | 100% | ||
80% | $ \alpha $ | 0.4214 | 0.2786 | 0.0776 | 0.2084 | 0.6345 | 0.4261 | 97.9% | |
$ \beta $ | 0.6361 | 0.1361 | 0.0243 | 0.4490 | 0.8231 | 0.3741 | 98.5% | ||
$ \gamma $ | 0.2297 | 0.0203 | 0.0010 | 0.1719 | 0.2874 | 0.1155 | 100% | ||
100% | $ \alpha $ | 0.4234 | 0.2766 | 0.0765 | 0.2497 | 0.5970 | 0.3473 | 98.3% | |
$ \beta $ | 0.5161 | 0.0161 | 0.0062 | 0.3649 | 0.6673 | 0.3025 | 97.6% | ||
$ \gamma $ | 0.2565 | 0.0065 | 0.0007 | 0.2044 | 0.3086 | 0.1043 | 100% | ||
100 | 70% | $ \alpha $ | 0.4210 | 0.2790 | 0.0779 | 0.0208 | 0.8212 | 0.8004 | 96.4% |
$ \beta $ | 0.7006 | 0.2006 | 0.0431 | 0.5593 | 0.8419 | 0.2826 | 98.0% | ||
$ \gamma $ | 0.2141 | 0.0359 | 0.0016 | 0.1720 | 0.2562 | 0.0842 | 100% | ||
80% | $ \alpha $ | 0.4215 | 0.2785 | 0.0776 | 0.2708 | 0.5721 | 0.3013 | 97.2% | |
$ \beta $ | 0.6357 | 0.1357 | 0.0214 | 0.5035 | 0.7680 | 0.2645 | 97.7% | ||
$ \gamma $ | 0.2270 | 0.0230 | 0.0008 | 0.1866 | 0.2673 | 0.0807 | 100% | ||
100% | $ \alpha $ | 0.4234 | 0.2766 | 0.0765 | 0.3006 | 0.5462 | 0.2456 | 97.3% | |
$ \beta $ | 0.5158 | 0.0158 | 0.0033 | 0.4088 | 0.6227 | 0.2140 | 98.2% | ||
$ \gamma $ | 0.2540 | 0.0040 | 0.0003 | 0.2176 | 0.2905 | 0.0729 | 100% | ||
150 | 70% | $ \alpha $ | 0.4212 | 0.2788 | 0.0778 | 0.0330 | 0.8093 | 0.7763 | 97.7% |
$ \beta $ | 0.7000 | 0.2000 | 0.0419 | 0.5847 | 0.8153 | 0.2306 | 97.7% | ||
$ \gamma $ | 0.2122 | 0.0378 | 0.0016 | 0.1781 | 0.2463 | 0.0682 | 100% | ||
80% | $ \alpha $ | 0.4215 | 0.2785 | 0.0776 | 0.2985 | 0.5445 | 0.2460 | 98.8% | |
$ \beta $ | 0.6350 | 0.1350 | 0.0203 | 0.5271 | 0.7430 | 0.2159 | 98.1% | ||
$ \gamma $ | 0.2259 | 0.0241 | 0.0008 | 0.1931 | 0.2587 | 0.0656 | 96.0% | ||
100% | $ \alpha $ | 0.4234 | 0.2766 | 0.0765 | 0.3232 | 0.5237 | 0.2005 | 97.2% | |
$ \beta $ | 0.5151 | 0.0151 | 0.0023 | 0.4278 | 0.6024 | 0.1746 | 97.0% | ||
$ \gamma $ | 0.2529 | 0.0029 | 0.0002 | 0.2232 | 0.2825 | 0.0593 | 95.4% | ||
200 | 70% | $ \alpha $ | 0.4209 | 0.2791 | 0.0779 | 0.0849 | 0.7569 | 0.6720 | 100% |
$ \beta $ | 0.6981 | 0.1981 | 0.0405 | 0.5984 | 0.7978 | 0.1994 | 97.2% | ||
$ \gamma $ | 0.2118 | 0.0382 | 0.0016 | 0.1823 | 0.2412 | 0.0589 | 97.3% | ||
80% | $ \alpha $ | 0.4215 | 0.2785 | 0.0776 | 0.3150 | 0.5280 | 0.2131 | 100% | |
$ \beta $ | 0.6331 | 0.1331 | 0.0191 | 0.5398 | 0.7265 | 0.1867 | 98.2% | ||
$ \gamma $ | 0.2256 | 0.0244 | 0.0007 | 0.1973 | 0.2540 | 0.0567 | 98.0% | ||
100% | $ \alpha $ | 0.4234 | 0.2766 | 0.0765 | 0.3366 | 0.5103 | 0.1737 | 100% | |
$ \beta $ | 0.5136 | 0.0136 | 0.0016 | 0.4381 | 0.5891 | 0.1510 | 98.8% | ||
$ \gamma $ | 0.2523 | 0.0023 | 0.0002 | 0.2267 | 0.2779 | 0.0512 | 100% | ||
n | r | Set3 ($ \alpha $ = 0.7, $ \beta $ = 0.7, $ \gamma $ = 0.5) | |||||||
MLE | Bias | MSE | LB | UB | AL | CP | |||
50 | 70% | $ \alpha $ | 0.4178 | 0.2822 | 0.0797 | 0.1937 | 0.6419 | 0.4482 | 96.2% |
$ \beta $ | 0.8994 | 0.1994 | 0.0477 | 0.6064 | 1.1923 | 0.5859 | 95.9% | ||
$ \gamma $ | 0.6151 | 0.1151 | 0.0239 | 0.3763 | 0.8540 | 0.4776 | 95.0% | ||
80% | $ \alpha $ | 0.4193 | 0.2807 | 0.0788 | 0.2255 | 0.6131 | 0.3875 | 95.9% | |
$ \beta $ | 0.8238 | 0.1238 | 0.0218 | 0.5471 | 1.1006 | 0.5535 | 95.9% | ||
$ \gamma $ | 0.5695 | 0.0695 | 0.0182 | 0.3447 | 0.7943 | 0.4496 | 96.7% | ||
100% | $ \alpha $ | 0.4211 | 0.2789 | 0.0778 | 0.2480 | 0.5942 | 0.3461 | 96.8% | |
$ \beta $ | 0.7612 | 0.0612 | 0.0104 | 0.5012 | 1.0213 | 0.5201 | 97.0% | ||
$ \gamma $ | 0.5425 | 0.0425 | 0.0163 | 0.3395 | 0.7456 | 0.4061 | 95.0% | ||
100 | 70% | $ \alpha $ | 0.4174 | 0.2826 | 0.0798 | 0.2439 | 0.5910 | 0.3470 | 95.0% |
$ \beta $ | 0.8787 | 0.1787 | 0.0353 | 0.6426 | 1.1148 | 0.4722 | 96.3% | ||
$ \gamma $ | 0.5696 | 0.0696 | 0.0201 | 0.3697 | 0.7696 | 0.3998 | 96.0% | ||
80% | $ \alpha $ | 0.4191 | 0.2809 | 0.0789 | 0.2690 | 0.5691 | 0.3001 | 95.5% | |
$ \beta $ | 0.7802 | 0.0802 | 0.0134 | 0.5600 | 1.0004 | 0.4404 | 95.7% | ||
$ \gamma $ | 0.5479 | 0.0479 | 0.0091 | 0.3814 | 0.7144 | 0.3330 | 96.0% | ||
100% | $ \alpha $ | 0.4206 | 0.2794 | 0.0781 | 0.2866 | 0.5546 | 0.2680 | 95.6% | |
$ \beta $ | 0.7146 | 0.0146 | 0.0068 | 0.5069 | 0.9223 | 0.4154 | 95.7% | ||
$ \gamma $ | 0.5597 | 0.0597 | 0.0074 | 0.4063 | 0.7131 | 0.3067 | 96.0% | ||
150 | 70% | $ \alpha $ | 0.4176 | 0.2824 | 0.0798 | 0.2949 | 0.5403 | 0.2454 | 95.8% |
$ \beta $ | 0.8697 | 0.1697 | 0.0305 | 0.7056 | 1.0338 | 0.3282 | 96.2% | ||
$ \gamma $ | 0.5924 | 0.0924 | 0.0174 | 0.4449 | 0.7399 | 0.2950 | 97.1% | ||
80% | $ \alpha $ | 0.4193 | 0.2807 | 0.0788 | 0.3131 | 0.5254 | 0.2122 | 96.2% | |
$ \beta $ | 0.8023 | 0.1023 | 0.0124 | 0.6492 | 0.9555 | 0.3063 | 96.1% | ||
$ \gamma $ | 0.5524 | 0.0524 | 0.0058 | 0.4332 | 0.6716 | 0.2384 | 97.0% | ||
100% | $ \alpha $ | 0.4209 | 0.2791 | 0.0779 | 0.3261 | 0.5156 | 0.1895 | 95.8% | |
$ \beta $ | 0.7374 | 0.0374 | 0.0032 | 0.5932 | 0.8817 | 0.2885 | 96.3% | ||
$ \gamma $ | 0.5507 | 0.0507 | 0.0041 | 0.4447 | 0.6568 | 0.2121 | 96.9% | ||
200 | 70% | $ \alpha $ | 0.4175 | 0.2825 | 0.0798 | 0.3173 | 0.5177 | 0.2004 | 96.1% |
$ \beta $ | 0.8406 | 0.1406 | 0.0265 | 0.7075 | 0.9736 | 0.2661 | 97.2% | ||
$ \gamma $ | 0.5683 | 0.0683 | 0.0069 | 0.4569 | 0.6798 | 0.2230 | 96.9% | ||
80% | $ \alpha $ | 0.4193 | 0.2807 | 0.0788 | 0.3327 | 0.5059 | 0.1733 | 96.3% | |
$ \beta $ | 0.8043 | 0.1043 | 0.0123 | 0.6790 | 0.9295 | 0.2505 | 96.6% | ||
$ \gamma $ | 0.5502 | 0.0502 | 0.0041 | 0.4533 | 0.6471 | 0.1938 | 97.0% | ||
100% | $ \alpha $ | 0.4209 | 0.2791 | 0.0779 | 0.3435 | 0.4983 | 0.1548 | 96.1% | |
$ \beta $ | 0.7411 | 0.0411 | 0.0030 | 0.6229 | 0.8592 | 0.2362 | 97.0% | ||
$ \gamma $ | 0.5463 | 0.0463 | 0.0041 | 0.4589 | 0.6338 | 0.1749 | 96.2% | ||
n | r | Set4 ($ \alpha $ = 0.6, $ \beta $ = 0.3, $ \gamma $ = 0.5) | |||||||
MLE | Bias | MSE | LB | UB | AL | CP | |||
50 | 70% | $ \alpha $ | 0.4197 | 0.1803 | 0.0325 | 0.2456 | 0.5939 | 0.3482 | 98.1% |
$ \beta $ | 0.5719 | 0.2719 | 0.0744 | 0.3742 | 0.7696 | 0.3954 | 97.0% | ||
$ \gamma $ | 0.2990 | 0.2010 | 0.0406 | 0.1769 | 0.4211 | 0.2442 | 98.0% | ||
80% | $ \alpha $ | 0.4221 | 0.1779 | 0.0317 | 0.2716 | 0.5725 | 0.3009 | 98.4% | |
$ \beta $ | 0.4934 | 0.1934 | 0.0384 | 0.3126 | 0.6742 | 0.3616 | 97.4% | ||
$ \gamma $ | 0.3593 | 0.1407 | 0.0200 | 0.2383 | 0.4803 | 0.2421 | 98.2% | ||
100% | $ \alpha $ | 0.4246 | 0.1754 | 0.0308 | 0.2903 | 0.5589 | 0.2686 | 97.9% | |
$ \beta $ | 0.4198 | 0.1198 | 0.0150 | 0.2576 | 0.5820 | 0.3244 | 97.4% | ||
$ \gamma $ | 0.4172 | 0.0828 | 0.0078 | 0.2968 | 0.5376 | 0.2408 | 98.4% | ||
100 | 70% | $ \alpha $ | 0.4198 | 0.1802 | 0.0325 | 0.2966 | 0.5429 | 0.2463 | 97.2% |
$ \beta $ | 0.5674 | 0.2674 | 0.0717 | 0.4278 | 0.7069 | 0.2791 | 97.7% | ||
$ \gamma $ | 0.3137 | 0.1863 | 0.0351 | 0.2278 | 0.3995 | 0.1717 | 98.0% | ||
80% | $ \alpha $ | 0.4222 | 0.1778 | 0.0316 | 0.3157 | 0.5286 | 0.2128 | 97.9% | |
$ \beta $ | 0.4857 | 0.1857 | 0.0350 | 0.3584 | 0.6130 | 0.2546 | 97.9% | ||
$ \gamma $ | 0.3669 | 0.1331 | 0.0179 | 0.2818 | 0.4519 | 0.1701 | 98.7% | ||
100% | $ \alpha $ | 0.4248 | 0.1752 | 0.0307 | 0.3242 | 0.5253 | 0.2011 | 98.1% | |
$ \beta $ | 0.4082 | 0.1082 | 0.0125 | 0.2944 | 0.5221 | 0.2277 | 97.8% | ||
$ \gamma $ | 0.4171 | 0.0829 | 0.0074 | 0.3349 | 0.4993 | 0.1644 | 98.3% | ||
150 | 70% | $ \alpha $ | 0.4198 | 0.1802 | 0.0325 | 0.3248 | 0.5148 | 0.1899 | 97.2% |
$ \beta $ | 0.5653 | 0.2653 | 0.0706 | 0.4518 | 0.6788 | 0.2270 | 98.3% | ||
$ \gamma $ | 0.3140 | 0.1860 | 0.0350 | 0.2421 | 0.3860 | 0.1439 | 99.3% | ||
80% | $ \alpha $ | 0.4222 | 0.1778 | 0.0316 | 0.3353 | 0.5091 | 0.1738 | 97.7% | |
$ \beta $ | 0.4832 | 0.1832 | 0.0338 | 0.3797 | 0.5866 | 0.2069 | 98.2% | ||
$ \gamma $ | 0.3677 | 0.1323 | 0.0178 | 0.2996 | 0.4358 | 0.1363 | 99.0% | ||
100% | $ \alpha $ | 0.4248 | 0.1752 | 0.0307 | 0.3377 | 0.5119 | 0.1742 | 97.3% | |
$ \beta $ | 0.4018 | 0.1018 | 0.0107 | 0.3032 | 0.5004 | 0.1972 | 97.9% | ||
$ \gamma $ | 0.4248 | 0.0752 | 0.0061 | 0.3608 | 0.4888 | 0.1280 | 99.7% | ||
200 | 70% | $ \alpha $ | 0.4198 | 0.1802 | 0.0325 | 0.3423 | 0.4973 | 0.1551 | 99.0% |
$ \beta $ | 0.5650 | 0.2650 | 0.0705 | 0.4738 | 0.6561 | 0.1823 | 99.1% | ||
$ \gamma $ | 0.3256 | 0.1744 | 0.0309 | 0.2637 | 0.3875 | 0.1239 | 98.7% | ||
80% | $ \alpha $ | 0.4222 | 0.1778 | 0.0316 | 0.3470 | 0.4975 | 0.1505 | 99.6% | |
$ \beta $ | 0.4781 | 0.1781 | 0.0321 | 0.3882 | 0.5679 | 0.1798 | 99.3% | ||
$ \gamma $ | 0.3733 | 0.1267 | 0.0173 | 0.3129 | 0.4336 | 0.1206 | 99.5% | ||
100% | $ \alpha $ | 0.4250 | 0.1750 | 0.0306 | 0.3578 | 0.4921 | 0.1343 | 98.7% | |
$ \beta $ | 0.3904 | 0.0904 | 0.0088 | 0.3106 | 0.4703 | 0.1596 | 99.6% | ||
$ \gamma $ | 0.4262 | 0.0738 | 0.0060 | 0.3681 | 0.4843 | 0.1162 | 100% |
Rank | Country | % Global Reserves | Rank | Country | % Global Reserves |
1 | Russia | 19.9 | 23 | Ukraine | 0.6 |
2 | Iran | 17.1 | 24 | Malaysia | 0.5 |
3 | Qatar | 13.1 | 25 | Uzbekistan | 0.4 |
4 | Turkmenistan | 7.2 | 26 | Oman | 0.4 |
5 | United States | 6.7 | 27 | Vietnam | 0.3 |
6 | China | 4.5 | 28 | Israel | 0.3 |
7 | Venezuela | 3.3 | 29 | Argentina | 0.2 |
8 | Saudi Arabia | 3.2 | 30 | Pakistan | 0.2 |
9 | United Arab Emirates | 3.2 | 31 | Trinidad | 0.2 |
10 | Nigeria | 2.9 | 32 | Brazil | 0.2 |
11 | Iraq | 1.9 | 33 | Myanmar | 0.2 |
12 | Canada | 1.3 | 34 | United Kingdom | 0.1 |
13 | Australia | 1.3 | 35 | Thailand | 0.1 |
14 | Azerbaijan | 1.3 | 36 | Mexico | 0.1 |
15 | Algeria | 1.2 | 37 | Bangladesh | 0.1 |
16 | Kazakhstan | 1.2 | 38 | Netherlands | 0.1 |
17 | Egypt | 1.1 | 39 | Bolivia | 0.1 |
18 | Kuwait | 0.9 | 40 | Brunei | 0.1 |
19 | Norway | 0.8 | 41 | Peru | 0.1 |
20 | Libya | 0.8 | 42 | Syria | 0.1 |
21 | Indonesia | 0.7 | 43 | Yemen | 0.1 |
22 | India | 0.7 | 44 | Papua New Guinea | 0.1 |
Rank | Country | reserves2020 | Rank | Country | reserves2020 |
1 | Venezuela | 303.8 | 11 | Nigeria | 36.9 |
2 | Saudi Arabia | 297.5 | 12 | Kazakhstan | 30 |
3 | Canada | 168.1 | 13 | China | 26 |
4 | Iran | 157.8 | 14 | Qatar | 25.2 |
5 | Iraq | 145 | 15 | Algeria | 12.2 |
6 | Russia | 107.8 | 16 | Brazil | 11.9 |
7 | Kuwait | 101.5 | 17 | Norway | 7.9 |
8 | United Arab Emirates | 97.8 | 18 | Angola | 7.8 |
9 | United States | 68.8 | 19 | Azerbaijan | 7 |
10 | Libya | 48.4 | 20 | Mexico | 6.1 |
Rank | Country | Reserves of Gold | Rank | Country | Reserves of Gold | Rank | Country | Reserves of Gold |
1 | USA | 8.1335 | 35 | LBY | 0.1166 | 68 | CYP | 0.0139 |
2 | DEU | 3.3585 | 36 | GRC | 0.1141 | 69 | CUW | 0.0131 |
3 | IMF | 2.814 | 37 | ROK | 0.1045 | 70 | MUS | 0.0124 |
4 | ITA | 2.4518 | 38 | ROU | 0.1036 | 71 | IRL | 0.012 |
5 | FRA | 2.4365 | 39 | BIS | 0.102 | 72 | CZE | 0.0109 |
6 | RUS | 2.2985 | 40 | IRQ | 0.0964 | 73 | KGZ | 0.0102 |
7 | CHN | 1.9483 | 41 | HUN | 0.0945 | 74 | GHA | 0.0087 |
8 | CHE | 1.04 | 42 | AUS | 0.0798 | 75 | PRY | 0.0082 |
9 | JPN | 0.846 | 43 | KWT | 0.079 | 76 | NPL | 0.008 |
10 | IND | 0.7604 | 44 | IDN | 0.0786 | 77 | MNG | 0.0076 |
11 | NLD | 0.6125 | 45 | DNK | 0.0666 | 78 | MMR | 0.0073 |
12 | ECB | 0.5048 | 46 | PAK | 0.0647 | 79 | GTM | 0.0069 |
13 | TUR | 0.4311 | 47 | ARG | 0.0617 | 80 | MKD | 0.0069 |
14 | TAI | 0.4236 | 48 | ARE | 0.0553 | 81 | TUN | 0.0068 |
15 | PRT | 0.3826 | 49 | BLR | 0.0535 | 82 | LVA | 0.0067 |
16 | KAZ | 0.3681 | 50 | QAT | 0.0513 | 83 | LTU | 0.0058 |
17 | UZB | 0.3375 | 51 | KHM | 0.0504 | 84 | COL | 0.0047 |
18 | SAU | 0.3231 | 52 | FIN | 0.049 | 85 | BHR | 0.0047 |
19 | GBR | 0.3103 | 53 | JOR | 0.0435 | 86 | BRN | 0.0046 |
20 | LBN | 0.2868 | 54 | BOL | 0.0425 | 87 | GIN | 0.0042 |
21 | ESP | 0.2816 | 55 | BGR | 0.0408 | 88 | MOZ | 0.0039 |
22 | AUT | 0.28 | 56 | MYS | 0.0389 | 89 | SVN | 0.0032 |
23 | THA | 0.2442 | 57 | SRB | 0.0378 | 90 | ABW | 0.0031 |
24 | POL | 0.2287 | 58 | WAEMU | 0.0365 | 91 | BIH | 0.003 |
25 | BEL | 0.2274 | 59 | PER | 0.0347 | 92 | ALB | 0.0028 |
26 | DZA | 0.1736 | 60 | SVK | 0.0317 | 93 | LUX | 0.0022 |
27 | VEN | 0.1612 | 61 | UKR | 0.0271 | 94 | HKG | 0.0021 |
28 | PHL | 0.1563 | 62 | SYR | 0.0258 | 95 | ISL | 0.002 |
29 | SGP | 0.1537 | 63 | MAR | 0.0221 | 96 | TTO | 0.0019 |
30 | BRA | 0.1297 | 64 | ECU | 0.0219 | 97 | HTI | 0.0018 |
31 | SWE | 0.1257 | 65 | AFG | 0.0219 | 98 | YEM | 0.0016 |
32 | ZAF | 0.1254 | 66 | NGA | 0.0215 | 99 | SUR | 0.0015 |
33 | EGY | 0.125 | 67 | BGD | 0.014 | 100 | SLV | 0.0014 |
34 | MEX | 0.1199 |
n | Mean | Median | Skewness | Kurtosis | Range | Min | Max | Sum | |
Data1 | 20 | 1.900 | 1.700 | 1.860 | 4.185 | 3.000 | 1.100 | 4.100 | 38.000 |
Data2 | 44 | 2.248 | 0.650 | 2.990 | 8.864 | 19.800 | 0.100 | 19.900 | 98.900 |
Data3 | 20 | 83.375 | 42.650 | 1.430 | 1.420 | 297.700 | 6.100 | 303.800 | 1667.500 |
Data4 | 100 | 0.347 | 0.050 | 5.590 | 38.257 | 8.130 | 0.001 | 8.133 | 34.676 |
Distributions | MLE and SE | ||||
$ \alpha $ | $ \beta $ | $ \gamma $ | $ \lambda $ | $ \theta $ | |
LBTLoW | 8.648 | 3.074 | 0.042 | ||
(3.545) | (0.474) | (0.025) | |||
ETGR | 0.103 | 0.692 | 23.539 | -0.342 | |
(0.436) | (0.086) | (105.137) | (1.971) | ||
BW | 0.831 | 0.613 | 29.947 | 11.632 | |
(0.954) | (0.340) | (40.414) | (21.900) | ||
T-Li | 0.665 | 0.359 | |||
(0.332) | (0.048) | ||||
McLL | 0.881 | 2.070 | 1.926 | 19.225 | 32.033 |
(0.109) | (3.693) | (5.165) | (22.341) | (43.081) | |
NMW | 0.121 | 2.784 | 2.787 | 0.003 | 0.008 |
(0.056) | (20.370) | (0.428) | (0.025) | (0.002) | |
W | 0.122 | 2.787 | |||
(0.056) | (0.427) |
Distributions | $ A_{IC} $ | $ C_{AIC} $ | $ B_{IC} $ | $ H_{QIC} $ | $ K_S $ | $ P_V $ |
LBTLoW | 40.140 | 41.640 | 38.040 | 40.720 | 0.146 | 0.790 |
ETGR | 44.860 | 47.520 | 42.060 | 45.630 | 0.190 | 0.465 |
BW | 42.400 | 45.060 | 39.600 | 43.170 | 0.160 | 0.683 |
T-Li | 65.730 | 66.440 | 64.330 | 66.120 | 0.380 | 0.006 |
McLL | 43.850 | 48.140 | 40.360 | 44.830 | 0.147 | 0.734 |
NMW | 51.170 | 55.460 | 47.680 | 52.150 | 0.190 | 0.501 |
W | 45.170 | 45.880 | 43.780 | 45.560 | 0.180 | 0.509 |
Distributions | MLE and SE | ||||
$ \alpha $ | $ \beta $ | $ \gamma $ | $ \lambda $ | $ \theta $ | |
LBTLoW | 6.268 | 0.623 | 0.484 | ||
(2.631) | (0.066) | (0.210) | |||
ETGR | 0.055 | 0.071 | 8.773 | 0.947 | |
(0.027) | (0.029) | (7.043) | (0.081) | ||
TCWG | 34.076 | 0.802 | 0.005 | 1.12 | |
(81.023) | (0.021) | (0.013) | (0.285) | ||
EKW | 0.221 | 400.298 | 5.215 | 1 | 3.823 |
(0.038) | (718.99) | (0.649) | (0.004) | (3.036) | |
TMW | 0.851 | 1.159 | -0.554 | 0.519 | |
(0.163) | (1.026) | (0.985) | (0.379) | ||
BW | 2.861 | 0.075 | 78.550 | 42.576 | |
(69.095) | (0.090) | (167.320) | (187.300) | ||
T-Li | 0.604 | 0.671 | |||
(0.155) | (0.074) | ||||
McLL | 0.181 | 1.565 | 1.286 | 21.234 | 28.124 |
(0.193) | (9.254) | (5.432) | (34.701) | (45.757) | |
NMW | 6.8 x $ 10^{-8} $ | 0.680 | 0.223 | 0.015 | 0.806 |
(0.623) | (0.110) | (617.48) | (0.015) | (0.418) | |
W | 0.799 | 0.621 | |||
(0.136) | (0.068) |
Distributions | $ A_{IC} $ | $ C_{AIC} $ | $ B_{IC} $ | $ H_{QIC} $ | $ K_S $ | $ P_V $ |
LBTLoW | 132.210 | 132.810 | 131.140 | 134.200 | 0.130 | 0.425 |
ETGR | 143.470 | 144.490 | 142.040 | 146.110 | 0.180 | 0.118 |
TCWG | 137.690 | 138.710 | 136.260 | 140.330 | 0.150 | 0.251 |
EKW | 133.890 | 135.470 | 132.110 | 140.330 | 0.140 | 0.355 |
TMW | 140.900 | 142.480 | 139.120 | 144.210 | 0.150 | 0.276 |
BW | 133.180 | 134.200 | 131.750 | 135.820 | 0.130 | 0.408 |
T-Li | 174.360 | 174.660 | 173.650 | 175.690 | 0.200 | 0.057 |
McLL | 134.830 | 136.410 | 133.040 | 138.130 | 0.130 | 0.419 |
NMW | 143.780 | 145.360 | 142.000 | 147.090 | 0.160 | 0.243 |
W | 138.650 | 138.940 | 137.940 | 139.970 | 0.170 | 0.139 |
Distributions | MLE and SE | ||||
$ \alpha $ | $ \beta $ | $ \gamma $ | $ \lambda $ | $ \theta $ | |
LBTLoW | 2.515 | 0.756 | 0.040 | ||
(3.877) | (0.166) | (0.048) | |||
WEIW | 0.909 | 0.871 | 7.225 | ||
(106700) | (0.152) | (384700) | |||
TMW | 0.998 | 0.459 | -0.443 | 0.202 | |
(0.081) | (18.537) | (18.537) | (0.769) | ||
T-Li | 0.021 | 0.384 | |||
(0.345) | (0.004) | ||||
McLL | 0.208 | 93.978 | 1.279 | 24.759 | 32.815 |
(0.499) | (1721) | (19.272) | (142.806) | (161.611) | |
NMW | 10.7 x $ 10^{-8} $ | 0.930 | 0.859 | 7.46 x $ 10^{-8} $ | 0.017 |
(0.001) | (0.250) | (1.216) | (0.002) | (0.017) | |
EKW | 0.167 | 261.64 | 45.725 | 1.201 | 2.138 |
(0.079) | (1709) | (219.725) | (0.741) | (7.209) |
Distributions | $ A_{IC} $ | $ C_{AIC} $ | $ B_{IC} $ | $ H_{QIC} $ | $ K_S $ | $ P_V $ |
LBTLoW | 221.690 | 223.190 | 219.600 | 222.280 | 0.135 | 0.857 |
WEIW | 223.400 | 224.900 | 221.300 | 223.980 | 0.157 | 0.708 |
TMW | 226.410 | 230.690 | 222.910 | 227.380 | 0.153 | 0.734 |
T-Li | 230.480 | 231.180 | 229.080 | 230.870 | 0.265 | 0.120 |
McLL | 225.990 | 230.280 | 222.500 | 222.500 | 0.146 | 0.789 |
NMW | 226.570 | 230.860 | 223.080 | 227.540 | 0.140 | 0.826 |
EKW | 226.290 | 230.570 | 222.790 | 229.300 | 0.148 | 0.776 |
Distributions | MLE and SE | ||||
$ \alpha $ | $ \beta $ | $ \gamma $ | $ \lambda $ | $ \theta $ | |
LBTLoW | 6.498 | 0.482 | 1.490 | ||
(2.301) | (0.034) | (0.573) | |||
EKW | 0.221 | 1096 | 4.424 | 1 | 1.717 |
(0.030) | (1376) | (1.817) | (0.001) | (0.901) | |
TMW | 0.596 | 2.612 | 0.588 | -0.523 | |
(0.057) | (0.689) | (0.256) | (0.346) | ||
BW | 134.832 | 0.073 | 49.149 | 22.930 | |
(956.622) | (0.060) | (74.497) | (46.500) | ||
WEIW | 27.512 | 0.549 | 0.094 | ||
(3272000) | (0.042) | (856.967) | |||
W | 2.648 | 0.489 | |||
(0.281) | (0.035) |
Distributions | $ A_{IC} $ | $ C_{AIC} $ | $ B_{IC} $ | $ H_{QIC} $ | $ K_S $ | $ P_V $ |
LBTLoW | –170.510 | –170.260 | –170.510 | –167.350 | 0.070 | 0.704 |
ETGR | –167.710 | –167.070 | –167.710 | –157.710 | 0.078 | 0.584 |
BW | –158.180 | –157.540 | –158.180 | –152.910 | 0.077 | 0.593 |
T-Li | –170.220 | –169.800 | –170.220 | –166.010 | 0.071 | 0.703 |
McLL | –170.200 | –169.950 | –170.200 | –167.040 | 0.091 | 0.383 |
W | –157.390 | –157.260 | –157.390 | –155.280 | 0.100 | 0.269 |