
The initial value problem in Cauchy-type under the (k,ψ)-Caputo proportional fractional operators was our focus in this paper. An extended Gronwall inequality and its properties were analyzed. The existence and uniqueness results were proven utilizing the fixed point theory of Banach's and Leray-Schauder's types. The qualitative analysis included results for Ulam-Mittag-Leffler stability, which was also investigated. Using a decomposition principle, a novel numerical technique was presented for the (k,ψ)-Caputo proportional fractional derivative operator. Finally, theoretical results were supported with numerical examples to demonstrate their practical application, especially to blood alcohol level problems.
Citation: Weerawat Sudsutad, Chatthai Thaiprayoon, Aphirak Aphithana, Jutarat Kongson, Weerapan Sae-dan. Qualitative results and numerical approximations of the (k,ψ)-Caputo proportional fractional differential equations and applications to blood alcohol levels model[J]. AIMS Mathematics, 2024, 9(12): 34013-34041. doi: 10.3934/math.20241622
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The initial value problem in Cauchy-type under the (k,ψ)-Caputo proportional fractional operators was our focus in this paper. An extended Gronwall inequality and its properties were analyzed. The existence and uniqueness results were proven utilizing the fixed point theory of Banach's and Leray-Schauder's types. The qualitative analysis included results for Ulam-Mittag-Leffler stability, which was also investigated. Using a decomposition principle, a novel numerical technique was presented for the (k,ψ)-Caputo proportional fractional derivative operator. Finally, theoretical results were supported with numerical examples to demonstrate their practical application, especially to blood alcohol level problems.
Modeling industrial data is now considered a substantial area of interest for researchers across different disciplines, including biometrics, engineering, survival, lifetime, reliability sciences, and numerous other areas. Different probability distributions are available, but they have some limitations in fitting these types of data sets. For example, see Verevka et al. [1] and Barskov et al. [2]. There is a constant increase in the range of count data, and the constraints of existing models make modeling these data challenging. Therefore, in the last few decades, many researchers have sought to propose adaptable models for modeling these types of data sets using different generalized approaches. For more details, see Alzaatreh and Famoye [3], Ieren et al. [4], Riad et al. [5], EL-Helbawy et al. [6], Altun et al. [7], Alotaibi et al. [8], Maya et al. [9], Meraou et al. [10,11,12,13], Alrweili et al. [14], Alrweili [15,16].
On the other side, several situations exist where the suggested extension distributions are unsuitable for analyzing different data sets. Additionally, the role and importance of generating a new family of distribution using various generators becomes a very important and compulsory task for researchers to accommodate the variety of data patterns being generated in every field of life. Data coming from different fields of study, specifically in the industrial field, all need a better model to fit their diverse data patterns and be motivated by the urgency of highly flexible statistical models. Consequently, Nofal et al. [17] introduced a new methodology to create a new distribution called the generalized Kumaraswamy (GK) family. It is a new concept of generalizing a given distribution, which introduces three additional parameters in a baseline distribution, and it has wider applications in industrial, engineering, survival, and other fields. The cumulative distribution function (cdf) and the probability density function (pdf) of the GK class of distributions can be defined as
Δ(x)=1−{1−α(F(x))β}γ1−(1−α)γ,x∈R,β,γ>0, 0<α≤1, | (1.1) |
and
δ(x)=αβγf(x)1−(1−α)γ {F(x)}β−1 {1−α(F(x))β}γ−1, | (1.2) |
where f(x) and F(x) are, respectively, the pdf and cdf of the basic distribution.
It is well documented that the Rayleigh distribution (RD) is frequently used to model diverse data sets drawn from different areas, especially for analyzing industrial data. Consequently, the RD is suitable for modeling industrial data. This is vital in the reliability analysis of industrial devices, such as 3D printing, drones, and robots. Let us consider the random variable X has RD with positive parameter θ, so its cdf and pdf, respectively are
Π(x)=1−e−θx2,x>0, | (1.3) |
and
π(x)=2θy e−θx2. | (1.4) |
The RD is widely used in reliability and survival analysis for mortality rates, especially when studying extreme events. Since it captures tail behavior effectively, it is particularly also useful in understanding the upper quantiles of life expectancy or survival time. Also, it has undoubtedly established itself as a crucial tool for data modeling across nearly all sectors, including survival, hydrology, insurance, and energy theory. However, despite its widespread use and advantages, the RD is constrained by its inherent limitations. One of the primary constraints of the RD is its capacity to represent solely monotonically increasing forms of hazard functions, as it can only model data where the hazard rate increases or decreases consistently over time. Also, the RD is regarded as a limiting model for residual lifetimes. For this, we have seen increased interest in studying the RD and its applications in various fields such as medicine, engineering, insurance, industry, and risk management. Some of these efforts are listed below by Chukwudi et al. [18], Muzamil et al. [19], Aijaz et al. [20], Abdulsalam et al. [21], Javed et al. [22].
The basic motivations for the recommended GKRD in practice are:
(1) The GKRD provides a crucial important role in analyzing numerous kinds of data sets. Its parameters provide a flexible way to manipulate the shape and characteristics of a probability distribution. This adaptability allows researchers and analysts to tailor the distribution to better fit real-world data, making it a valuable tool in diverse fields such as statistics, engineering, biology, etc. Further, the four parameters of the proposed GKRD make the underlying patterns more interpretable. This enhanced interpretation ability can lead to deeper insights and a better understanding of the factors influencing the data.
(2) With adding three additional parameters, the GKRD has the ability to represent the unimodal or bimodal probability distribution.
(3) Another motivation is the ability to induce skewness in symmetrical and asymmetrical distributions. This capability is particularly valuable in fields where skewed distributions are prevalent, such as finance, economics, and insurance.
The following are the key objectives:
● The primary objective that must be fulfilled is introducing a novel model, and the new distribution is named the generalized Kumaraswamy Rayleigh distribution (GKRD). We also determined its various statistical properties, including moments, the moment-generating function, and order statistics.
● Derive and discuss its reliability characteristics.
● Estimate the model parameters using the maximum likelihood and Bayesian approaches under the square error loss function (SELF) and illustrate the pattern of these derived estimators using a comprehensive simulation study.
● Check the validity and flexibility of the GKRD using industrial and financial data sets.
The rest of the study is organized as follows: The recommended GKRD is defined in Section 2 with some distributional properties. In Section 3, we present several mathematical properties including moments, the quantile function, and the moment generating function. Section 4 demonstrates the estimation of the model parameters based on two different proposed methods. The effectiveness of the proposed estimation tools are studied using some simulation studies in Section 5, and three distinct real data sets are applied to show the results of the application of the GKRD. The concluding report is given in Section 7.
Here, in this part of the study, several distributional properties such as the pdf, cdf, survival, and hazard rate function of the GKRD are derived.
Based on Eqs (1.1)–(1.2) and by replacing the classical distribution with the RD, the cdf and pdf of the proposed GKRD are
Ξ(t)=1−{1−α(1−e−θt2)β}γ1−(1−α)γ, t>0, β,γ,θ>0, 0<α≤1, | (2.1) |
ξ(t)=2αβγθ te−θt21−(1−α)γ {1−e−θt2}β−1 {1−α(1−e−θt2)β}γ−1. | (2.2) |
The pdf curves of the GKRD are explored using several parametric values of parameters and displayed in Figure 1. As shown in Figure 1, the proposed GKRD is unimodal and has decreasing behavior.
The survival function (sf) and hazard rate function (hrf) can be obtained from the following equations:
S(t)={1−α(1−e−θt2)β}γ−(1−α)γ1−(1−α)γ, | (2.3) |
and
h(t)=2αβγθ te−θt2{1−α(1−e−θt2)β}γ−(1−α)γ {1−e−θt2}β−1 {1−α(1−e−θt2)β}γ−1. | (2.4) |
Figure 2 reports the hrf curves of the GKRD using several parameter values. It is upside down and increasing depending on the parameter values.
The proposed GKRD has several specialized sub-models, which confirm its importance in modeling various types of data sets. The specialized sub-models are displayed in Table 1.
Parameter | Model | ||||
α | β | γ | θ | ||
1 | KRD | ||||
1 | 1 | GRD | |||
1 | 1 | 1 | RD |
In this section, some key mathematical characteristics of the GKRD are investigated.
The quantile function Qp of the proposed GKRD is given by
Qp={−1θlog(1−[1−{1−p[1−(1−α)γ]}1γ]1β)}12,0<p<1, | (3.1) |
where p∈(0,1) represents the probability level. Further, with p=12, the value of the median is obtained, and it is
Q12={−1θlog(1−[1−{1−12[1−(1−α)γ]}1γ]1β)}12. | (3.2) |
The skewness (S) and kurtosis (K) coefficients can be obtained using the formula:
S(T)=Q0.25+Q0.75−2Q0.5Q0.75−Q0.25, | (3.3) |
and
K(T)=Q0.875−Q0.625+Q0.375−Q0.125Q0.75−Q0.25. |
Let us define the following series as:
(1−z)j=j∑l=0(−1)l(jl)zl. | (3.4) |
Thus, by applying Eq (3.4) in (2.2), the pdf of the GKRD becomes
ξ(t)=2βγθ t1−(1−α)γ∞∑l=0(−1)2l(β−1l)(γ−1l) αl+1e−θ(l+1)t2 (1−e−θt2)βl=2βγθ t1−(1−α)γ∞∑l=0ηl(α,β,γ) e−θ(l+1)t2 (1−e−θt2)βl, | (3.5) |
where ηl(α,β,γ)=(−1)2l(β−1l)(γ−1l) αl+1.
The kth ordinary moment of T that follows the GKRD is defined as follows:
μ′k=2βγθ1−(1−α)γ∞∑l=0ηl(α,β,γ) ωk,l(t,β,θ), | (3.6) |
with ωk,l(t,β,θ)=∫∞0tk+1e−θ(l+1)t2 (1−e−θt2)βl dt.
Taking k=1 and 2 in Eq (3.6), the first and second moments of origin of the GKRD can be obtained as
μ′1=2βγθ1−(1−α)γ∞∑l=0ηl(α,β,γ) ω1,l(t,β,θ), | (3.7) |
and
μ′2=2βγθ1−(1−α)γ∞∑l=0ηl(α,β,γ) ω2,l(t,β,θ). | (3.8) |
Next, the variance and coefficient of variance (CV) for the GKRD can be found as follows:
Var=μ′2−(μ′1)2, |
and
CV=√Varμ′1. |
The moment-generating function (MGF) of the GKRD can be derived as
M(y)=2βγθ1−(1−α)γ∞∑l=0∞∑k=0ykk!ηl(α,β,γ) ωk,l(t,β,θ). | (3.9) |
We have numerically assessed several statistical summary measures such as Mean, Variance, CV(T), S(T), and K(T) for different parametric values and posted them in Tables 2 and 3. The same can easily be observed for these quantities from the plots presented in Figures 3 and 4. Henceforth, the GKRD is an option to model the positively skewed and leptokurtic data sets.
θ | γ | μ′1 | Var | CV(T) | S(T) | K(T) |
0.4 | 0.3 | 0.6879 | 0.1257 | 0.5155 | 0.3580 | -0.5424 |
0.55 | 0.4798 | 0.0629 | 0.5227 | 0.3875 | -0.517 | |
0.8 | 0.3828 | 0.0406 | 0.5263 | 0.4025 | -0.5025 | |
1.2 | 0.2992 | 0.0251 | 0.5294 | 0.4160 | -0.4885 | |
0.8 | 0.3 | 0.4864 | 0.0629 | 0.5155 | 0.3580 | -0.5424 |
0.55 | 0.3392 | 0.0314 | 0.5227 | 0.3875 | -0.517 | |
0.8 | 0.2707 | 0.0203 | 0.5263 | 0.4025 | -0.5025 | |
1.2 | 0.2116 | 0.0125 | 0.5294 | 0.4160 | -0.4885 | |
1.2 | 0.3 | 0.3971 | 0.0419 | 0.5155 | 0.3580 | -0.5424 |
0.55 | 0.2770 | 0.0210 | 0.5227 | 0.3875 | -0.517 | |
0.8 | 0.2210 | 0.0135 | 0.5263 | 0.4025 | -0.5025 | |
1.2 | 0.1728 | 0.0084 | 0.5294 | 0.4160 | -0.4885 | |
1.6 | 0.3 | 0.3439 | 0.0314 | 0.5155 | 0.3580 | -0.5424 |
0.55 | 0.2399 | 0.0157 | 0.5227 | 0.3875 | -0.517 | |
0.8 | 0.1914 | 0.0101 | 0.5263 | 0.4025 | -0.5025 | |
1.2 | 0.1496 | 0.0063 | 0.5294 | 0.4160 | -0.4885 |
θ | γ | μ′1 | Var | CV(T) | S(T) | K(T) |
0.4 | 0.3 | 1.9210 | 0.5316 | 0.3795 | 0.2499 | -0.3336 |
0.55 | 1.5114 | 0.3015 | 0.3633 | 0.1814 | -0.353 | |
0.8 | 1.3110 | 0.2162 | 0.3546 | 0.1402 | -0.3631 | |
1.2 | 1.1289 | 0.1530 | 0.3465 | 0.0988 | -0.3706 | |
0.8 | 0.3 | 1.3583 | 0.2658 | 0.3795 | 0.2499 | -0.3336 |
0.55 | 1.0687 | 0.1507 | 0.3633 | 0.1814 | -0.353 | |
0.8 | 0.9270 | 0.1081 | 0.3546 | 0.1402 | -0.3631 | |
1.2 | 0.7982 | 0.0765 | 0.3465 | 0.0988 | -0.3706 | |
1.2 | 0.3 | 1.1091 | 0.1772 | 0.3795 | 0.2499 | -0.3336 |
0.55 | 0.8726 | 0.1005 | 0.3633 | 0.1814 | -0.353 | |
0.8 | 0.7569 | 0.0721 | 0.3546 | 0.1402 | -0.3631 | |
1.2 | 0.6517 | 0.0510 | 0.3465 | 0.0988 | -0.3706 | |
1.6 | 0.3 | 0.9605 | 0.1329 | 0.3795 | 0.2499 | -0.3336 |
0.55 | 0.7557 | 0.0754 | 0.3633 | 0.1814 | -0.353 | |
0.8 | 0.6555 | 0.0540 | 0.3546 | 0.1402 | -0.3631 | |
1.2 | 0.5644 | 0.0383 | 0.3465 | 0.0988 | -0.3706 |
Let a random sample t1,t2,…,tm represent a continuous GKRD. Based on Arnold et al. [23] and David et al. [24], the density function of the kth order statistic is as follows:
ϕ(k:m)(t)=m!(k−1)!(m−k)! ξ(t)[Ξ(t)]k−1 [1−Ξ(t)]m−k. | (3.10) |
Next,
ϕ(k:m)(t)=m!(k−1)!(m−k)!2αβγθ te−θt2(1−(1−α)γ)m {1−e−θt2}β−1 {1−α(1−e−θt2)β}γ−1×[1−{1−α(1−e−θt2)β}γ]k−1 [{1−α(1−e−θt2)β}γ−(1−α)γ]m−k. | (3.11) |
The ϕ(1:m)(t) minimum-order statistics is obtained by substituting m=1 in equation:
ϕ(1:m)(t)=2mαβγθ te−θt2(1−(1−α)γ)m {1−e−θt2}β−1 {1−α(1−e−θt2)β}γ−1×[{1−α(1−e−θt2)β}γ−(1−α)γ]m−1. | (3.12) |
Similarly, we will get the expression of the mth order-statistic by replacing k=m,
ϕ(m:m)(t)=2mαβγθ te−θt2(1−(1−α)γ)m {1−e−θt2}β−1 {1−α(1−e−θt2)β}γ−1×[1−{1−α(1−e−θt2)β}γ]m−1. | (3.13) |
In this estimation section, we estimate the parameters of the GKRD using two estimation methods. These methods, which include maximum likelihood (MLE) and Bayesian estimators under SELF, are crucial in enhancing ecological studies.
Consider {t1,t2,…,tm} a random sample of size m is taken from the GKRD and its associated log likelihood function LL is
LL(t;ρ)=m∑i=1logξ(ti)=m∑i=1log{2αβγθ tie−θt2i1−(1−α)γ (1−e−θt2i)β−1 (1−α(1−e−θt2i)β)γ−1}=m(logα+logβ+logγ+logθ)−mlog[1−(1−α)γ]+(β−1)m∑i=1log(1−e−θt2i)+(γ−1)m∑i=1log(1−α(1−e−θt2i)β). | (4.1) |
Now differentiate the above equation for ρ=(α,β,γ,θ)
∂LL(t;ρ)∂α=mα−mγ(1−α)γ−11−(1−α)γ−(γ−1)m∑i=1(1−e−θt2i)β1−α(1−e−θt2i)β, | (4.2) |
∂LL(t;ρ)∂β=mβ+m∑i=1log(1−e−θt2i)−α(γ−1)m∑i=1log(1−e−θt2i)(1−e−θt2i)β1−α(1−e−θt2i)β, | (4.3) |
∂LL(t;ρ)∂γ=mγ+mlog(1−α)(1−α)γ1−(1−α)γ+m∑i=1log(1−α(1−e−θt2i)β), | (4.4) |
and
∂LL(t;ρ)∂θ=mθ+(β−1)m∑i=1t2ie−θt2i1−e−θt2i−(γ−1)m∑i=1βαt2ie−θt2i(1−e−θt2i)β−11−α(1−e−θt2i)β. | (4.5) |
The parameter estimates of ρ=(α,β,γ,θ) are the solution of the above non-linear equations (4.2)–(4.5). Because these normal equations lack closed-form solutions, we use numerical methods to effectively solve them and derive ML estimates such as the Newton-Raphson, fixed point, or secant methods. To achieve this goal, we used the optim function in R software for the estimation process.
Compared to the maximum likelihood estimation approach, Bayesian estimation is a more current and efficient approximation. Considering past data and samples, we can make the Bayesian estimation.
We consider the independent informative type of priors for the parameters ρ=(α,β,γ,θ) as
π(ρ)∝αa1−1 βa2−1 γa3−1 θa4−1 e−b1α−b2β−b3γ−b4θ. |
The posterior density of ρ has the below form:
π∗(ρ∣t)=L(t;ρ) π(ρ∣t)=2αm+a1−1βm+a2−1γm+a3−1θm+a4−1(1−(1−α)γ)m e−b1α−b2β−b3γ−b4θ×m∏i=1ti e−θt2i {1−e−θt2i}β−1 {1−α(1−e−θt2i)β}γ−1. | (4.6) |
Hence, the Bayes estimation based on SELF
B=(ρ−ˆρ)2 |
is obtained to be:
ˆBSELF=∫ρB π∗(ρ∣t)dρ. | (4.7) |
By obtaining the joint prior, the posterior function can be determined, and it can be applied to the Metropolis-Hasting method.
Here, we discuss the performance of the two proposed estimators, MLE and Bayes, considering a finite number of samples. We do a simulation study with various samples (α,β,γ,θ) (Scenario 1: ρ=(0.75,1,2,1.5), Scenario 2: ρ=(0.8,1.1,2.3,1.8), Scenario 3: ρ=(0.9,1.2,2.5,2)) from the GKRD. We calculated the mean estimates (Mean), average bias (Bias), root mean square error (RMSE), and the efficiency (Eff) to weigh the MLEs and Bayes accuracy. Additionally, we present the mean of the number of iterations (NIT) required for convergence in each method (the Metropolis-Hasting technique for Bayes estimator and Newton Raphson for the MLE technique), showing that convergence occurs within number=1000 steps.
The computations were obtained employing the R program with the function optim for Newton Raphson technique and optim for the Metropolis-Hasting procedure by taking the values of ρ as Scenario 1, Scenario 2, and Scenario 3 respectively. Recall that, for the Bayesian estimation, we choose the gamma informative prior to obtaining the final estimate ˆρ. The algorithm for computing the unknown parameters for the GKRD is presented in details in Appendix. The results of the simulation are presented in Tables 4–6. The following expression is utilized to generate random samples from the suggested model:
t={−1θlog(1−[1−{1−q[1−(1−α)γ]}1γ]1β)}12,0<q<1. |
m | Par | MLE | Bayes | Eff | |||||||||
Mean | Bias | RMSE | NIT | Mean | Bias | RMSE | NIT | ||||||
75 | α | 0.8997 | 0.1497 | 0.2497 | 11 | 0.5874 | 0.1626 | 0.0398 | 5 | 6.2738 | |||
β | 1.1629 | 0.1629 | 0.2249 | 1.1549 | 0.1549 | 0.0461 | 4.8785 | ||||||
γ | 2.8214 | 0.8214 | 1.2828 | 1.9501 | 0.0499 | 0.0236 | 54.355 | ||||||
θ | 1.7158 | 0.2158 | 0.7419 | 1.3143 | 0.8143 | 0.2897 | 2.5609 | ||||||
100 | α | 0.7990 | 0.0490 | 0.2491 | 10 | 0.7387 | 0.0113 | 0.0094 | 3 | 26.5 | |||
β | 1.1617 | 0.1617 | 0.2118 | 1.0496 | 0.0496 | 0.0284 | 7.4577 | ||||||
γ | 2.9147 | 0.9147 | 1.1374 | 2.0204 | 0.0204 | 0.0172 | 66.127 | ||||||
θ | 1.5592 | 0.0592 | 0.4687 | 1.2876 | 0.7876 | 0.2178 | 3.4063 | ||||||
200 | α | 0.7985 | 0.0485 | 0.2487 | 11 | 0.7459 | 0.0041 | 0.0056 | 6 | 44.410 | |||
β | 1.1332 | 0.1332 | 0.1607 | 1.0271 | 0.0271 | 0.0034 | 47.264 | ||||||
γ | 2.2124 | 0.2124 | 0.8976 | 2.0704 | 0.0704 | 0.0088 | 102 | ||||||
θ | 1.4794 | 0.0206 | 0.3462 | 1.7081 | 0.2081 | 0.1401 | 2.4710 | ||||||
300 | α | 0.7698 | 0.0198 | 0.2398 | 10 | 0.7655 | 0.0155 | 0.0027 | 5 | 88.814 | |||
β | 1.1399 | 0.1399 | 0.1590 | 1.0221 | 0.0221 | 0.0025 | 63.6 | ||||||
γ | 2.0489 | 0.0489 | 0.4968 | 1.9358 | 0.0642 | 0.0079 | 62.886 | ||||||
θ | 1.4971 | 0.0029 | 0.2711 | 1.7054 | 0.2054 | 0.1134 | 2.3906 |
m | Par | MLE | Bayes | Eff | |||||||||
Mean | Bias | RMSE | NIT | Mean | Bias | RMSE | NIT | ||||||
75 | α | 0.9976 | 0.1976 | 0.2981 | 8 | 0.5797 | 0.2203 | 0.0703 | 3 | 4.2403 | |||
β | 1.2336 | 0.1336 | 0.2737 | 1.0340 | 0.0660 | 0.0750 | 3.6493 | ||||||
γ | 2.9946 | 0.6946 | 2.1815 | 2.6245 | 0.3245 | 0.1171 | 18.629 | ||||||
θ | 2.0733 | 0.2733 | 1.3915 | 2.3758 | 0.5758 | 0.6553 | 2.1234 | ||||||
100 | α | 0.8955 | 0.0955 | 0.2669 | 9 | 0.6414 | 0.1586 | 0.0449 | 5 | 5.9443 | |||
β | 1.1852 | 0.0852 | 0.1615 | 0.9583 | 0.1417 | 0.0271 | 5.9594 | ||||||
γ | 2.6917 | 0.3917 | 1.0912 | 2.2717 | 0.0283 | 0.0203 | 53.753 | ||||||
θ | 1.9411 | 0.1411 | 0.7705 | 2.3332 | 0.5332 | 0.5777 | 1.3337 | ||||||
200 | α | 0.8557 | 0.0557 | 0.2581 | 11 | 0.7418 | 0.0582 | 0.0131 | 4 | 19.702 | |||
β | 1.1900 | 0.0900 | 0.1305 | 1.1555 | 0.0555 | 0.0121 | 10.785 | ||||||
γ | 2.6241 | 0.3241 | 1.0744 | 2.4736 | 0.1736 | 0.0201 | 53.452 | ||||||
θ | 1.9438 | 0.1438 | 0.4761 | 2.4321 | 0.6321 | 0.3764 | 1.3337 | ||||||
300 | α | 0.8382 | 0.0382 | 0.2485 | 10 | 0.7586 | 0.0414 | 0.0089 | 5 | 27.921 | |||
β | 1.1816 | 0.0816 | 0.1120 | 1.1336 | 0.0336 | 0.0063 | 17.777 | ||||||
γ | 2.4060 | 0.1060 | 0.6505 | 2.3139 | 0.0161 | 0.0133 | 48.909 | ||||||
θ | 1.9205 | 0.1205 | 0.3338 | 2.1268 | 0.3268 | 0.2595 | 0.1302 |
m | Par | MLE | Bayes | Eff | |||||||||
Mean | Bias | RMSE | NIT | Mean | Bias | RMSE | NIT | ||||||
75 | α | 0.9091 | 0.0091 | 0.2196 | 12 | 0.7811 | 0.1189 | 0.0396 | 7 | 5.5454 | |||
β | 1.2678 | 0.0678 | 0.1892 | 1.1170 | 0.0830 | 0.0208 | 9.0961 | ||||||
γ | 3.7911 | 1.2911 | 3.3738 | 2.6981 | 0.1981 | 0.0485 | 69.562 | ||||||
θ | 2.2781 | 0.2781 | 1.1318 | 2.5410 | 0.5410 | 0.6393 | 1.7703 | ||||||
100 | α | 0.9252 | 0.0252 | 0.2017 | 8 | 0.8040 | 0.0960 | 0.0122 | 3 | 16.532 | |||
β | 1.2727 | 0.0727 | 0.1859 | 1.3039 | 0.1039 | 0.0138 | 13.471 | ||||||
γ | 3.3419 | 0.8419 | 2.4688 | 2.3542 | 0.1458 | 0.0300 | 82.293 | ||||||
θ | 2.2186 | 0.2186 | 1.0014 | 2.2183 | 0.2183 | 0.3614 | 2.7708 | ||||||
200 | α | 0.9825 | 0.0825 | 0.1190 | 9 | 0.8509 | 0.0591 | 0.0108 | 4 | 9.2523 | |||
β | 1.2317 | 0.0317 | 0.1059 | 1.2016 | 0.0016 | 0.0121 | 8.7520 | ||||||
γ | 2.9719 | 0.4719 | 1.8267 | 2.4405 | 0.0595 | 0.0200 | 91.335 | ||||||
θ | 2.1325 | 0.1325 | 0.8462 | 2.2748 | 0.2748 | 0.2861 | 2.9577 | ||||||
300 | α | 0.9944 | 0.0944 | 0.0990 | 12 | 0.9227 | 0.0273 | 0.0107 | 5 | 9.2523 | |||
β | 1.2275 | 0.0275 | 0.0860 | 1.2325 | 0.0325 | 0.0069 | 12.463 | ||||||
γ | 2.8578 | 0.3578 | 1.3202 | 2.5359 | 0.0359 | 0.0120 | 110.01 | ||||||
θ | 2.0783 | 0.0783 | 0.7752 | 2.1816 | 0.1816 | 0.2164 | 3.5822 |
Based on the findings presented in Tables 4–6, we can concluded the following points:
● As m increases the parameter estimates become closer to the true parameter. It appears obvious that the estimates of ρ are generally unbiased for the two methods of estimates.
● The RMSEs also show a decreasing pattern with an increase in m for the two methods of estimates.
● The results show that the Bayes estimator under SELF achieves an excellent performance among ML estimators. This is evident from the consistently low values of RMSE observed across all cases.
● Based on the NIT for the two proposed estimation procedures, it indicates the excellent performance of the Bayes estimator under SELF compared to the MLE method.
● The performance Bayes under SELF estimator is better than the MLE technique in all procedure scenarios because all efficiency values are greater than 1.
The most important part of statistical inference is the application of actual data and mathematical modeling. Data is frequently modeled under established probability distributions in the manufacturing and quality control industries in order to discover faults, guarantee product quality, and keep standards consistent. So we always need a new distribution.
Here in this study, the proposed GKRD is used to model two industrial data sets taken from Saudi Arabia (KSA) and one from financial data, and the resulting fits are compared to the competitive continuous distributions including the Kumaraswamu Gul alpha power transformed Rayleigh distribution (KGAPRD), Poisson generalized Rayleigh distribution (PGRD), new generalized Rayleigh distribution (NGRD), generalized Rayleigh distribution (GRD), Rayleigh distribution (RD), gamma distribution (GD), Weibull distribution (WD), and log-normal distribution (LND). Using the conventional criteria of the lowest values of the Akaike information criterion (AIC) and Bayesian information criterion (BIC), the fits given by the GKRD and the other examined models were compared. Furthermore, the comparison of the fitted distributions was assessed using the Kolmogorov-Smirnov (KS) test with its associated P-values and Anderson-Darling test (AD). For more information about recently financial applications see Atchadé et al. [25], and Kamal et al. [26].
The first real life data set used in this study was studied by Yu et al. [27], and it is based on the efficiency of the construction industry and their pure technical between 2013 and 2022 in the KSA. The values of the proposed data set are shown in Table 7.
Zone | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 |
Mecca | 3.58 | 4.81 | 5.95 | 6.71 | 7.81 | 8.68 | 8.91 | 9.97 | 10.011 | 9.80 |
Eastern | 2.55 | 3.90 | 4.59 | 6.37 | 7.11 | 7.38 | 7.55 | 7.17 | 7.89 | 8.54 |
Al Madinah | 3.26 | 3.46 | 3.47 | 4.99 | 6.38 | 6.42 | 6.81 | 6.16 | 6.77 | 7.21 |
Asir | 3.41 | 3.81 | 3.98 | 4.65 | 5.47 | 5.74 | 5.92 | 6.17 | 6.13 | 6.53 |
Jizan | 3.42 | 3.39 | 3.62 | 4.46 | 5.37 | 5.71 | 5.56 | 5.49 | 5.64 | 5.80 |
Al-Qassim | 3.43 | 3.45 | 3.37 | 4.11 | 4.46 | 4.81 | 5.10 | 5.07 | 5.24 | 5.45 |
Tabuk | 2.99 | 2.78 | 2.96 | 3.96 | 4.48 | 4.96 | 4.82 | 4.75 | 4.89 | 5.13 |
Ha'il | 2.89 | 2.59 | 2.73 | 3.59 | 4.19 | 4.59 | 4.52 | 4.50 | 4.70 | 4.75 |
Al Jawf | 2.29 | 2.75 | 2.48 | 3.35 | 4.22 | 4.42 | 4.55 | 4.44 | 4.63 | 4.71 |
Najran | 2.83 | 2.92 | 2.62 | 3.33 | 4.02 | 4.38 | 4.47 | 4.44 | 4.61 | 4.8 |
Northern Borders | 1.51 | 1.51 | 1.6 | 2.79 | 3.95 | 4.04 | 3.99 | 4.08 | 4.4 | 4.48 |
Al Bahah | 1.96 | 2.17 | 2 | 2.97 | 3.63 | 4.07 | 3.76 | 3.68 | 3.85 | 4.13 |
Here are the values provided about the construction industry and their scale efficiency in the KSA from 2013 to 2022. The suggested data set was considered by Yu et al. [27] and is shown in Table 8.
Zone | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 |
Mecca | 9.39 | 9.71 | 9.83 | 9.96 | 9.97 | 9.95 | 9.98 | 9.97 | 10.005 | 9.96 |
Eastern | 8.92 | 9.23 | 9.43 | 9.56 | 9.58 | 9.71 | 9.78 | 9.72 | 9.82 | 9.87 |
Al-Madinah | 7.46 | 7.47 | 7.81 | 8.52 | 8.62 | 8.61 | 8.73 | 8.43 | 8.74 | 8.77 |
Asir | 7.49 | 7.77 | 7.93 | 8.34 | 8.29 | 8.32 | 8.42 | 8.36 | 8.47 | 8.51 |
Jizan | 6.66 | 6.69 | 6.84 | 7.64 | 7.71 | 7.75 | 7.75 | 7.68 | 7.82 | 7.82 |
Al-Qassim | 6.6 | 6.62 | 6.67 | 7.47 | 7.51 | 7.53 | 7.67 | 7.6 | 7.73 | 7.73 |
Tabuk | 5.31 | 5.46 | 5.66 | 6.41 | 6.54 | 6.52 | 6.54 | 6.43 | 6.67 | 6.6 |
Ha'il | 4.23 | 4.27 | 4.29 | 5.31 | 5.47 | 5.47 | 5.59 | 5.14 | 5.62 | 5.72 |
In this subsection, the data set is drawn from group medical insurance. It is defined as the total loss for all the claim amounts exceeding 25,000 USD during 1991. The values of the data set are given in euros (EUR), and it is available at http://www.soa.org as well as used by Meraou et al. [10].
Based on the three proposed data sets, various non-parametric plots including the kernel density, fitted histogram, scaled total time on the test (TTT), probability-probability (PP), QQ normal, and box plots are plotted in Figures 5–7.
Table 9 summarized the obtained results of goodness-of-fit test with the ML estimates of all fitted model using the industrial and financial data sets. It is well documented that the GKRD is recognized as the optimal choice for the three data sets, with the following:
Data set | Model | ˆα | ˆβ | ˆδ | ˆθ | KS | KS (P-value) | AIC | BIC | AD |
GKRD | 0.2024 (0.4282) | 2.385 (0.3231) | 10.851 (24.654) | 0.0437 (0.0134) | 0.0663 | 0.6658 | 467.253 | 474.403 | 0.3239 | |
KGAPRD | 1.9052 (3.0495) | 1.5923 (0.4020) | 0.2816 (0.1602) | 0.2534 (1.2086) | 0.0768 | 0.4783 | 468.424 | 479.574 | 0.4321 | |
PGRD | 3.9205 (0.1949) | 0.0434 (0.0246) | 52.541 (2.9280) | 0.0902 | 0.2820 | 469.007 | 477.370 | 0.6984 | ||
I | NGRD | 3.9341 (0.2011) | 2.2256 (0.3212) | 42.308 (10.120) | 0.0937 | 0.2426 | 468.777 | 477.139 | 0.6817 | |
GRD | 2.2389 (0.3197) | 2.7901 (0.1413) | 0.0930 | 0.2501 | 469.633 | 475.208 | 0.6680 | |||
RD | 0.0401 (0.0036) | 0.1758 | 0.0011 | 494.084 | 496.871 | 0.8479 | ||||
GD | 5.1841 (0.6488) | 1.1019 (0.1448) | 0.0932 | 0.2479 | 471.134 | 476.709 | 0.4459 | |||
WD | 2.7961 (0.1863) | 5.2474 (0.1815) | 0.1128 | 0.0941 | 475.208 | 480.783 | 0.7885 | |||
LND | 4.6723 (0.1601) | 1.7548 (0.1132) | 0.1248 | 0.0475 | 479.517 | 485.091 | 0.8003 | |||
GKRD | 0.8638 (1.1574) | 3.7510 (0.5461) | 17.464 (31.538) | 0.0093 (0.0033) | 0.0926 | 0.5853 | 271.904 | 280.898 | 0.9785 | |
KGAPRD | 2.9305 (0.3225) | 74.310 (93.608) | 45.282 (9.2674) | 42.954 (1.4238) | 0.0953 | 0.5477 | 275.190 | 284.184 | 1.1391 | |
PGRD | 4.9536 (0.2978) | 0.1952 (0.8534) | 38.003 (0.0298) | 0.1197 | 0.2684 | 280.548 | 287.293 | 1.6369 | ||
II | NGRD | 4.9852 (0.2595) | 6.9592 (1.5763) | 51.870 (3.6785) | 0.1232 | 0.2385 | 280.780 | 287.525 | 1.6594 | |
GRD | 6.957 (1.5724) | 3.5342 (0.1866) | 0.1236 | 0.2346 | 278.894 | 283.391 | 1.6678 | |||
RD | 0.0156 (1.1864) | 0.3000 | 6.7×10−06 | 342.372 | 344.621 | 1.3876 | ||||
GD | 20.254 (3.3957) | 2.5916 (0.4399) | 0.1195 | 0.2703 | 277.583 | 282.080 | 1.5974 | |||
WD | 5.7702 (0.5667) | 8.4693 (0.1843) | 0.0948 | 0.5551 | 272.030 | 281.527 | 0.9818 | |||
LND | 7.8151 (0.1968) | 1.6465 (0.1391) | 0.0991 | 0.4965 | 272.459 | 281.956 | 1.2196 | |||
GKRD | 0.1193 (0.2206) | 1.7833 (0.3249) | 27.478 (3.1088) | 0.0251 (0.0087) | 0.1124 | 0.5540 | 206.203 | 213.604 | 1.006 | |
KGAPRD | 2.3725 (0.7543) | 1.7809 (0.0039) | 0.2262 (0.0365) | 1.3333 (0.0046) | 0.1365 | 0.3155 | 208.619 | 216.019 | 1.3328 | |
PGRD | 4.8427 (0.4424) | 0.0094 (0.0085) | 159.821 (1.4759) | 0.1439 | 0.2588 | 212.353 | 217.903 | 1.5645 | ||
III | NGRD | 4.8419 (0.4452) | 1.4818 (0.3299) | 32.342 (9.8564) | 0.1455 | 0.2473 | 212.470 | 218.020 | 1.5853 | |
GRD | 1.5009 (0.3300) | 3.4326 (0.3156) | 0.1442 | 0.2562 | 210.189 | 213.889 | 1.5751 | |||
RD | 0.0326 (0.0047) | 0.1935 | 0.0513 | 211.470 | 213.320 | 1.6496 | ||||
GD | 5.1224 (1.0241) | 1.0364 (0.2177) | 0.1184 | 0.4881 | 210.977 | 214.677 | 1.0809 | |||
WD | 2.1042 (0.2110) | 5.6024 (0.4133) | 0.1758 | 0.0964 | 223.267 | 226.967 | 1.7173 | |||
LND | 4.9392 (0.3637) | 2.4936 (0.2571) | 0.1706 | 0.1149 | 213.222 | 216.923 | 1.6214 |
(1) Data 1: α=0.2, β=2.4, γ=10, θ=0.05.
(2) Data 2: α=0.9, β=3.75, γ=20, θ=0.01.
(3) Data 3: α=0.1, β=1.8, γ=30, θ=0.02.
Clearly, from the obtained results in Table 9, for the three data sets, the GKRD is an efficient, superior model among competing models based on all AIC and BIC measures. This ensures that the GKRD is the most appropriate model among the choices. Specifically, in terms of P-value and AD, the GKRD outperforms all other models considered, confirming its status as the most optimal distribution for the three data sets when compared to alternative models. Additionally, the plots for the estimated pdf versus fitted histogram and estimated cdf versus empirical cdf are plotted in Figures 8–10. The various plots presented also confirm a good fit for the GKRD to the considered data sets.
For modeling industrial and financial data sets that many models lack, to model, we defined a novel four-parameter probabilistic model. The new model was created using the generalized Kumaraswamy technique, resulting in the generalized Kumaraswamy Rayleigh model. Several important distributional and statistical characteristics have been determined and analyzed. By using a wide range of methods, including the classical MLE and Bayesian techniques, we are able to handle statistical analysis of the GKRD distribution and its unknown parameters. Therefore, we came to the conclusion that the Bayes approach is superior to the conventional estimating method since it consistently produces lower values for the MSE. Additionally, using three real-life data sets taken from industrial and financial domains, the results indicate that the proposed GKRD distribution effectively analyzes both data sets compared to competing distributions.
In future work, this study may attract the bivariate case of the GKRD. In addition, this study may contribute to the estimation of model parameters in censored samples based on several cases. In addition, the proposed model can attract a wider set of applications, such as in engineering and environmental fields.
Alanazi Talal Abdulrahman, Tariq S. Alshammari and Ramlah H Albayyat worked on mathematics; Eslam Hussam, Amirah Saeed Alharthi and Khudhayr A. Rashedi worked on english and programming. All the authors have read and approved the final version of the manuscript for publication.
This research has been funded by the Scientific Research Deanship at the University of Ha'il, Saudi Arabia, through project number (RG-24 068).
All data exists in the paper with their related references.
This research has been funded by the Scientific Research Deanship at the University of Ha'il, Saudi Arabia, through project number RG-24 068.
The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.
All authors declare no conflicts of interest in this paper.
pdf.GKRD=function(star,x){
alpha=star[1]
beta=star[2]
delta=star[3]
theta=star[4]
2*beta*delta*alpha*theta*x*exp(-theta*x^2)/(1-(1-alpha)^delta)*
(1-exp(-theta*x^2))^(beta-1)*(1-alpha*(1-exp(-theta*x^2))^beta)^(delta-1)
}
t=seq(0,2,len=1000)
plot(t,pdfGKRD(c(0.95,0.45,3,1.2),t),col="red",type="l",lwd=2,lty=4)
cdf.GKRD=function(star,x){
alpha=star[1]
beta=star[2]
delta=star[3]
theta=star[4]
(1-(1-alpha*(1-exp(-theta*x^2))^beta)^delta)/(1-(1-alpha)^delta)
}
t=seq(0,2,len=1000)
plot(t,cdf.GKRD(c(0.2,1.3,1.5,4.5),t),col="red",type="l",lwd=2,lty=4)
hrf.GKRD <- function(star,x){
alpha=star[1]
beta=star[2]
delta=star[3]
theta=star[4]
2*beta*delta*alpha*theta*x*exp(-theta*x^2)/(1-(1-alpha)^delta)*
(1-exp(-theta*x^2))^(beta-1)*(1-alpha*(1-exp(-theta*x^2))^beta)^(delta-1)/
(1-(1-(1-alpha*(1-exp(-theta*x^2))^beta)^delta)/(1-(1-alpha)^delta))
}
t=seq(0,2,len=1000)
plot(t,hrf.GKRD(c(0.95,0.3,0.6,3.5),t),col="red",type="l",lwd=2,lty=4)
## Estimation
fMLE<-function(star,x){
alpha=star[1]
beta=star[2]
delta=star[3]
theta=star[4]
-sum(log(beta*delta*alpha*theta*exp(-theta*x^2)/(1-(1-alpha)^delta)*
(1-exp(-theta*x^2))^(beta-1)*(1-alpha*(1-exp(-theta*x^2))^beta)^(delta-1)
))
}
NB=100; nb=100; res.alpha=numeric(NB);res.beta=numeric(NB)
res.delta=numeric(NB);res.theta=numeric(NB)
for(i in 1:NB){
alpha=0.75;delta=2; beta=1.0;theta=1.5
u=runif(nb,0,1)
X=sqrt(-1/theta*log(1-(1-(1-u*(1-(1-alpha)^delta))^(1/delta))^(1/beta)))
res.alpha[i]=optim(c(alpha,beta,delta,theta),fMLE,method="N",x=X)$par[1]
res.beta[i]=optim(c(alpha,beta,delta,theta),fMLE,method="N",x=X)$par[2]
res.delta[i]=optim(c(alpha,beta,delta,theta),fMLE,method="N",x=X)$par[3]
res.theta[i]=optim(c(alpha,beta,delta,theta),fMLE,method="N",x=X)$par[4]
}
AEMLE.alpha=mean(res.alpha); AEMLE.beta=mean(res.beta)
AEMLE.delta=mean(res.delta); AEMLE.theta=mean(res.theta)
AB.alpha=abs(mean(res.alpha-alpha)); AB.beta=abs(mean(res.beta-beta))
AB.delta=abs(mean(res.delta-delta)); AB.theta=abs(mean(res.theta-theta))
MSEMLE.alpha=mean((alpha-res.alpha)**2); MSEMLE.beta=mean((beta-res.beta)**2)
MSEMLE.delta=mean((delta-res.delta)**2); MSEMLE.theta=mean((theta-res.theta)**2)
## Application
result=goodness.fit(pdf = pdf.GKRD, cdf =cdf.GKRD, method = "BFGS",
starts = c(alpha,beta,delta,theta), data = data, domain = c(0,Inf),mle = NULL)
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Parameter | Model | ||||
α | β | γ | θ | ||
1 | KRD | ||||
1 | 1 | GRD | |||
1 | 1 | 1 | RD |
θ | γ | μ′1 | Var | CV(T) | S(T) | K(T) |
0.4 | 0.3 | 0.6879 | 0.1257 | 0.5155 | 0.3580 | -0.5424 |
0.55 | 0.4798 | 0.0629 | 0.5227 | 0.3875 | -0.517 | |
0.8 | 0.3828 | 0.0406 | 0.5263 | 0.4025 | -0.5025 | |
1.2 | 0.2992 | 0.0251 | 0.5294 | 0.4160 | -0.4885 | |
0.8 | 0.3 | 0.4864 | 0.0629 | 0.5155 | 0.3580 | -0.5424 |
0.55 | 0.3392 | 0.0314 | 0.5227 | 0.3875 | -0.517 | |
0.8 | 0.2707 | 0.0203 | 0.5263 | 0.4025 | -0.5025 | |
1.2 | 0.2116 | 0.0125 | 0.5294 | 0.4160 | -0.4885 | |
1.2 | 0.3 | 0.3971 | 0.0419 | 0.5155 | 0.3580 | -0.5424 |
0.55 | 0.2770 | 0.0210 | 0.5227 | 0.3875 | -0.517 | |
0.8 | 0.2210 | 0.0135 | 0.5263 | 0.4025 | -0.5025 | |
1.2 | 0.1728 | 0.0084 | 0.5294 | 0.4160 | -0.4885 | |
1.6 | 0.3 | 0.3439 | 0.0314 | 0.5155 | 0.3580 | -0.5424 |
0.55 | 0.2399 | 0.0157 | 0.5227 | 0.3875 | -0.517 | |
0.8 | 0.1914 | 0.0101 | 0.5263 | 0.4025 | -0.5025 | |
1.2 | 0.1496 | 0.0063 | 0.5294 | 0.4160 | -0.4885 |
θ | γ | μ′1 | Var | CV(T) | S(T) | K(T) |
0.4 | 0.3 | 1.9210 | 0.5316 | 0.3795 | 0.2499 | -0.3336 |
0.55 | 1.5114 | 0.3015 | 0.3633 | 0.1814 | -0.353 | |
0.8 | 1.3110 | 0.2162 | 0.3546 | 0.1402 | -0.3631 | |
1.2 | 1.1289 | 0.1530 | 0.3465 | 0.0988 | -0.3706 | |
0.8 | 0.3 | 1.3583 | 0.2658 | 0.3795 | 0.2499 | -0.3336 |
0.55 | 1.0687 | 0.1507 | 0.3633 | 0.1814 | -0.353 | |
0.8 | 0.9270 | 0.1081 | 0.3546 | 0.1402 | -0.3631 | |
1.2 | 0.7982 | 0.0765 | 0.3465 | 0.0988 | -0.3706 | |
1.2 | 0.3 | 1.1091 | 0.1772 | 0.3795 | 0.2499 | -0.3336 |
0.55 | 0.8726 | 0.1005 | 0.3633 | 0.1814 | -0.353 | |
0.8 | 0.7569 | 0.0721 | 0.3546 | 0.1402 | -0.3631 | |
1.2 | 0.6517 | 0.0510 | 0.3465 | 0.0988 | -0.3706 | |
1.6 | 0.3 | 0.9605 | 0.1329 | 0.3795 | 0.2499 | -0.3336 |
0.55 | 0.7557 | 0.0754 | 0.3633 | 0.1814 | -0.353 | |
0.8 | 0.6555 | 0.0540 | 0.3546 | 0.1402 | -0.3631 | |
1.2 | 0.5644 | 0.0383 | 0.3465 | 0.0988 | -0.3706 |
m | Par | MLE | Bayes | Eff | |||||||||
Mean | Bias | RMSE | NIT | Mean | Bias | RMSE | NIT | ||||||
75 | α | 0.8997 | 0.1497 | 0.2497 | 11 | 0.5874 | 0.1626 | 0.0398 | 5 | 6.2738 | |||
β | 1.1629 | 0.1629 | 0.2249 | 1.1549 | 0.1549 | 0.0461 | 4.8785 | ||||||
γ | 2.8214 | 0.8214 | 1.2828 | 1.9501 | 0.0499 | 0.0236 | 54.355 | ||||||
θ | 1.7158 | 0.2158 | 0.7419 | 1.3143 | 0.8143 | 0.2897 | 2.5609 | ||||||
100 | α | 0.7990 | 0.0490 | 0.2491 | 10 | 0.7387 | 0.0113 | 0.0094 | 3 | 26.5 | |||
β | 1.1617 | 0.1617 | 0.2118 | 1.0496 | 0.0496 | 0.0284 | 7.4577 | ||||||
γ | 2.9147 | 0.9147 | 1.1374 | 2.0204 | 0.0204 | 0.0172 | 66.127 | ||||||
θ | 1.5592 | 0.0592 | 0.4687 | 1.2876 | 0.7876 | 0.2178 | 3.4063 | ||||||
200 | α | 0.7985 | 0.0485 | 0.2487 | 11 | 0.7459 | 0.0041 | 0.0056 | 6 | 44.410 | |||
β | 1.1332 | 0.1332 | 0.1607 | 1.0271 | 0.0271 | 0.0034 | 47.264 | ||||||
γ | 2.2124 | 0.2124 | 0.8976 | 2.0704 | 0.0704 | 0.0088 | 102 | ||||||
θ | 1.4794 | 0.0206 | 0.3462 | 1.7081 | 0.2081 | 0.1401 | 2.4710 | ||||||
300 | α | 0.7698 | 0.0198 | 0.2398 | 10 | 0.7655 | 0.0155 | 0.0027 | 5 | 88.814 | |||
β | 1.1399 | 0.1399 | 0.1590 | 1.0221 | 0.0221 | 0.0025 | 63.6 | ||||||
γ | 2.0489 | 0.0489 | 0.4968 | 1.9358 | 0.0642 | 0.0079 | 62.886 | ||||||
θ | 1.4971 | 0.0029 | 0.2711 | 1.7054 | 0.2054 | 0.1134 | 2.3906 |
m | Par | MLE | Bayes | Eff | |||||||||
Mean | Bias | RMSE | NIT | Mean | Bias | RMSE | NIT | ||||||
75 | α | 0.9976 | 0.1976 | 0.2981 | 8 | 0.5797 | 0.2203 | 0.0703 | 3 | 4.2403 | |||
β | 1.2336 | 0.1336 | 0.2737 | 1.0340 | 0.0660 | 0.0750 | 3.6493 | ||||||
γ | 2.9946 | 0.6946 | 2.1815 | 2.6245 | 0.3245 | 0.1171 | 18.629 | ||||||
θ | 2.0733 | 0.2733 | 1.3915 | 2.3758 | 0.5758 | 0.6553 | 2.1234 | ||||||
100 | α | 0.8955 | 0.0955 | 0.2669 | 9 | 0.6414 | 0.1586 | 0.0449 | 5 | 5.9443 | |||
β | 1.1852 | 0.0852 | 0.1615 | 0.9583 | 0.1417 | 0.0271 | 5.9594 | ||||||
γ | 2.6917 | 0.3917 | 1.0912 | 2.2717 | 0.0283 | 0.0203 | 53.753 | ||||||
θ | 1.9411 | 0.1411 | 0.7705 | 2.3332 | 0.5332 | 0.5777 | 1.3337 | ||||||
200 | α | 0.8557 | 0.0557 | 0.2581 | 11 | 0.7418 | 0.0582 | 0.0131 | 4 | 19.702 | |||
β | 1.1900 | 0.0900 | 0.1305 | 1.1555 | 0.0555 | 0.0121 | 10.785 | ||||||
γ | 2.6241 | 0.3241 | 1.0744 | 2.4736 | 0.1736 | 0.0201 | 53.452 | ||||||
θ | 1.9438 | 0.1438 | 0.4761 | 2.4321 | 0.6321 | 0.3764 | 1.3337 | ||||||
300 | α | 0.8382 | 0.0382 | 0.2485 | 10 | 0.7586 | 0.0414 | 0.0089 | 5 | 27.921 | |||
β | 1.1816 | 0.0816 | 0.1120 | 1.1336 | 0.0336 | 0.0063 | 17.777 | ||||||
γ | 2.4060 | 0.1060 | 0.6505 | 2.3139 | 0.0161 | 0.0133 | 48.909 | ||||||
θ | 1.9205 | 0.1205 | 0.3338 | 2.1268 | 0.3268 | 0.2595 | 0.1302 |
m | Par | MLE | Bayes | Eff | |||||||||
Mean | Bias | RMSE | NIT | Mean | Bias | RMSE | NIT | ||||||
75 | α | 0.9091 | 0.0091 | 0.2196 | 12 | 0.7811 | 0.1189 | 0.0396 | 7 | 5.5454 | |||
β | 1.2678 | 0.0678 | 0.1892 | 1.1170 | 0.0830 | 0.0208 | 9.0961 | ||||||
γ | 3.7911 | 1.2911 | 3.3738 | 2.6981 | 0.1981 | 0.0485 | 69.562 | ||||||
θ | 2.2781 | 0.2781 | 1.1318 | 2.5410 | 0.5410 | 0.6393 | 1.7703 | ||||||
100 | α | 0.9252 | 0.0252 | 0.2017 | 8 | 0.8040 | 0.0960 | 0.0122 | 3 | 16.532 | |||
β | 1.2727 | 0.0727 | 0.1859 | 1.3039 | 0.1039 | 0.0138 | 13.471 | ||||||
γ | 3.3419 | 0.8419 | 2.4688 | 2.3542 | 0.1458 | 0.0300 | 82.293 | ||||||
θ | 2.2186 | 0.2186 | 1.0014 | 2.2183 | 0.2183 | 0.3614 | 2.7708 | ||||||
200 | α | 0.9825 | 0.0825 | 0.1190 | 9 | 0.8509 | 0.0591 | 0.0108 | 4 | 9.2523 | |||
β | 1.2317 | 0.0317 | 0.1059 | 1.2016 | 0.0016 | 0.0121 | 8.7520 | ||||||
γ | 2.9719 | 0.4719 | 1.8267 | 2.4405 | 0.0595 | 0.0200 | 91.335 | ||||||
θ | 2.1325 | 0.1325 | 0.8462 | 2.2748 | 0.2748 | 0.2861 | 2.9577 | ||||||
300 | α | 0.9944 | 0.0944 | 0.0990 | 12 | 0.9227 | 0.0273 | 0.0107 | 5 | 9.2523 | |||
β | 1.2275 | 0.0275 | 0.0860 | 1.2325 | 0.0325 | 0.0069 | 12.463 | ||||||
γ | 2.8578 | 0.3578 | 1.3202 | 2.5359 | 0.0359 | 0.0120 | 110.01 | ||||||
θ | 2.0783 | 0.0783 | 0.7752 | 2.1816 | 0.1816 | 0.2164 | 3.5822 |
Zone | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 |
Mecca | 3.58 | 4.81 | 5.95 | 6.71 | 7.81 | 8.68 | 8.91 | 9.97 | 10.011 | 9.80 |
Eastern | 2.55 | 3.90 | 4.59 | 6.37 | 7.11 | 7.38 | 7.55 | 7.17 | 7.89 | 8.54 |
Al Madinah | 3.26 | 3.46 | 3.47 | 4.99 | 6.38 | 6.42 | 6.81 | 6.16 | 6.77 | 7.21 |
Asir | 3.41 | 3.81 | 3.98 | 4.65 | 5.47 | 5.74 | 5.92 | 6.17 | 6.13 | 6.53 |
Jizan | 3.42 | 3.39 | 3.62 | 4.46 | 5.37 | 5.71 | 5.56 | 5.49 | 5.64 | 5.80 |
Al-Qassim | 3.43 | 3.45 | 3.37 | 4.11 | 4.46 | 4.81 | 5.10 | 5.07 | 5.24 | 5.45 |
Tabuk | 2.99 | 2.78 | 2.96 | 3.96 | 4.48 | 4.96 | 4.82 | 4.75 | 4.89 | 5.13 |
Ha'il | 2.89 | 2.59 | 2.73 | 3.59 | 4.19 | 4.59 | 4.52 | 4.50 | 4.70 | 4.75 |
Al Jawf | 2.29 | 2.75 | 2.48 | 3.35 | 4.22 | 4.42 | 4.55 | 4.44 | 4.63 | 4.71 |
Najran | 2.83 | 2.92 | 2.62 | 3.33 | 4.02 | 4.38 | 4.47 | 4.44 | 4.61 | 4.8 |
Northern Borders | 1.51 | 1.51 | 1.6 | 2.79 | 3.95 | 4.04 | 3.99 | 4.08 | 4.4 | 4.48 |
Al Bahah | 1.96 | 2.17 | 2 | 2.97 | 3.63 | 4.07 | 3.76 | 3.68 | 3.85 | 4.13 |
Zone | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 |
Mecca | 9.39 | 9.71 | 9.83 | 9.96 | 9.97 | 9.95 | 9.98 | 9.97 | 10.005 | 9.96 |
Eastern | 8.92 | 9.23 | 9.43 | 9.56 | 9.58 | 9.71 | 9.78 | 9.72 | 9.82 | 9.87 |
Al-Madinah | 7.46 | 7.47 | 7.81 | 8.52 | 8.62 | 8.61 | 8.73 | 8.43 | 8.74 | 8.77 |
Asir | 7.49 | 7.77 | 7.93 | 8.34 | 8.29 | 8.32 | 8.42 | 8.36 | 8.47 | 8.51 |
Jizan | 6.66 | 6.69 | 6.84 | 7.64 | 7.71 | 7.75 | 7.75 | 7.68 | 7.82 | 7.82 |
Al-Qassim | 6.6 | 6.62 | 6.67 | 7.47 | 7.51 | 7.53 | 7.67 | 7.6 | 7.73 | 7.73 |
Tabuk | 5.31 | 5.46 | 5.66 | 6.41 | 6.54 | 6.52 | 6.54 | 6.43 | 6.67 | 6.6 |
Ha'il | 4.23 | 4.27 | 4.29 | 5.31 | 5.47 | 5.47 | 5.59 | 5.14 | 5.62 | 5.72 |
Data set | Model | ˆα | ˆβ | ˆδ | ˆθ | KS | KS (P-value) | AIC | BIC | AD |
GKRD | 0.2024 (0.4282) | 2.385 (0.3231) | 10.851 (24.654) | 0.0437 (0.0134) | 0.0663 | 0.6658 | 467.253 | 474.403 | 0.3239 | |
KGAPRD | 1.9052 (3.0495) | 1.5923 (0.4020) | 0.2816 (0.1602) | 0.2534 (1.2086) | 0.0768 | 0.4783 | 468.424 | 479.574 | 0.4321 | |
PGRD | 3.9205 (0.1949) | 0.0434 (0.0246) | 52.541 (2.9280) | 0.0902 | 0.2820 | 469.007 | 477.370 | 0.6984 | ||
I | NGRD | 3.9341 (0.2011) | 2.2256 (0.3212) | 42.308 (10.120) | 0.0937 | 0.2426 | 468.777 | 477.139 | 0.6817 | |
GRD | 2.2389 (0.3197) | 2.7901 (0.1413) | 0.0930 | 0.2501 | 469.633 | 475.208 | 0.6680 | |||
RD | 0.0401 (0.0036) | 0.1758 | 0.0011 | 494.084 | 496.871 | 0.8479 | ||||
GD | 5.1841 (0.6488) | 1.1019 (0.1448) | 0.0932 | 0.2479 | 471.134 | 476.709 | 0.4459 | |||
WD | 2.7961 (0.1863) | 5.2474 (0.1815) | 0.1128 | 0.0941 | 475.208 | 480.783 | 0.7885 | |||
LND | 4.6723 (0.1601) | 1.7548 (0.1132) | 0.1248 | 0.0475 | 479.517 | 485.091 | 0.8003 | |||
GKRD | 0.8638 (1.1574) | 3.7510 (0.5461) | 17.464 (31.538) | 0.0093 (0.0033) | 0.0926 | 0.5853 | 271.904 | 280.898 | 0.9785 | |
KGAPRD | 2.9305 (0.3225) | 74.310 (93.608) | 45.282 (9.2674) | 42.954 (1.4238) | 0.0953 | 0.5477 | 275.190 | 284.184 | 1.1391 | |
PGRD | 4.9536 (0.2978) | 0.1952 (0.8534) | 38.003 (0.0298) | 0.1197 | 0.2684 | 280.548 | 287.293 | 1.6369 | ||
II | NGRD | 4.9852 (0.2595) | 6.9592 (1.5763) | 51.870 (3.6785) | 0.1232 | 0.2385 | 280.780 | 287.525 | 1.6594 | |
GRD | 6.957 (1.5724) | 3.5342 (0.1866) | 0.1236 | 0.2346 | 278.894 | 283.391 | 1.6678 | |||
RD | 0.0156 (1.1864) | 0.3000 | 6.7×10−06 | 342.372 | 344.621 | 1.3876 | ||||
GD | 20.254 (3.3957) | 2.5916 (0.4399) | 0.1195 | 0.2703 | 277.583 | 282.080 | 1.5974 | |||
WD | 5.7702 (0.5667) | 8.4693 (0.1843) | 0.0948 | 0.5551 | 272.030 | 281.527 | 0.9818 | |||
LND | 7.8151 (0.1968) | 1.6465 (0.1391) | 0.0991 | 0.4965 | 272.459 | 281.956 | 1.2196 | |||
GKRD | 0.1193 (0.2206) | 1.7833 (0.3249) | 27.478 (3.1088) | 0.0251 (0.0087) | 0.1124 | 0.5540 | 206.203 | 213.604 | 1.006 | |
KGAPRD | 2.3725 (0.7543) | 1.7809 (0.0039) | 0.2262 (0.0365) | 1.3333 (0.0046) | 0.1365 | 0.3155 | 208.619 | 216.019 | 1.3328 | |
PGRD | 4.8427 (0.4424) | 0.0094 (0.0085) | 159.821 (1.4759) | 0.1439 | 0.2588 | 212.353 | 217.903 | 1.5645 | ||
III | NGRD | 4.8419 (0.4452) | 1.4818 (0.3299) | 32.342 (9.8564) | 0.1455 | 0.2473 | 212.470 | 218.020 | 1.5853 | |
GRD | 1.5009 (0.3300) | 3.4326 (0.3156) | 0.1442 | 0.2562 | 210.189 | 213.889 | 1.5751 | |||
RD | 0.0326 (0.0047) | 0.1935 | 0.0513 | 211.470 | 213.320 | 1.6496 | ||||
GD | 5.1224 (1.0241) | 1.0364 (0.2177) | 0.1184 | 0.4881 | 210.977 | 214.677 | 1.0809 | |||
WD | 2.1042 (0.2110) | 5.6024 (0.4133) | 0.1758 | 0.0964 | 223.267 | 226.967 | 1.7173 | |||
LND | 4.9392 (0.3637) | 2.4936 (0.2571) | 0.1706 | 0.1149 | 213.222 | 216.923 | 1.6214 |
Parameter | Model | ||||
α | β | γ | θ | ||
1 | KRD | ||||
1 | 1 | GRD | |||
1 | 1 | 1 | RD |
θ | γ | μ′1 | Var | CV(T) | S(T) | K(T) |
0.4 | 0.3 | 0.6879 | 0.1257 | 0.5155 | 0.3580 | -0.5424 |
0.55 | 0.4798 | 0.0629 | 0.5227 | 0.3875 | -0.517 | |
0.8 | 0.3828 | 0.0406 | 0.5263 | 0.4025 | -0.5025 | |
1.2 | 0.2992 | 0.0251 | 0.5294 | 0.4160 | -0.4885 | |
0.8 | 0.3 | 0.4864 | 0.0629 | 0.5155 | 0.3580 | -0.5424 |
0.55 | 0.3392 | 0.0314 | 0.5227 | 0.3875 | -0.517 | |
0.8 | 0.2707 | 0.0203 | 0.5263 | 0.4025 | -0.5025 | |
1.2 | 0.2116 | 0.0125 | 0.5294 | 0.4160 | -0.4885 | |
1.2 | 0.3 | 0.3971 | 0.0419 | 0.5155 | 0.3580 | -0.5424 |
0.55 | 0.2770 | 0.0210 | 0.5227 | 0.3875 | -0.517 | |
0.8 | 0.2210 | 0.0135 | 0.5263 | 0.4025 | -0.5025 | |
1.2 | 0.1728 | 0.0084 | 0.5294 | 0.4160 | -0.4885 | |
1.6 | 0.3 | 0.3439 | 0.0314 | 0.5155 | 0.3580 | -0.5424 |
0.55 | 0.2399 | 0.0157 | 0.5227 | 0.3875 | -0.517 | |
0.8 | 0.1914 | 0.0101 | 0.5263 | 0.4025 | -0.5025 | |
1.2 | 0.1496 | 0.0063 | 0.5294 | 0.4160 | -0.4885 |
θ | γ | μ′1 | Var | CV(T) | S(T) | K(T) |
0.4 | 0.3 | 1.9210 | 0.5316 | 0.3795 | 0.2499 | -0.3336 |
0.55 | 1.5114 | 0.3015 | 0.3633 | 0.1814 | -0.353 | |
0.8 | 1.3110 | 0.2162 | 0.3546 | 0.1402 | -0.3631 | |
1.2 | 1.1289 | 0.1530 | 0.3465 | 0.0988 | -0.3706 | |
0.8 | 0.3 | 1.3583 | 0.2658 | 0.3795 | 0.2499 | -0.3336 |
0.55 | 1.0687 | 0.1507 | 0.3633 | 0.1814 | -0.353 | |
0.8 | 0.9270 | 0.1081 | 0.3546 | 0.1402 | -0.3631 | |
1.2 | 0.7982 | 0.0765 | 0.3465 | 0.0988 | -0.3706 | |
1.2 | 0.3 | 1.1091 | 0.1772 | 0.3795 | 0.2499 | -0.3336 |
0.55 | 0.8726 | 0.1005 | 0.3633 | 0.1814 | -0.353 | |
0.8 | 0.7569 | 0.0721 | 0.3546 | 0.1402 | -0.3631 | |
1.2 | 0.6517 | 0.0510 | 0.3465 | 0.0988 | -0.3706 | |
1.6 | 0.3 | 0.9605 | 0.1329 | 0.3795 | 0.2499 | -0.3336 |
0.55 | 0.7557 | 0.0754 | 0.3633 | 0.1814 | -0.353 | |
0.8 | 0.6555 | 0.0540 | 0.3546 | 0.1402 | -0.3631 | |
1.2 | 0.5644 | 0.0383 | 0.3465 | 0.0988 | -0.3706 |
m | Par | MLE | Bayes | Eff | |||||||||
Mean | Bias | RMSE | NIT | Mean | Bias | RMSE | NIT | ||||||
75 | α | 0.8997 | 0.1497 | 0.2497 | 11 | 0.5874 | 0.1626 | 0.0398 | 5 | 6.2738 | |||
β | 1.1629 | 0.1629 | 0.2249 | 1.1549 | 0.1549 | 0.0461 | 4.8785 | ||||||
γ | 2.8214 | 0.8214 | 1.2828 | 1.9501 | 0.0499 | 0.0236 | 54.355 | ||||||
θ | 1.7158 | 0.2158 | 0.7419 | 1.3143 | 0.8143 | 0.2897 | 2.5609 | ||||||
100 | α | 0.7990 | 0.0490 | 0.2491 | 10 | 0.7387 | 0.0113 | 0.0094 | 3 | 26.5 | |||
β | 1.1617 | 0.1617 | 0.2118 | 1.0496 | 0.0496 | 0.0284 | 7.4577 | ||||||
γ | 2.9147 | 0.9147 | 1.1374 | 2.0204 | 0.0204 | 0.0172 | 66.127 | ||||||
θ | 1.5592 | 0.0592 | 0.4687 | 1.2876 | 0.7876 | 0.2178 | 3.4063 | ||||||
200 | α | 0.7985 | 0.0485 | 0.2487 | 11 | 0.7459 | 0.0041 | 0.0056 | 6 | 44.410 | |||
β | 1.1332 | 0.1332 | 0.1607 | 1.0271 | 0.0271 | 0.0034 | 47.264 | ||||||
γ | 2.2124 | 0.2124 | 0.8976 | 2.0704 | 0.0704 | 0.0088 | 102 | ||||||
θ | 1.4794 | 0.0206 | 0.3462 | 1.7081 | 0.2081 | 0.1401 | 2.4710 | ||||||
300 | α | 0.7698 | 0.0198 | 0.2398 | 10 | 0.7655 | 0.0155 | 0.0027 | 5 | 88.814 | |||
β | 1.1399 | 0.1399 | 0.1590 | 1.0221 | 0.0221 | 0.0025 | 63.6 | ||||||
γ | 2.0489 | 0.0489 | 0.4968 | 1.9358 | 0.0642 | 0.0079 | 62.886 | ||||||
θ | 1.4971 | 0.0029 | 0.2711 | 1.7054 | 0.2054 | 0.1134 | 2.3906 |
m | Par | MLE | Bayes | Eff | |||||||||
Mean | Bias | RMSE | NIT | Mean | Bias | RMSE | NIT | ||||||
75 | α | 0.9976 | 0.1976 | 0.2981 | 8 | 0.5797 | 0.2203 | 0.0703 | 3 | 4.2403 | |||
β | 1.2336 | 0.1336 | 0.2737 | 1.0340 | 0.0660 | 0.0750 | 3.6493 | ||||||
γ | 2.9946 | 0.6946 | 2.1815 | 2.6245 | 0.3245 | 0.1171 | 18.629 | ||||||
θ | 2.0733 | 0.2733 | 1.3915 | 2.3758 | 0.5758 | 0.6553 | 2.1234 | ||||||
100 | α | 0.8955 | 0.0955 | 0.2669 | 9 | 0.6414 | 0.1586 | 0.0449 | 5 | 5.9443 | |||
β | 1.1852 | 0.0852 | 0.1615 | 0.9583 | 0.1417 | 0.0271 | 5.9594 | ||||||
γ | 2.6917 | 0.3917 | 1.0912 | 2.2717 | 0.0283 | 0.0203 | 53.753 | ||||||
θ | 1.9411 | 0.1411 | 0.7705 | 2.3332 | 0.5332 | 0.5777 | 1.3337 | ||||||
200 | α | 0.8557 | 0.0557 | 0.2581 | 11 | 0.7418 | 0.0582 | 0.0131 | 4 | 19.702 | |||
β | 1.1900 | 0.0900 | 0.1305 | 1.1555 | 0.0555 | 0.0121 | 10.785 | ||||||
γ | 2.6241 | 0.3241 | 1.0744 | 2.4736 | 0.1736 | 0.0201 | 53.452 | ||||||
θ | 1.9438 | 0.1438 | 0.4761 | 2.4321 | 0.6321 | 0.3764 | 1.3337 | ||||||
300 | α | 0.8382 | 0.0382 | 0.2485 | 10 | 0.7586 | 0.0414 | 0.0089 | 5 | 27.921 | |||
β | 1.1816 | 0.0816 | 0.1120 | 1.1336 | 0.0336 | 0.0063 | 17.777 | ||||||
γ | 2.4060 | 0.1060 | 0.6505 | 2.3139 | 0.0161 | 0.0133 | 48.909 | ||||||
θ | 1.9205 | 0.1205 | 0.3338 | 2.1268 | 0.3268 | 0.2595 | 0.1302 |
m | Par | MLE | Bayes | Eff | |||||||||
Mean | Bias | RMSE | NIT | Mean | Bias | RMSE | NIT | ||||||
75 | α | 0.9091 | 0.0091 | 0.2196 | 12 | 0.7811 | 0.1189 | 0.0396 | 7 | 5.5454 | |||
β | 1.2678 | 0.0678 | 0.1892 | 1.1170 | 0.0830 | 0.0208 | 9.0961 | ||||||
γ | 3.7911 | 1.2911 | 3.3738 | 2.6981 | 0.1981 | 0.0485 | 69.562 | ||||||
θ | 2.2781 | 0.2781 | 1.1318 | 2.5410 | 0.5410 | 0.6393 | 1.7703 | ||||||
100 | α | 0.9252 | 0.0252 | 0.2017 | 8 | 0.8040 | 0.0960 | 0.0122 | 3 | 16.532 | |||
β | 1.2727 | 0.0727 | 0.1859 | 1.3039 | 0.1039 | 0.0138 | 13.471 | ||||||
γ | 3.3419 | 0.8419 | 2.4688 | 2.3542 | 0.1458 | 0.0300 | 82.293 | ||||||
θ | 2.2186 | 0.2186 | 1.0014 | 2.2183 | 0.2183 | 0.3614 | 2.7708 | ||||||
200 | α | 0.9825 | 0.0825 | 0.1190 | 9 | 0.8509 | 0.0591 | 0.0108 | 4 | 9.2523 | |||
β | 1.2317 | 0.0317 | 0.1059 | 1.2016 | 0.0016 | 0.0121 | 8.7520 | ||||||
γ | 2.9719 | 0.4719 | 1.8267 | 2.4405 | 0.0595 | 0.0200 | 91.335 | ||||||
θ | 2.1325 | 0.1325 | 0.8462 | 2.2748 | 0.2748 | 0.2861 | 2.9577 | ||||||
300 | α | 0.9944 | 0.0944 | 0.0990 | 12 | 0.9227 | 0.0273 | 0.0107 | 5 | 9.2523 | |||
β | 1.2275 | 0.0275 | 0.0860 | 1.2325 | 0.0325 | 0.0069 | 12.463 | ||||||
γ | 2.8578 | 0.3578 | 1.3202 | 2.5359 | 0.0359 | 0.0120 | 110.01 | ||||||
θ | 2.0783 | 0.0783 | 0.7752 | 2.1816 | 0.1816 | 0.2164 | 3.5822 |
Zone | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 |
Mecca | 3.58 | 4.81 | 5.95 | 6.71 | 7.81 | 8.68 | 8.91 | 9.97 | 10.011 | 9.80 |
Eastern | 2.55 | 3.90 | 4.59 | 6.37 | 7.11 | 7.38 | 7.55 | 7.17 | 7.89 | 8.54 |
Al Madinah | 3.26 | 3.46 | 3.47 | 4.99 | 6.38 | 6.42 | 6.81 | 6.16 | 6.77 | 7.21 |
Asir | 3.41 | 3.81 | 3.98 | 4.65 | 5.47 | 5.74 | 5.92 | 6.17 | 6.13 | 6.53 |
Jizan | 3.42 | 3.39 | 3.62 | 4.46 | 5.37 | 5.71 | 5.56 | 5.49 | 5.64 | 5.80 |
Al-Qassim | 3.43 | 3.45 | 3.37 | 4.11 | 4.46 | 4.81 | 5.10 | 5.07 | 5.24 | 5.45 |
Tabuk | 2.99 | 2.78 | 2.96 | 3.96 | 4.48 | 4.96 | 4.82 | 4.75 | 4.89 | 5.13 |
Ha'il | 2.89 | 2.59 | 2.73 | 3.59 | 4.19 | 4.59 | 4.52 | 4.50 | 4.70 | 4.75 |
Al Jawf | 2.29 | 2.75 | 2.48 | 3.35 | 4.22 | 4.42 | 4.55 | 4.44 | 4.63 | 4.71 |
Najran | 2.83 | 2.92 | 2.62 | 3.33 | 4.02 | 4.38 | 4.47 | 4.44 | 4.61 | 4.8 |
Northern Borders | 1.51 | 1.51 | 1.6 | 2.79 | 3.95 | 4.04 | 3.99 | 4.08 | 4.4 | 4.48 |
Al Bahah | 1.96 | 2.17 | 2 | 2.97 | 3.63 | 4.07 | 3.76 | 3.68 | 3.85 | 4.13 |
Zone | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 |
Mecca | 9.39 | 9.71 | 9.83 | 9.96 | 9.97 | 9.95 | 9.98 | 9.97 | 10.005 | 9.96 |
Eastern | 8.92 | 9.23 | 9.43 | 9.56 | 9.58 | 9.71 | 9.78 | 9.72 | 9.82 | 9.87 |
Al-Madinah | 7.46 | 7.47 | 7.81 | 8.52 | 8.62 | 8.61 | 8.73 | 8.43 | 8.74 | 8.77 |
Asir | 7.49 | 7.77 | 7.93 | 8.34 | 8.29 | 8.32 | 8.42 | 8.36 | 8.47 | 8.51 |
Jizan | 6.66 | 6.69 | 6.84 | 7.64 | 7.71 | 7.75 | 7.75 | 7.68 | 7.82 | 7.82 |
Al-Qassim | 6.6 | 6.62 | 6.67 | 7.47 | 7.51 | 7.53 | 7.67 | 7.6 | 7.73 | 7.73 |
Tabuk | 5.31 | 5.46 | 5.66 | 6.41 | 6.54 | 6.52 | 6.54 | 6.43 | 6.67 | 6.6 |
Ha'il | 4.23 | 4.27 | 4.29 | 5.31 | 5.47 | 5.47 | 5.59 | 5.14 | 5.62 | 5.72 |
Data set | Model | ˆα | ˆβ | ˆδ | ˆθ | KS | KS (P-value) | AIC | BIC | AD |
GKRD | 0.2024 (0.4282) | 2.385 (0.3231) | 10.851 (24.654) | 0.0437 (0.0134) | 0.0663 | 0.6658 | 467.253 | 474.403 | 0.3239 | |
KGAPRD | 1.9052 (3.0495) | 1.5923 (0.4020) | 0.2816 (0.1602) | 0.2534 (1.2086) | 0.0768 | 0.4783 | 468.424 | 479.574 | 0.4321 | |
PGRD | 3.9205 (0.1949) | 0.0434 (0.0246) | 52.541 (2.9280) | 0.0902 | 0.2820 | 469.007 | 477.370 | 0.6984 | ||
I | NGRD | 3.9341 (0.2011) | 2.2256 (0.3212) | 42.308 (10.120) | 0.0937 | 0.2426 | 468.777 | 477.139 | 0.6817 | |
GRD | 2.2389 (0.3197) | 2.7901 (0.1413) | 0.0930 | 0.2501 | 469.633 | 475.208 | 0.6680 | |||
RD | 0.0401 (0.0036) | 0.1758 | 0.0011 | 494.084 | 496.871 | 0.8479 | ||||
GD | 5.1841 (0.6488) | 1.1019 (0.1448) | 0.0932 | 0.2479 | 471.134 | 476.709 | 0.4459 | |||
WD | 2.7961 (0.1863) | 5.2474 (0.1815) | 0.1128 | 0.0941 | 475.208 | 480.783 | 0.7885 | |||
LND | 4.6723 (0.1601) | 1.7548 (0.1132) | 0.1248 | 0.0475 | 479.517 | 485.091 | 0.8003 | |||
GKRD | 0.8638 (1.1574) | 3.7510 (0.5461) | 17.464 (31.538) | 0.0093 (0.0033) | 0.0926 | 0.5853 | 271.904 | 280.898 | 0.9785 | |
KGAPRD | 2.9305 (0.3225) | 74.310 (93.608) | 45.282 (9.2674) | 42.954 (1.4238) | 0.0953 | 0.5477 | 275.190 | 284.184 | 1.1391 | |
PGRD | 4.9536 (0.2978) | 0.1952 (0.8534) | 38.003 (0.0298) | 0.1197 | 0.2684 | 280.548 | 287.293 | 1.6369 | ||
II | NGRD | 4.9852 (0.2595) | 6.9592 (1.5763) | 51.870 (3.6785) | 0.1232 | 0.2385 | 280.780 | 287.525 | 1.6594 | |
GRD | 6.957 (1.5724) | 3.5342 (0.1866) | 0.1236 | 0.2346 | 278.894 | 283.391 | 1.6678 | |||
RD | 0.0156 (1.1864) | 0.3000 | 6.7×10−06 | 342.372 | 344.621 | 1.3876 | ||||
GD | 20.254 (3.3957) | 2.5916 (0.4399) | 0.1195 | 0.2703 | 277.583 | 282.080 | 1.5974 | |||
WD | 5.7702 (0.5667) | 8.4693 (0.1843) | 0.0948 | 0.5551 | 272.030 | 281.527 | 0.9818 | |||
LND | 7.8151 (0.1968) | 1.6465 (0.1391) | 0.0991 | 0.4965 | 272.459 | 281.956 | 1.2196 | |||
GKRD | 0.1193 (0.2206) | 1.7833 (0.3249) | 27.478 (3.1088) | 0.0251 (0.0087) | 0.1124 | 0.5540 | 206.203 | 213.604 | 1.006 | |
KGAPRD | 2.3725 (0.7543) | 1.7809 (0.0039) | 0.2262 (0.0365) | 1.3333 (0.0046) | 0.1365 | 0.3155 | 208.619 | 216.019 | 1.3328 | |
PGRD | 4.8427 (0.4424) | 0.0094 (0.0085) | 159.821 (1.4759) | 0.1439 | 0.2588 | 212.353 | 217.903 | 1.5645 | ||
III | NGRD | 4.8419 (0.4452) | 1.4818 (0.3299) | 32.342 (9.8564) | 0.1455 | 0.2473 | 212.470 | 218.020 | 1.5853 | |
GRD | 1.5009 (0.3300) | 3.4326 (0.3156) | 0.1442 | 0.2562 | 210.189 | 213.889 | 1.5751 | |||
RD | 0.0326 (0.0047) | 0.1935 | 0.0513 | 211.470 | 213.320 | 1.6496 | ||||
GD | 5.1224 (1.0241) | 1.0364 (0.2177) | 0.1184 | 0.4881 | 210.977 | 214.677 | 1.0809 | |||
WD | 2.1042 (0.2110) | 5.6024 (0.4133) | 0.1758 | 0.0964 | 223.267 | 226.967 | 1.7173 | |||
LND | 4.9392 (0.3637) | 2.4936 (0.2571) | 0.1706 | 0.1149 | 213.222 | 216.923 | 1.6214 |