
We present a novel digital twin model that implements advanced artificial intelligence techniques to robustly link news and stock market uncertainty. On the basis of central results in financial economics, our model efficiently identifies, quantifies, and forecasts the uncertainty encapsulated in the news by mirroring the human mind's information processing mechanisms. After obtaining full statistical descriptions of the timeline and contextual patterns of the appearances of specific words, the applied data mining techniques lead to the definition of regions of homogeneous knowledge. The absence of a clear assignment of informative elements to specific knowledge regions is regarded as uncertainty, which is then measured and quantified using Shannon Entropy. As compared with standard models, the empirical analyses demonstrate the effectiveness of this approach in anticipating stock market uncertainty, thus showcasing a meaningful integration of natural language processing, artificial intelligence, and information theory to comprehend the perception of uncertainty encapsulated in the news by market agents and its subsequent impact on stock markets.
Citation: Pedro J. Gutiérrez-Diez, Jorge Alves-Antunes. Stock market uncertainty determination with news headlines: A digital twin approach[J]. AIMS Mathematics, 2024, 9(1): 1683-1717. doi: 10.3934/math.2024083
[1] | Ahmad Mohammed Alghamdi, Sadek Gala, Maria Alessandra Ragusa . A regularity criterion of weak solutions to the 3D Boussinesq equations. AIMS Mathematics, 2017, 2(3): 451-457. doi: 10.3934/Math.2017.2.451 |
[2] | Oussama Melkemi, Mohammed S. Abdo, Wafa Shammakh, Hadeel Z. Alzumi . Yudovich type solution for the two dimensional Euler-Boussinesq system with critical dissipation and general source term. AIMS Mathematics, 2023, 8(8): 18566-18580. doi: 10.3934/math.2023944 |
[3] | Xinli Wang, Haiyang Yu, Tianfeng Wu . Global well-posedness and optimal decay rates for the n-D incompressible Boussinesq equations with fractional dissipation and thermal diffusion. AIMS Mathematics, 2024, 9(12): 34863-34885. doi: 10.3934/math.20241660 |
[4] | Wei Zhang . A priori estimates for the free boundary problem of incompressible inviscid Boussinesq and MHD-Boussinesq equations without heat diffusion. AIMS Mathematics, 2023, 8(3): 6074-6094. doi: 10.3934/math.2023307 |
[5] | Muhammad Shakeel, Amna Mumtaz, Abdul Manan, Marouan Kouki, Nehad Ali Shah . Soliton solutions of the nonlinear dynamics in the Boussinesq equation with bifurcation analysis and chaos. AIMS Mathematics, 2025, 10(5): 10626-10649. doi: 10.3934/math.2025484 |
[6] | Shang Mengmeng . Large time behavior framework for the time-increasing weak solutions of bipolar hydrodynamic model of semiconductors. AIMS Mathematics, 2017, 2(1): 102-110. doi: 10.3934/Math.2017.1.102 |
[7] | Ailing Ban . Asymptotic behavior of non-autonomous stochastic Boussinesq lattice system. AIMS Mathematics, 2025, 10(1): 839-857. doi: 10.3934/math.2025040 |
[8] | Hamood-Ur-Rahman, Muhammad Imran Asjad, Nayab Munawar, Foroud parvaneh, Taseer Muhammad, Ahmed A. Hamoud, Homan Emadifar, Faraidun K. Hamasalh, Hooshmand Azizi, Masoumeh Khademi . Traveling wave solutions to the Boussinesq equation via Sardar sub-equation technique. AIMS Mathematics, 2022, 7(6): 11134-11149. doi: 10.3934/math.2022623 |
[9] | Ruihong Ji, Ling Tian . Stability of the 3D incompressible MHD equations with horizontal dissipation in periodic domain. AIMS Mathematics, 2021, 6(11): 11837-11849. doi: 10.3934/math.2021687 |
[10] | Ali Althobaiti . Novel wave solutions for the sixth-order Boussinesq equation arising in nonlinear lattice dynamics. AIMS Mathematics, 2024, 9(11): 30972-30988. doi: 10.3934/math.20241494 |
We present a novel digital twin model that implements advanced artificial intelligence techniques to robustly link news and stock market uncertainty. On the basis of central results in financial economics, our model efficiently identifies, quantifies, and forecasts the uncertainty encapsulated in the news by mirroring the human mind's information processing mechanisms. After obtaining full statistical descriptions of the timeline and contextual patterns of the appearances of specific words, the applied data mining techniques lead to the definition of regions of homogeneous knowledge. The absence of a clear assignment of informative elements to specific knowledge regions is regarded as uncertainty, which is then measured and quantified using Shannon Entropy. As compared with standard models, the empirical analyses demonstrate the effectiveness of this approach in anticipating stock market uncertainty, thus showcasing a meaningful integration of natural language processing, artificial intelligence, and information theory to comprehend the perception of uncertainty encapsulated in the news by market agents and its subsequent impact on stock markets.
Probability distributions are very important in modeling and fitting random phenomena in all areas of life. In the literature on distribution theory, there are various probability distributions for analyzing and predicting multiple kinds of data in many sectors, including life, biology, medical science, insurance, finance, engineering, and industry [1,2,3,4,5,6,7,8,9]. Based on existing findings, industrial data often exhibits a thick right tail, and many authors have developed several well-known right-skewed families. Afify et al. [10] defined the power-modified Kies-exponential distribution. Coşkun et al. [11] introduced the modified-Lindley distribution, and Gómez et al. [12] proposed the power piecewise exponential model. In addition, Dhungana and Kumar [13] proposed an exponentiated odd Lomax exponential distribution, while Hassan et al. [14] introduced the alpha power transformed extended exponential distribution. In the same line, Karakaya et al. [15], presented a unit-Lindley distribution, and Tung et al. [16] developed the Arcsine-X family of distributions.
To bring further flexibility to these generated distributions, various approaches of well-known models have been defined and used in several applied sciences to allow the smoothing parameter to vary across different locations in the data space. One of the new model-generating techniques is the error function (EF) transformation, which was first proposed by Fernández and De Andrade [17]. The cumulative distribution function (CDF) and the corresponding probability distribution function (PDF) of the EF transformation are as follows:
Δ(y)=erf(H(y)1−H(y)),y∈R, | (1.1) |
and
δ(y)=2h(y)√π(1−H(y))2 exp{−(H(y)1−H(y))2}. | (1.2) |
The EF transformation is a novel method for generalizing a given model, which transforms a distribution without adding any parameters. It is a modified version of traditional probability distributions for the relative importance or worth of data points. This strategy improves flexibility, allowing analysts to better explain real-world scenarios in which traditional random sampling fails to capture the underlying data structure. The derivation of the new attractive EF transformation to modify the existing distribution helps the fitting power of the existing distributions. The proposed method has many applications that extend to fitting, especially in industrial domains. However, recent works considering the EF technique, such as [18,19].
The inverse Weibull (IW) distribution is widely used in reliability and lifetime modeling for mortality rates, especially when studying extreme events. Since it captures tail behavior effectively, it is effective in understanding the upper quantiles of life expectancy or survival time. The CDF of the IW distribution, denoted as G(x), is defined as follows:
G(x)=e−θx−β,; x, θ,β>0. | (1.3) |
In reference to G(x) as stated in Eq (1.3), the PDF g(x) is formulated as:
g(x)=θβ x−(β+1) e−θx−β. | (1.4) |
The IW model has undoubtedly established itself as a crucial tool for data modeling across nearly all sectors. However, despite its widespread use and advantages, the IW distribution is constrained by its inherent limitations. One of the primary constraints of the IW distribution is its capacity to represent solely monotonic forms of hazard functions, as it can only model situations where the hazard rate increases or decreases consistently over time. More papers have used the IW model for many different statistical models, such as the following : Alzeley et al. [20] discussed statistical inference under censored data for IW model, Hussam et al. [21] discussed fuzzy vs. traditional reliability models, Ahmad et al. [22] derived the new cotangent IW model, Mohamed et al. [23] discussed Bayesian and E-Bayesian estimation for an odd generalized exponential IW model. Abdelall et al. [24] introduced a new extension of the odd IW model. Al Mutairi et al. [25] obtained Bayesian and non-Bayesian inference based on a jointly type-Ⅱ hybrid censoring model. Hassan et al. [26] discussed the statistical analysis of IW based on step-stress partially accelerated life testing. Alsadat et al. [27] presented novel Kumaraswamy power IW distribution with data analysis related to diverse scientific areas.
In this paper, we focus on providing a new form of the IW distribution for analyzing the datasets of different areas and highlighting specific characteristics. We extend this distribution by using the approach discussed in equation (1.1), and the resultant distribution is named the error function inverse Weibull (EF-IW) model. This heightened flexibility allows for a better fit to datasets with diverse kurtotic characteristics, enhancing the model's applicability across various scenarios. Further, the key objectives of the current study are as follows.
(1) The primary objective was extending the EF-IW distribution using the error function method, allowing for the derivation and investigation of its essential mathematical characteristics.
(2) The second main goal was to estimate the models' parameters using two different estimation methods, such as the maximum likelihood estimator (MLE) and Bayesian estimator, under different loss functions via Metropolis-Hastings (MH) algorithms. We conduct a detailed simulation study to demonstrate the behavior of derived estimators and pinpoint the most efficient estimation method.
(3) Two data sets from the industry field are utilized to illustrate the applicability and utilization of the proposed distribution.
The following is the organization of the study. Section 2 introduces the model description and the extension distribution, while Section 3 discusses various statistical properties such as moments, quantiles, and moment-generating functions. In Section 4, parameters are estimated using two different estimation methods. The performance of the EF-IW distribution using simulation is carried out and illustrated using three real industrial data sets in Sections 5 and 6, respectively. Finally, Section 7 presents the concluding remark of the paper.
Here, we provide the inverse Weibull distribution as a classical distribution. Plugging Eqs (1.3) and (1.4) into Eqs (1.1) and (1.2) gives the CDF and PDF of the new EF-IW model:
Ξ(z)=erf(e−θz−β1−e−θz−β),z,θ,β>0, | (2.1) |
and
ξ(z)=2θβ z−(β+1) e−θz−β√π(1−e−θz−β)2 exp{−(e−θz−β1−e−θz−β)2}, | (2.2) |
where erf(x)=2√π∫x0e−z2dz. The plots of the EF-IW PDF for some parameter values given in Figure 1 reveal that this function can be decreasing, unimodal, and skewed depending on the parameter values.
Suppose the random variable Z has a CDF denoted by Ξ(z). Then, its survival function (SF) and hazard rate function (HRF) can then be expressed as
S(z)=1−erf(e−θz−β1−e−θz−β), | (2.3) |
and
h(z)=2θβ z−(β+1) e−θz−β√π(1−e−θz−β)2[1−erf(t)] exp{−t}, | (2.4) |
with t=(e−θz−β1−e−θz−β)2.
Next, the cumulative hazard rate function (CHRF) and reversed hazard rate function (RHRF) of the random variable Z can be expressed as
H(z)=−log[1−erf(e−θz−β1−e−θz−β)], | (2.5) |
and
R(z)=2θβ z−(β+1) e−θz−β√π(1−e−θz−β)2 erf(e−θz−β1−e−θz−β). | (2.6) |
Figure 2 shows HRF plots of EF-IW for different sets of parameter values. It has increasing, unimodal, and decreasing shapes.
The quantile function Ξ−1(u) holds significant importance in simulation studies across various disciplines due to its ability to generate random variables with desired distribution characteristics.The quantile function of the new EF-IW model can be expressed as
Ξ−1(u)=[−1θlog(erf−1(u)1+erf−1(u))]−1/β,0≤u≤1, | (3.1) |
where erf−1(x)=Φ−1(x) is the standard normal quantile function.
Proof. By setting the Eq (2.1) equal u, we get
erf(e−θz−β1−e−θz−β)=u,e−θz−β1−e−θz−β=erf−1(u),e−θz−β(1+erf−1(u))=erf−1(u),e−θz−β=erf−1(u)1+erf−1(u),θz−β=−log(erf−1(u)1+erf−1(u)),z=[−1θlog(erf−1(u)1+erf−1(u))]−1/β. |
The quantile function can be used to compute the first, second, and third quantiles by replacing u with 14, 12, and 34.
Additionally, the Bowleys skewness (N) and Moors kurtosis (M) of the EF-IW model are described as
N=Ξ−1(1/4)+Ξ−1(3/4)−2Ξ−1(1/2)Ξ−1(3/4)−Ξ−1(1/4), |
and
M=Ξ−1(7/8)−Ξ−1(5/8)+Ξ−1(3/8)−Ξ−1(1/8)Ξ−1(6/8)−Ξ−1(2/8). |
In this part, we provide a series representation of the EF-IW CDF and PDF by employing the erf series, see Fernández and De Andrade [17] and Ajongba et al. [18],
erf(t)=2√(π)∞∑l=0(−1)lt2l+1l!(2l+1), |
and by applying the expansion
t1−t=∞∑j=0tj, |t|<1, |
the corresponding CDF of the EF-IW distribution can be rewritten as:
Ξ(z)=2√(π)∞∑l=0(−1)ll!(2l+1) [∞∑j=0 e−θz−β]2l+1. |
Now, consider the series expansion
[∞∑j=0ajtj]k=∞∑n=0Dk,n tn, |
where Dk,0=ak0 and Dk,n=1n a0n∑s=1 (sk−n+s) as Dk,n−s, n≥1.
Consequently, the EF-IW CDF takes the expression
Ξ(z)=2√π∞∑l=0∞∑n=0(−1)lD2l+1,nl!(2l+1) e−θ(n+2l+1)z−β=∞∑l=0∞∑n=0Cl,n e−θ(n+2l+1)z−β, |
with Cl,n=2(−1)lD2l+1,n√πl!(2l+1), D2l+1,n=1nn∑s=1[2s(l+1)−n] D2l+1,n−s and D2l+1,0=1.
Similarly, the density of the recommended EF-IW model becomes
ξ(z)=θβ∞∑l=0∞∑n=0 Hl,n z−β−1 e−θ(n+2l)z−β, |
with Hl,n=Cl,n(n+2l+1).
One of the efficient statistical criteria that can calculate symmetry, spread-ness, and asymmetry is the ordinary moment. The r-th moment of the EF-IW distribution, whose PDF is given in Eq (2.2), can be determined as follows:
μ′r=θ∞∑l=0∞∑n=0Hl,n Γ(1−rβ)[θ(2l+n)]1−rβ, | (3.2) |
where Γ(.) represents the gamma function.
Thus, for r=1 and r=2, the mean (μ′1) and second moment (μ′2) of the EF-IW distribution are defined, respectively, as
μ′1=θ∞∑l=0∞∑n=0Hl,n Γ(1−1β)[θ(2l+n)]1−1β, |
and
μ′2=θ∞∑l=0∞∑n=0Hl,n Γ(1−2β)[θ(2l+n)]1−2β. |
The variance (Varz) with a corresponding coefficient of variation (CV) for the EF-IW model are obtained to be
VarZ=μ′2−μ′21, |
and
CV=VarZμ′1. |
Table 1 defined various proposed mathematical characteristics of the suggested EF-IW. In addition, Figure 3 shows the 3D plots of these statistical properties.
β | μ1 | VarZ | CV | N | M | |
θ=0.4 | 0.3 | 0.0874 | 0.0234 | 1.7508 | 4.3227 | 30.106 |
0.6 | 0.2287 | 0.0351 | 0.8190 | 1.5817 | 3.3285 | |
0.9 | 0.3488 | 0.0369 | 0.5506 | 0.8949 | 0.7335 | |
1.2 | 0.4412 | 0.034 | 0.4179 | 0.5665 | 0.0143 | |
θ=0.6 | 0.3 | 0.3375 | 0.3492 | 1.7508 | 4.3227 | 30.106 |
0.6 | 0.4495 | 0.1355 | 0.8190 | 1.5817 | 3.3285 | |
0.9 | 0.5473 | 0.0908 | 0.5506 | 0.8949 | 0.7335 | |
1.2 | 0.6186 | 0.0668 | 0.4179 | 0.5665 | 0.0143 | |
θ=0.8 | 0.3 | 0.8806 | 2.3770 | 1.7508 | 4.3227 | 30.106 |
0.6 | 0.7260 | 0.3535 | 0.8190 | 1.5817 | 3.3285 | |
0.9 | 0.7534 | 0.1721 | 0.5506 | 0.8949 | 0.7335 | |
1.2 | 0.7862 | 0.1080 | 0.4179 | 0.5665 | 0.0143 | |
θ=1.2 | 0.3 | 3.4022 | 35.479 | 1.7508 | 4.3227 | 30.106 |
0.6 | 1.4270 | 1.3659 | 0.8190 | 1.5817 | 3.3285 | |
0.9 | 1.1822 | 0.4238 | 0.5506 | 0.8949 | 0.7335 | |
1.2 | 1.1022 | 0.2122 | 0.4179 | 0.5665 | 0.0143 | |
θ=1.5 | 0.3 | 7.1581 | 157.05 | 1.7508 | 4.3227 | 30.106 |
0.6 | 2.0699 | 2.8737 | 0.8190 | 1.5817 | 3.3285 | |
0.9 | 1.5149 | 0.6958 | 0.5506 | 0.8949 | 0.7335 | |
1.2 | 1.3274 | 0.3078 | 0.4179 | 0.5665 | 0.0143 |
The moment generating function (MGF), M(t) of the KMIW model is derived as
M(t)=θ∞∑l=0∞∑n=0∞∑r=0Hl,n trr! Γ(1−rβ)[θ(2l+n)]1−rβ. |
The PDF of the rth-order statistics for a sample of size m taken from the EF-IW model is expressed as follows:
k(r)(z)=m!ξ(z)(r−1)!(m−r)![Ξ(z)]r−1[1−Ξ(z)]m−r=m!(r−1)!(m−r)! θβ∞∑l=0∞∑n=0 Hl,n z−β−1 e−θ(n+2l)z−β [∞∑l=0∞∑n=0Cl,n e−θ(n+2l+1)z−β]r−1×[1−∞∑l=0∞∑n=0Cl,n e−θ(n+2l+1)z−β]m−r. |
In a special case, the PDF of the minimum 1th and maximum mth order statistics of the EF-IW distribution can be given below as
k(1)(z)=mθβ∞∑l=0∞∑n=0 Hl,n z−β−1 e−θ(n+2l)z−β [1−∞∑l=0∞∑n=0Cl,n e−θ(n+2l+1)z−β]1−r, |
and
k(m)(z)=mθβ∞∑l=0∞∑n=0 Hl,n z−β−1 e−θ(n+2l)z−β [∞∑l=0∞∑n=0Cl,n e−θ(n+2l+1)z−β]m−1. |
The corresponding CDF of the EF-IW model can be written as
K(r)(z)=m∑k=0Ξk(z)[1−Ξ(z)]m−k=m∑k=0[∞∑l=0∞∑n=0Cl,n e−θ(n+2l+1)z−β]k [1−∞∑l=0∞∑n=0Cl,n e−θ(n+2l+1)z−β]m−k. |
In this part of the study, we estimate the models' parameters η=(β,θ) using two different estimation methods. For this purpose, the maximum likelihood and Bayesian estimators are the estimation methods used.
Assuming {z1,z2,…,zm} are the observed values of a random sample {Z1,Z2,…,Zm} from the EF-IW distribution with vector of parameters η=(β,θ), the log-likelihood function can be obtained to be
LL(z)=m∑i=1logξ(z)=m∑i=1log(2θβ z−(β+1) e−θz−β√π(1−e−θz−β)2 exp{−(e−θz−β1−e−θz−β)2})∝mlogθ+mlogβ−2m∑i=1log(1−e−θz−βi)−θm∑i=1z−βi−(β+1)m∑i=1logzi−m∑i=1(e−θz−βi1−e−θz−βi)2. | (4.1) |
With the vector of the parameters η=(β,θ), the corresponding partial derivatives of Eq (4.1) are obtained as:
∂LL(z;ϑ)∂θ=mθ−2m∑i=1z−βie−θz−βi1−e−θz−βi−m∑i=1z−βi+2[m∑i=1z−βi e−3θz−βi(1−e−θz−βi)3+m∑i=1z−βi e−2θz−βi(1−e−θz−βi)2], | (4.2) |
and
∂LL(z;ϑ)∂β=mβ−m∑i=1logz−βi−θm∑i=1z−βilogzi+2θ[m∑i=1z−βilogzi e−3θz−βi(1−e−θz−βi)3+m∑i=1z−βilogzi e−2θz−βi(1−e−θz−βi)2]. | (4.3) |
The parameter estimates for the parameters η=(β,θ) can be obtained by solving the above non-linear equations with respect to the parameters. It might be difficult to obtain a precise solution to the derived equations, and thus one option to optimize them is to use techniques like the Newton-Raphson algorithm. We used the R software's optimize function in this case.
We proceed based on the information available on the unknown parameters obtained from the opinions of the researchers. The interpretation of the informative prior is rarely precise enough to determine a single prior distribution. However, there are laws calibrated according to the distribution of observations, called the conjugate prior or the gamma prior. For more details, see Xu [28] and Zhuang [29]. Assuming that the unknown parameters β and θ are random variables that have a Gamma distribution with PDF expressed as
π1(θ)=ba11Γ(a1) θa1−1 e−b1θ, θ,a1,b1>0, |
and
π1(β)=ba22Γ(a2s) βa2−1 e−b2β, β,a2,b2>0. |
Henceforth, the joint prior PDF of η=(β,θ) can be derived as
π(ϑ)∝θa1−1 βa2−1 e−b1θ−b2β. |
Next, the joint posterior PDF of η=(β,θ) is
π∗(ϑ∣z)=L(ϑ))π(ϑ)∣z)∝θm+a1−1 βm+a2−1 eb1θ−b2βm∏i=1=z−(β+1)i e−θz−βi(1−e−θz−βi)2 exp{−(e−θz−βi1−e−θz−βi)2}. |
The Bayes estimates of the parametric function η=(β,θ) under the assumption of the square error loss function (BSE) is the posterior mean of η. The BSE is
ˆfSE=∫ηf π∗(ϑ∣z)dη. | (4.4) |
Now, the Bayes estimator under linear exponential loss function (BLI), can be written f=eδ(η−ˆη)−δ(η−ˆη). The BLI is
ˆfLI=−1δlog(∫ηe−δf π∗(η∣z)dη). | (4.5) |
In the end, the Bayes estimator under general entropy loss function (BGE), defined as f=(ˆηη)δ−δlog(ˆηη)−1, is
ˆfGE=(∫ηf−δ π∗(η∣z)dη)−1/δ, | (4.6) |
with δ≠0. It is difficult to obtain analytical expressions of Eqs (4.4)–(4.6). To solve this issue, we have considered the Metropolis Hasting (MH) algorithm for this purpose.
In this section, a detailed simulation study is carried out to examine the behavior of two derived estimators using the R software to evaluate the efficiency of the recommended estimators. The results are presented for various sample sizes m={30,60,80,100} from the proposed EF-IW distribution and several parameter values of η=(β,θ) (Set 1: (0.5, 0.75), Set 2: (0.8, 1.25), and Set 3: (1.2, 1.5)) to provide more accurate and comprehensive results. The Monte Carlo simulations are repeated 1000 times, and the estimates are assessed based on the mean estimate (AEs) and mean squared errors (MSEs). The empirical results are illustrated in Tables (2)–(4), and in this simulation, we choose δ=1.5 to compute the BLI and BGE. To check that the iterative non-linear method converges to the MLEs, we have applied the Newton Raphson technique with some other initial estimates, and it converges to the same set of estimates, which ensures that the estimates obtained via the suggested Newton Raphson method converges to the MLEs. The following conclusions are drawn from these tables.
m | MLE | BSE | BLI | BGE | |||||
Mean | MSE | Mean | MSE | Mean | MSE | Mean | MSE | ||
30 | θ | 0.4919 | 0.0039 | 0.4502 | 0.0035 | 0.4505 | 0.0037 | 0.4496 | 0.0039 |
β | 0.7802 | 0.0115 | 0.7496 | 0.0008 | 0.7497 | 0.0101 | 0.7494 | 0.0103 | |
60 | θ | 0.4928 | 0.0018 | 0.5093 | 0.0010 | 0.5093 | 0.0013 | 0.5085 | 0.0015 |
β | 0.7681 | 0.0051 | 0.7395 | 0.0007 | 0.7396 | 0.0009 | 0.7393 | 0.0101 | |
80 | θ | 0.5007 | 0.0011 | 0.4888 | 0.0006 | 0.4888 | 0.0008 | 0.4886 | 0.0010 |
β | 0.7586 | 0.0034 | 0.7882 | 0.0005 | 0.7883 | 0.0008 | 0.7880 | 0.0009 | |
100 | θ | 0.4943 | 0.0010 | 0.5091 | 0.0004 | 0.5093 | 0.0005 | 0.5087 | 0.0008 |
β | 0.7591 | 0.0024 | 0.7577 | 0.0004 | 0.7581 | 0.0006 | 0.7573 | 0.0008 |
m | MLE | BSE | BLI | BGE | |||||
Mean | MSE | Mean | MSE | Mean | MSE | Mean | MSE | ||
30 | θ | 0.7998 | 0.0055 | 0.8643 | 0.0039 | 0.8645 | 0.0041 | 0.4816 | 0.0043 |
β | 1.2981 | 0.0309 | 1.3373 | 0.0121 | 1.3384 | 0.0123 | 1.3364 | 0.0125 | |
60 | θ | 0.8001 | 0.0024 | 0.8358 | 0.0015 | 0.8359 | 0.0017 | 0.8357 | 0.0019 |
β | 1.2646 | 0.0148 | 1.1742 | 0.0075 | 1.1746 | 0.0078 | 1.1738 | 0.0079 | |
80 | θ | 0.7993 | 0.0023 | 0.7856 | 0.0014 | 0.7859 | 0.0016 | 0.7852 | 0.0018 |
β | 1.2641 | 0.0075 | 1.2241 | 0.0017 | 1.2244 | 0.0020 | 1.2239 | 0.0021 | |
100 | θ | 0.7948 | 0.0014 | 0.8046 | 0.0007 | 0.8048 | 0.0009 | 0.8044 | 0.0011 |
β | 1.2673 | 0.0072 | 1.2320 | 0.0011 | 1.2322 | 0.0013 | 1.2319 | 0.0015 |
m | MLE | BSE | BLI | BGE | ||||||
Mean | MSE | Mean | MSE | Mean | MSE | Mean | MSE | |||
30 | θ | 1.2240 | 0.0188 | 1.2206 | 0.0051 | 1.2217 | 0.0053 | 1.2196 | 0.0055 | |
β | 1.5799 | 0.0559 | 1.6200 | 0.0245 | 1.6225 | 0.0248 | 1.6184 | 0.0249 | ||
60 | θ | 1.2049 | 0.0065 | 1.2310 | 0.0032 | 1.2316 | 0.0034 | 1.2306 | 0.0036 | |
β | 1.5556 | 0.0215 | 1.5530 | 0.0058 | 1.5538 | 0.0061 | 1.5526 | 0.0062 | ||
80 | θ | 1.2035 | 0.0040 | 1.2276 | 0.0023 | 1.2280 | 0.0025 | 1.2272 | 0.0028 | |
β | 1.5524 | 0.0159 | 1.4679 | 0.0032 | 1.4685 | 0.0034 | 1.4676 | 0.0035 | ||
100 | θ | 1.2139 | 0.0041 | 1.2160 | 0.0019 | 1.2165 | 0.0022 | 1.2157 | 0.0024 | |
β | 1.5158 | 0.0113 | 1.4818 | 0.0022 | 1.4822 | 0.0023 | 1.4815 | 0.0025 |
(1) All estimation approaches produce estimates that converge toward the true parameter values as the sample size increases, which confirm that they are consistent and asymptotically unbiased.
(2) In most cases, the value of MSEs decreases as the value of m increases.
(3) As m increases, the Bayes estimates tends to perform efficiently based on MSE as an optimal criterion. On the contrary, BSE is more appropriate than BLI and BGE.
(4) Figure (4) ensures the same conclusion.
In this section, we utilized two data sets from the industrial field to show the EF-IW model introduced in Section 2. We demonstrate the flexibility of this new distribution by analyzing two real-world datasets drawn from industrial areas in the Kingdom of Saudi Arabia (KSA).
The data set represents the quarterly evolution of the number of foreign licenses in the construction sector in KSA. It was obtained from https://datasaudi.sa/en/sector/construction#real-sector-indicators. The values of the data set are summarized in Table (5).
8 | 6 | 8 | 16 | 23 | 20 | 28 | 40 |
43 | 50 | 54 | 32 | 52 | 29 | 33 | 42 |
41 | 52 | 56 | 79 | 155 | 84 | 95 | 111 |
136 | 161 | 204 | 241 |
The second application introduced the scale efficiency of the construction industry in KSA between 2013 and 2022. The suggested data set was considered by Yu et al. [30], and the values are presented in Table (6).
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 |
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 |
This recommended data set is about the efficiency of the pure technical construction industry between 2013 and 2022 in the KSA. The proposed data was considered by Yu et al. [30], and its records can be reported in Table 7.
Zone | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 |
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 |
Table (8) presents a statistical summary of the three data sets. Furthermore, Figure 5 shows several significant plots (scaled total time on test (TTT), quantile-quantile (Q-Q), and box plots) derived from the three industrial datasets. These plots help analyze the historical performance of the industrial sectors.
Data | Q1 | Q2 | μ′1 | Q3 | CV | N | M |
1 | 28.75 | 46.50 | 67.82 | 27.60 | 54.57 | 1.31 | 0.83 |
2 | 6.55 | 7.72 | 7.67 | 9.35 | 0.35 | -0.18 | -0.96 |
3 | 3.445 | 4.475 | 4.521 | 5.39 | 0.35 | 0.3589 | -0.0846 |
Additionally, we would like to select the more appropriate fitting model for the two proposed data sets. We consider several renowned competitive probability distributions to compare with the results of EF-IW, including the inverse Weibull (IW), error function Weibull (EF-W), error function exponential (EF-E), power Burr X (PBX), and generalized exponential (GE) models.
Akaike information criterion (A), Bayesian information criterion (B), Hannan-Quin information criterion (C), correction Akaike information criterion (D), Kolmogorov-Smirnov (KS) statistics with its associated P-values are considered when comparing the model and recommending the best model. By calculating and comparing the proposed measures, we gain a clear understanding of the relative performance of each model. Models with the lowest values for these statistics will be considered for the best fit of the given data set. This approach reflects the strengths of the new distribution in terms of its suitability for different data structures and ensures that the model selection process takes into account both the complexity of the model and the goodness of fit across multiple aspects of the data distribution. Table (9) summarizes the final estimates of the unknown parameters with their corresponding log-likelihood (LL). Consequently, the recommended EF-IW model emerges as the most favorable distribution for modeling the three data sets. Henceforth, the empirical v.s. the fitted (PDF and CDF) plots for the proposed model with its competitors are generated and reported in Figures (6)–(8) using the two data sets. These visual plots demonstrate that the EF-IW distribution works well with the three data sets.
Data | Model | ˆθ | ˆβ | KS | P-value | LL | A | B | C | D |
EF-IW | 10.609 | 0.5686 | 0.1425 | 0.6197 | -144.601 | 293.202 | 295.866 | 294.016 | 293.682 | |
IW | 8.5601 | 0.6934 | 0.2135 | 0.1557 | -152.624 | 309.269 | 311.934 | 300.958 | 300.624 | |
1 | EF-W | 224.821 | 0.7022 | 0.1961 | 0.2316 | -147.318 | 298.637 | 301.301 | 299.451 | 299.117 |
EF-E | 0.0047 | 0.3475 | 0.0023 | -149.955 | 301.911 | 303.243 | 302.318 | 302.064 | ||
PBX | 0.0771 | 0.5983 | 0.1575 | 0.4901 | -145.356 | 294.713 | 297.378 | 295.603 | 295.269 | |
GE | 0.0188 | 1.4887 | 0.1495 | 0.5582 | -145.041 | 294.082 | 296.746 | 294.897 | 294.562 | |
EF-IW | 169.011 | 2.4633 | 0.1114 | 0.3500 | -132.473 | 268.947 | 273.444 | 270.734 | 269.126 | |
IW | 2168.36 | 4.0579 | 0.1697 | 0.0354 | -146.885 | 297.771 | 302.268 | 299.557 | 297.950 | |
2 | EF-W | 10.415 | 3.6140 | 0.1468 | 0.0978 | -132.736 | 269.472 | 273.969 | 271.259 | 269.651 |
EF-E | 0.0709 | 0.4251 | <0.0001 | -179.247 | 360.494 | 362.743 | 361.387 | 360.553 | ||
PBX | 0.0407 | 1.5378 | 0.2102 | 0.0041 | -146.950 | 297.936 | 302.433 | 299.723 | 298.115 | |
GE | 0.6226 | 69.883 | 0.1399 | 0.1290 | -139.298 | 282.596 | 287.093 | 284.382 | 282.775 | |
EF-IW | 11.193 | 1.5481 | 0.0756 | 0.6168 | -177.432 | 358.864 | 364.074 | 360.973 | 358.988 | |
IW | 25.075 | 2.5284 | 0.1395 | 0.0408 | -198.317 | 400.634 | 405.845 | 402.743 | 400.758 | |
3 | EF-W | 1.9167 | 7.5766 | 0.1301 | 0.0676 | -185.737 | 375.475 | 380.686 | 377.584 | 375.599 |
EF-E | 0.1131 | 0.3139 | <0.0001 | -210.347 | 422.694 | 425.300 | 423.749 | 422.735 | ||
PBX | 0.0639 | 1.7021 | 0.0909 | 0.3791 | -177.673 | 359.347 | 364.557 | 361.455 | 359.470 | |
GE | 0.7435 | 16.358 | 0.1060 | 0.2107 | -180.239 | 364.478 | 69.688 | 366.587 | 364.602 |
Finally, the estimates of the model parameters using the Bayesian technique under several loss functions of the EF-IW distribution by applying the three data sets are computed and reported in Table (10). Also, Figures (9)–(11) show the histogram and trace plots of MH results.
Data | Par | Bayes | ||
BSE | BLI | BGE | ||
1 | θ | 10.468 | 10.470 | 10.468 |
β | 0.5765 | 0.5766 | 0.5764 | |
2 | θ | 168.989 | 168.990 | 168.989 |
β | 2.457 | 2.457 | 2.457 | |
3 | θ | 11.692 | 11.689 | 11.694 |
β | 1.5635 | 1.5637 | 1.5634 |
This study introduces a new probability distribution, and its mathematical properties are thoroughly explored. The new model is named the error function inverse Weibull distribution. The model parameters are estimated using two different estimation methods, and extensive simulation studies are conducted to identify the most efficient estimation technique. To demonstrate the versatility and practical usefulness of the EF-IW distribution, the new distribution is applied to three datasets, demonstrating its ability to adapt to varied data properties. The findings of these applications show that the EF-IW distribution surpasses considered competitive probability distributions previously studied in the literature, giving more accurate and efficient outcomes in terms of fit and prediction. These findings show the novel distribution's potential as a robust tool for modeling data across several domains, providing a promising alternative to established models.
Future work on the EF-IW distribution may include expanding modifications, estimation, and applications. Some potential directions include the following
(1) New extended forms of the EF-IW distribution can be proposed, such as truncation, zero-inflation, and Neutrosophic extension for imprecise datasets.
(2) The progressive censoring type may also be used to obtain the model parameter estimations.
(3) Future studies should focus on the utilization of the EF-IW distribution to handle ranked set sampling data, which is frequently seen in survival and reliability analysis studies. Enhancing the distribution applicability and usefulness will require developing parameter estimation approaches for censored and uncensored data with a cure fraction.
All authors contributed equally to this paper. Badr Aloraini and Abdulaziz S. Alghamdi did the writing and mathematics, Mohammad Zaid Alaskar and Maryam Ibrahim Habadi did the revising, editing, and validating.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
This study is supported via funding from Prince Sattam bin Abdulaziz University project number (PSAU/2025/R/1446).
All authors declare no conflicts of interest in this paper.
[1] | G. Cardano, Liber de ludo aleae, In: C. Sponius (ed.), Hieronymi Cardani Mediolanensis Opera Omnia, Lyons, 1663, 1564. |
[2] | B. Pascal, P. Fermat, Letters, In: Pascal Fermat Correspondence, 1654. Available from: http://www.york.ac.uk/depts/maths/histstat/pascal.pdf. |
[3] | J. Bernoulli, The art of conjecturing, together with letter to a friend on sets in court tennis, English translation by Edith Sylla, Baltimore: Johns Hopkins Univ Press, 2005, 1713. https://doi.org/10.1111/j.1600-0498.2008.00117.x |
[4] | F. P. Ramsey, Truth and probability, In: The Foundations of Mathematics and other Logical Essays, ed. R. B. Braithwaite, London: Routledge & Kegan Paul Ltd, 1926. https://doi.org/10.1007/978-3-319-20451-2_3 |
[5] | L. J. Savage, The foundations of statistics, New York: John Wiley & Sons, 1954. https://doi.org/10.1002/nav.3800010316 |
[6] | J. M. Keynes, A treatise on probability, Macmillan & Co., 1921. https://doi.org/10.2307/2178916 |
[7] | F. H. Knight, Risk, uncertainty and profit, Chicago University Press, 31 (1921). https://doi.org/10.1017/CBO9780511817410.005 |
[8] |
M. Kurz, M. Motolese, Endogenous uncertainty and market volatility, Econ. Theory, 17 (2001), 497–544. http://dx.doi.org/10.2139/ssrn.159608 doi: 10.2139/ssrn.159608
![]() |
[9] |
M. B. Beck, Water quality modeling: A review of the analysis of uncertainty, Water Resour. Res., 23 (1987), 1393–1442. https://doi.org/10.1029/WR023i008p01393 doi: 10.1029/WR023i008p01393
![]() |
[10] | S. O. Funtowicz, J. R. Ravetz, Uncertainty and quality in science for policy, Springer Science & Business Media, 1990. http://dx.doi.org/10.1007/978-94-009-0621-1 |
[11] |
M. B. A. van Asselt, J. Rotmans, Uncertainty in integrated assessment modelling, Climatic Change, 54 (2002), 75–105. https://doi.org/10.1023/A:1015783803445 doi: 10.1023/A:1015783803445
![]() |
[12] |
E. F. Fama, The behavior of stock-market prices, J. Bus., 38 (1965), 34–105. http://dx.doi.org/10.1086/294743 doi: 10.1086/294743
![]() |
[13] |
A. Alchian, Uncertainty, evolution and economic theory, J. Polit. Econ., 58 (1950), 211–221. http://dx.doi.org/10.1086/256940 doi: 10.1086/256940
![]() |
[14] |
A. Sandroni, Do Markets favor agents able to make accurate predictions? Econometrica, 68 (2000), 1303–1341. http://dx.doi.org/10.1111/1468-0262.00163 doi: 10.1111/1468-0262.00163
![]() |
[15] |
A. Sandroni, Efficient markets and Bayes' rule, Econ. Theory, 26 (2005) 741–764. http://dx.doi.org/10.1007/s00199-004-0567-4 doi: 10.1007/s00199-004-0567-4
![]() |
[16] |
L. Blume, D. Easley, Evolution and market behavior, J. Econ. Theory, 58 (1992), 9–40. http://dx.doi.org/10.1016/0022-0531(92)90099-4 doi: 10.1016/0022-0531(92)90099-4
![]() |
[17] |
L. Blume, D. Easley, If you're so smart, why aren't you rich? Belief selection in complete and incomplete markets, Econometrica, 74 (2006), 929–966. http://dx.doi.org/10.1111/j.1468-0262.2006.00691.x doi: 10.1111/j.1468-0262.2006.00691.x
![]() |
[18] |
O. San, The digital twin revolution, Nat. Comput. Sci., 1 (2021), 307–308. https://doi.org/10.1038/s43588-021-00077-0 doi: 10.1038/s43588-021-00077-0
![]() |
[19] |
G. Caldarelli, E. Arcaute, M. Barthelemy, M. Batty, C. Gershenson, D. Helbing, et al., The role of complexity for digital twins of cities, Nat. Comput. Sci., 3 (2023), 374–381. https://doi.org/10.1038/s43588-023-00431-4 doi: 10.1038/s43588-023-00431-4
![]() |
[20] |
H. M. Markowitz, Portfolio selection, J. Financ., 7 (1952) 77–91. http://dx.doi.org/10.2307/2975974 doi: 10.2307/2975974
![]() |
[21] |
Z. Y. Guo, Heavy-tailed distributions and risk management of equity market tail events, J. Risk Control, 4 (2017), 31–41. http://dx.doi.org/10.2139/ssrn.3013749 doi: 10.2139/ssrn.3013749
![]() |
[22] |
R. E. Lucas, Asset prices in an exchange economy, Econometrica, 46 (1978), 1429–1445. https://doi.org/10.2307/1913837 doi: 10.2307/1913837
![]() |
[23] | J. H. Cochrane, Asset pricing, Princeton University Press, 2005. https://doi.org/10.1016/j.jebo.2005.08.001 |
[24] |
D. Ellsberg, Risk, ambiguity, and the savage axioms, Quart. J. Econ., 75 (1961), 643–669. http://dx.doi.org/10.2307/1884324 doi: 10.2307/1884324
![]() |
[25] | H. R. Varian, Differences of opinion in financial markets, In: C. C. Stone, (eds) Financial Risk: Theory, Evidence and Implications, Springer, Dordrecht., 1989. https://doi.org/10.1007/978-94-009-2665-3_1 |
[26] | B. Liu, Uncertainty theory, In: Uncertainty Theory, Studies in Fuzziness and Soft Computing, Berlin: Springer, 154 (2007). https://doi.org/10.1007/978-3-540-73165-8_5 |
[27] | B. Liu, Fuzzy process, hybrid process and uncertain process, J. Uncertain Syst., 2 (2008), 3–16. |
[28] |
B. Liu, Toward uncertain finance theory, J. Uncertain. Anal. Appl., 1 (2013), 1–15. http://dx.doi.org/10.1186/2195-5468-1-1 doi: 10.1186/2195-5468-1-1
![]() |
[29] | M. Segoviano, C. A. Goodhart, Banking stability measures, International Monetary Fund, 2009. https://doi.org/10.5089/9781451871517.001 |
[30] |
L. Liu, T. Zhang, Economic policy uncertainty and stock market volatility, Financ. Res. Lett., 15 (2015), 99–105. https://doi.org/10.1016/j.frl.2015.08.009 doi: 10.1016/j.frl.2015.08.009
![]() |
[31] |
H. Asgharian, C. Christiansen, A. J. Hou, The effect of uncertainty on stock market volatility and correlation, J. Bank. Financ., 154 (2023), 106929. https://doi.org/10.1016/j.jbankfin.2023.106929 doi: 10.1016/j.jbankfin.2023.106929
![]() |
[32] |
T. Simin, The poor predictive performance of asset pricing models, J. Financ. Quant. Anal., 43 (2008), 355–380. http://dx.doi.org/10.1017/S0022109000003550 doi: 10.1017/S0022109000003550
![]() |
[33] |
J. H. Boyd, J. Hu, R. Jagannathan, The stock market's reaction to unemployment news: Why bad news is usually good for stocks, J. Financ., 60 (2005), 649–672. http://dx.doi.org/10.1111/j.1540-6261.2005.00742.x doi: 10.1111/j.1540-6261.2005.00742.x
![]() |
[34] |
R. P. Schumaker, H. Chen, Textual analysis of stock market prediction using breaking financial news: The AZFin text system, ACM Trans. Inform. Syst., 27 (2009), 1–19. https://doi.org/10.1145/1462198.1462204 doi: 10.1145/1462198.1462204
![]() |
[35] |
M. T. Suleman, Stock market reaction to good and bad political news, Asian J. Financ. Account., 4 (2012), 299–312. https://doi.org/10.5296/ajfa.v4i1.1705 doi: 10.5296/ajfa.v4i1.1705
![]() |
[36] |
C. O. Cepoi, Asymmetric dependence between stock market returns and news during COVID-19 financial turmoil, Financ. Res. Lett., 36 (2020), 101658. https://doi.org/10.1016/j.frl.2020.101658 doi: 10.1016/j.frl.2020.101658
![]() |
[37] |
A. Caruso, Macroeconomic news and market reaction: Surprise indexes meet nowcasting, Int. J. Forecasting, 35 (2019), 1725–1734. https://doi.org/10.1016/j.ijforecast.2018.12.005 doi: 10.1016/j.ijforecast.2018.12.005
![]() |
[38] |
E. F. Fama, Efficient capital markets: Ⅱ, J. Financ., 46 (1991), 1575–1617. https://doi.org/10.2307/2328565 doi: 10.2307/2328565
![]() |
[39] | J. D. Thomas, K. Sycara, Integrating genetic algorithms and text learning for financial prediction, In: Proceedings of GECCO '00 Workshop on Data Mining with Evolutionary Algorithms, 2000, 72–75. |
[40] |
P. C. Tetlock, Giving content to investor sentiment: The role of media in the stock market, J. Financ. Forthcoming, 62 (2007), 1139–1168. https://dx.doi.org/10.2139/ssrn.685145 doi: 10.2139/ssrn.685145
![]() |
[41] | S. Kogan, D. Levin, B. R. Routledge, J. S. Sagi, N. A. Smith, Predicting risk from financial reports with regression, In: Proceedings of Human Language Technologies: The 2009 Annual Conference of the North American Chapter of the Association for Computational Linguistics on - NAACL '09, 2009,272–280. http://dx.doi.org/10.3115/1620754.1620794 |
[42] |
J. L. Rogers, D. J. Skinner, A. Van Buskirk, Earnings guidance and market uncertainty, J. Account. Econ., 48 (2009), 90–109. https://doi.org/10.1016/j.jacceco.2009.07.001 doi: 10.1016/j.jacceco.2009.07.001
![]() |
[43] | J. Si, A. Mukherjee, B. Liu, Q. Li, H. Li, X. Deng, Exploiting topic based twitter sentiment for stock prediction, In: Proceedings of the 51st Annual Meeting of the Association for Computational Linguistics, 2 (2013), 24–29. |
[44] | X. Ding, Y. Zhang, T. Liu, J. Duan, Using structured events to predict stock price movement: An empirical investigation, In: Proceedings of the 2014 conference on empirical methods in natural language processing, 2014, 1415–1425. http://dx.doi.org/10.3115/v1/D14-1148 |
[45] | W. Y. Wang, Z. Hua, A semiparametric gaussian copula regression model for predicting financial risks from earnings calls, In: Proceedings of the 52nd Annual Meeting of the Association for Computational Linguistics, 1 (2014), 1155–1165. http://dx.doi.org/10.3115/v1/P14-1109 |
[46] |
R. Luss, A. d'Aspremont, Predicting abnormal returns from news using text classification, Quant. Financ., 15 (2015), 999–1012. https://doi.org/10.1080/14697688.2012.672762 doi: 10.1080/14697688.2012.672762
![]() |
[47] |
P. K. Narayan, D. Bannigidadmath, Does financial news predict stock returns? New evidence from Islamic and non-Islamic stocks, Pac.-Basin Financ. J., 42 (2017) 24–45. https://doi.org/10.1016/j.pacfin.2015.12.009 doi: 10.1016/j.pacfin.2015.12.009
![]() |
[48] | F. Larkin, C. Ryan, Good news: Using news feeds with genetic programming to predict stock prices, In: European Conference on Genetic Programming, 4971 (2008), 49–60. https://doi.org/10.1007/978-3-540-78671-9_5 |
[49] | Y. Kim, S. R. Jeong, I. Ghani, Text opinion mining to analyze news for stock market prediction, Int. J. Adv. Soft Comput. Appl., 6 (2014), 2074–8523. |
[50] |
A. E. Khedr, S. E. Salama, N. Yaseen, Predicting stock market behavior using data mining technique and news sentiment analysis, Int. J. Adv. Soft Comput. Appl., 9 (2017), 22–30. https://doi.org/10.5815/ijisa.2017.07.03 doi: 10.5815/ijisa.2017.07.03
![]() |
[51] |
X. Zhou, H. Zhou, H. Long, Forecasting the equity premium: Do deep neural network models work? Mod. Financ., 1 (2023), 1–11. https://doi.org/10.61351/mf.v1i1.2 doi: 10.61351/mf.v1i1.2
![]() |
[52] |
X. Dong, Y. Li, D. E. Rapach, G. Zhou, Anomalies and the expected market return, J. Financ., 77 (2022), 639–681. https://doi.org/10.1111/jofi.13099 doi: 10.1111/jofi.13099
![]() |
[53] |
N. Cakici, C. Fieberg, D. Metko, A. Zaremba, Do anomalies really predict market returns? New data and new evidence, Rev. Financ., 2023, rfad025. https://doi.org/10.1093/rof/rfad025 doi: 10.1093/rof/rfad025
![]() |
[54] |
W. Shengli, Is human digital twin possible? Comput. Method. Prog. Biomed. Update, 1 (2021), 100014. https://doi.org/10.1016/j.cmpbup.2021.100014 doi: 10.1016/j.cmpbup.2021.100014
![]() |
[55] |
M. Singh, E. Fuenmayor, E. P. Hinchy, Y. Qiao, N. Murray, D. Devine, Digital twin: Origin to future, Appl. Syst. Inno., 4 (2021), 36. https://doi.org/10.3390/asi4020036 doi: 10.3390/asi4020036
![]() |
[56] |
H. D. Critchley, C. J. Mathias, R. J. Dolan, Neural activity in the human brain relating to uncertainty and arousal during anticipation, Neuron, 29 (2001), 537–545. https://doi.org/10.1016/s0896-6273(01)00225-2 doi: 10.1016/s0896-6273(01)00225-2
![]() |
[57] | H. A. Simon, Rational decision-making in business organizations, Am. Econ. Rev., 69 (1979), 493–513. |
[58] |
R. M. Hogarth, N. Karelaia, Regions of rationality: Maps for bounded agents, Decis. Anal., 3 (2006), 124–144. http://dx.doi.org/10.1287/deca.1060.0063 doi: 10.1287/deca.1060.0063
![]() |
[59] |
Y. Wang, N. Zhang, Uncertainty analysis of knowledge reductions in rough sets, The Scientific World J., 2014 (2014), 576409. https://doi.org/10.1155/2014/576409 doi: 10.1155/2014/576409
![]() |
[60] |
K. Erk, Understanding the combined meaning of words, Nat. Comput. Sci., 2 (2022), 701–702. https://doi.org/10.1038/s43588-022-00338-6 doi: 10.1038/s43588-022-00338-6
![]() |
[61] |
M. Toneva, T. M. Mitchell, L. Wehbe, Combining computational controls with natural text reveals aspects of meaning composition, Nat. Comput. Sci., 2 (2022), 745–757. https://doi.org/10.1038/s43588-022-00354-6 doi: 10.1038/s43588-022-00354-6
![]() |
[62] |
S. Hochreiter, J. Schmidhuber, Long short-term memory, Neural Comput., 9 (1997), 1735–1780. https://doi.org/10.1162/neco.1997.9.8.1735 doi: 10.1162/neco.1997.9.8.1735
![]() |
[63] | G. E. Hinton, Distributed representations, Carnegie Mellon University, 1984. |
[64] |
D. E. Rumelhart, G. E. Hinton, R. J. Williams, Learning representations by back-propagating errors, Nature, 323 (1986), 533–536. http://dx.doi.org/10.1038/323533a0 doi: 10.1038/323533a0
![]() |
[65] |
J. L. Elman, Finding structure in time, Cognitive Sci., 14 (1990), 179–211. http://dx.doi.org/10.1207/s15516709cog1402_1 doi: 10.1207/s15516709cog1402_1
![]() |
[66] | Y. Bengio, H. Schwenk, F. Morin, J. L. Gauvain, Neural probabilistic language models, In: Innovations in Machine Learning: Theory and Applications, 2006,137–186. https://doi.org/10.1007/3-540-33486-6_6 |
[67] | R. Collobert, J. Weston, A unified architecture for natural language processing: Deep neural networks with multitask learning, In: Proceedings of the 25th international conference on Machine learning - ICML '08, 2008,160–167. https://doi.org/10.1145/1390156.1390177 |
[68] |
A. Mnih, G. E. Hinton, A scalable hierarchical distributed language model, Adv. Neural Inform. Process. Syst., 21 (2008), 1081–1088. https://dl.acm.org/doi/10.5555/2981780.2981915 doi: 10.5555/2981780.2981915
![]() |
[69] | T. Mikolov, J. Kopecky, L. Burget, O. Glembek, J. Cernocky, Neural network based language models for highly inflective languages, In: 2009 IEEE international conference on acoustics, speech and signal processing, 2009, 4725–4728. https://doi.org/10.1109/ICASSP.2009.4960686 |
[70] |
R. Collobert, J. Weston, L. Bottou, M. Karlen, K. Kavukcuoglu, P. Kuksa, Natural language processing (almost) from scratch, J. Mach. Learn. Res., 12 (2011), 2493–2537. https://dl.acm.org/doi/10.5555/1953048.2078186 doi: 10.5555/1953048.2078186
![]() |
[71] | E. H. Huang, R. Socher, C. D. Manning, A. Y. Ng, Improving word representations via global context and multiple word prototypes, In: Proceedings of the 50th Annual Meeting of the Association for Computational Linguistics, 1 (2012), 873–882. |
[72] |
Y. Zhang, R. Jin, Z. H. Zhou, Understanding bag-of-words model: a statistical framework, Int. J. Mach. Learn. Cyb., 1 (2010), 43–52. http://dx.doi.org/10.1007/s13042-010-0001-0 doi: 10.1007/s13042-010-0001-0
![]() |
[73] | The lifespan of news stories, How the news enters (and exits) the public consciousness, Schema Design and Google Trends, 2019. Available from: https://newslifespan.com/. |
[74] |
N. Bloom, Fluctuations in uncertainty, J. Econ. Perspect., 28 (2014), 153–176. http://dx.doi.org/10.1257/jep.28.2.153 doi: 10.1257/jep.28.2.153
![]() |
[75] |
S. R. Baker, S. J. Davis, J. A. Levy, State-level economic policy uncertainty, J. Monetary Econ., 132 (2022), 81–99. http://dx.doi.org/10.1016/j.jmoneco.2022.08.004 doi: 10.1016/j.jmoneco.2022.08.004
![]() |
[76] |
S. Newcomb, A generalized theory of the combination of observations so as to obtain the best result, Am. J. Math., 1886,343–366. http://dx.doi.org/10.2307/2369392 doi: 10.2307/2369392
![]() |
[77] |
D. Böhning, E. Dietz, P. Schlattmann, Recent developments in computer-assisted analysis of mixtures, Biometrics, 54 (1998), 525–536. http://dx.doi.org/10.2307/3109760 doi: 10.2307/3109760
![]() |
[78] |
A. P. Dempster, N. M. Laird, D. B. Rubin, Maximum likelihood from incomplete data via the EM algorithm, J. Roy. Stat. Soc. Ser. B, 39 (1977), 1–22. http://dx.doi.org/10.1111/j.2517-6161.1977.tb01600.x doi: 10.1111/j.2517-6161.1977.tb01600.x
![]() |
[79] |
T. Heskes, Self-organizing maps, vector quantization, and mixture modeling, IEEE T. Neur. Net., 12 (2001), 1299–1305. http://dx.doi.org/10.1109/72.963766 doi: 10.1109/72.963766
![]() |
[80] | A. Gepperth, B. Pfülb, A rigorous link between self-organizing maps and gaussian mixture models, In: Artificial Neural Networks and Machine Learning-ICANN 2020, Springer, Cham, 2020,863–872. http://dx.doi.org/10.1007/978-3-030-61616-8_69 |
[81] |
D. Povey, L. Burget, M. Agarwal, P. Akyazi, F. Kai, A. Ghoshal, et al., The subspace Gaussian mixture model–-A structured model for speech recognition, Comput. Speech Lang., 25 (2011), 404–439. http://dx.doi.org/10.1016/j.csl.2010.06.003 doi: 10.1016/j.csl.2010.06.003
![]() |
[82] | J. Yin, J. Wang, A dirichlet multinomial mixture model-based approach for short text clustering, In: Proceedings of the 20th ACM SIGKDD international conference on knowledge discovery and data mining, 2014,233–242. http://dx.doi.org/10.1145/2623330.2623715 |
[83] | F. Najar, S. Bourouis, N. Bouguila, S. Belghith, A comparison between different Gaussian-based mixture models, In: 2017 IEEE/ACS 14th International Conference on Computer Systems and Applications (AICCSA), 2017,704–708. http://dx.doi.org/10.1109/AICCSA.2017.108 |
[84] |
G. Bordogna, G. Pasi, Soft clustering for information retrieval applications, WIRES Data Min. Knowl., 1 (2011), 138–146. http://dx.doi.org/10.1002/widm.3 doi: 10.1002/widm.3
![]() |
[85] |
N. F. G. Martin, J. W. England, R. Baierlein, Mathematical theory of entropy, Phys. Today, 36 (1983), 66–67. http://dx.doi.org/10.1063/1.2915804 doi: 10.1063/1.2915804
![]() |
[86] |
S. R. Bentes, R. Menezes, Entropy: A new measure of stock market volatility? J. Phys. Conf. Ser., 394 (2012), 012033. http://dx.doi.org/10.1088/1742-6596/394/1/012033 doi: 10.1088/1742-6596/394/1/012033
![]() |
[87] |
K. Ahn, D. Lee, S. Sohn, B. Yang, Stock market uncertainty and economic fundamentals: An entropy-based approach, Quant. Financ., 19 (2019), 1151–1163. http://dx.doi.org/10.1080/14697688.2019.1579922 doi: 10.1080/14697688.2019.1579922
![]() |
[88] |
T. Kohonen, The self-organizing map, Neurocomputing, 21 (1998), 1–6. http://dx.doi.org/10.1016/S0925-2312(98)00030-7 doi: 10.1016/S0925-2312(98)00030-7
![]() |
[89] |
M. Y. Kiang, Extending the Kohonen self-organizing map networks for clustering analysis, Comput. Stat. Data Anal., 38 (2001), 161–180. http://dx.doi.org/10.1016/S0167-9473(01)00040-8 doi: 10.1016/S0167-9473(01)00040-8
![]() |
[90] | T. Kohonen, Self-organization and associative memory, Springer Science & Business Media, 8 (2012). https://doi.org/10.1007/978-3-642-88163-3 |
[91] | B. Szmrecsanyi, Grammatical variation in British English dialects: A study in corpus-based dialectometry, Cambridge University Press, 2012. http://dx.doi.org/10.1017/CBO9780511763380 |
[92] | Dow Jones & CO WSJ.COM audience profile, comScore Media Metrix Q1, 2021. Available from: https://images.dowjones.com/wp-content/uploads/sites/183/2018/05/09164150/WSJ.com-Audience-Profile.pdf. |
[93] | VIX volatility suite, Cboe Global Markets, Inc., 2021. Available from: https://www.cboe.com/tradable_products/vix/. |
[94] | A. Elder, Trading for a living: Psychology, trading tactics, money management, John Wiley & Sons, 31 (1993). |
[95] | SCAYLE Supercomputación Castilla y León, 2021. Available from: https://www.scayle.es. |
[96] |
O. A. M. Salem, F. Liu, A. S. Sherif, W. Zhang, X. Chen, Feature selection based on fuzzy joint mutual information maximization, Math. Biosci. Eng., 18 (2020), 305–327. http://dx.doi.org/10.3934/mbe.2021016 doi: 10.3934/mbe.2021016
![]() |
[97] |
P. V. Balakrishnan, M. C. Cooper, V. S. Jacob, P. A. Lewis, A study of the classification capabilities of neural networks using unsupervised learning: A comparison with K-means clustering, Psychometrika, 59 (1994), 509–525. https://doi.org/10.1007/BF02294390 doi: 10.1007/BF02294390
![]() |
[98] | A. Flexer, Limitations of self-organizing maps for vector quantization and multidimensional scaling, Adv. Neur. Inform. Process. Syst., 9 (1996), 445–451. |
[99] | U. A. Kumar, Y. Dhamija, Comparative analysis of SOM neural network with K-means clustering algorithm, In: 2010 IEEE International Conference on Management of Innovation & Technology, 2010, 55–59. http://dx.doi.org/10.1109/ICMIT.2010.5492838 |
[100] | J. Han, M. Kamber, J. Pei, Data mining: Concepts and techniques, 3 Eds., Morgan Kauffman, 2012. https://doi.org/10.1016/C2009-0-61819-5 |
[101] |
H. M. Hodges, Arbitrage bounds of the implied volatility strike and term structures of European-style options, J. Deriv., 3 (1996), 23–35. http://dx.doi.org/10.3905/jod.1996.407950 doi: 10.3905/jod.1996.407950
![]() |
[102] |
A. M. Malz, A simple and reliable way to compute option-based risk-neutral distributions, FRB New York Staff Rep., 677 (2014). http://dx.doi.org/10.2139/ssrn.2449692 doi: 10.2139/ssrn.2449692
![]() |
[103] | B. Judge, 26 May 1896: Charles Dow launches the Dow Jones industrial average, 2015. Available from: https://moneyweek.com/392888/26-may-1896-charles-dow-launches-the-dow-jones-industrial-average/. |
[104] | A. C. MacKinlay, Event studies in economics and finance, J. Econ. Lit., 35 (1997), 13–39. |
[105] |
Z. Önder, C. Şimga-Mugan, How do political and economic news affect emerging markets? Evidence from Argentina and Turkey, Emerg. Mark. Financ. Tr., 42 (2006), 50–77. http://dx.doi.org/10.2753/REE1540-496X420403 doi: 10.2753/REE1540-496X420403
![]() |
[106] |
N. Aktas, E. de Bodt, J. G. Cousin, Event studies with a contaminated estimation period, J. Corp. Financ., 13 (2007), 129–145. http://dx.doi.org/10.1016/j.jcorpfin.2006.09.001 doi: 10.1016/j.jcorpfin.2006.09.001
![]() |
[107] |
O. Arslan, W. Xing, F. A. Inan, H. Du, Understanding topic duration in Twitter learning communities using data mining, J. Comput. Assist. Learn., 38 (2022), 513–525. http://dx.doi.org/10.1111/jcal.12633 doi: 10.1111/jcal.12633
![]() |
[108] |
T. Fawcett, An introduction to ROC analysis, Pattern Recogn. Lett., 27 (2006), 861–874. http://dx.doi.org/10.1016/j.patrec.2005.10.010 doi: 10.1016/j.patrec.2005.10.010
![]() |
[109] | D. W. Hosmer, S. Lemeshow, Applied logistic regression, 2 Eds., New York: John Wiley and Sons, 2000,160–164. http://dx.doi.org/10.1002/0471722146 |
[110] | T. Fawcett, ROC graphs: Notes and practical considerations for researchers, Mach. Learn., 31 (2004), 1–38. |
[111] |
F. Melo, Area under the ROC curve, Encyclopedia Syst. Biol., 2013 (2013). http://dx.doi.org/10.1007/978-1-4419-9863-7_209 doi: 10.1007/978-1-4419-9863-7_209
![]() |
[112] |
J. Cragg, R. Uhler, The demand for automobiles, Can. J. Econ., 3 (1970), 386–406. http://dx.doi.org/10.2307/133656 doi: 10.2307/133656
![]() |
[113] | G. Maddala, Limited dependent and qualitative variables in econometrics, New York: Cambridge University Press, 1983. http://dx.doi.org/10.1017/CBO9780511810176 |
[114] |
D. R. Cox, N. Wermuth, A comment on the coefficient of determination for binary responses, Am. Stat., 46 (1992), 1–4. http://dx.doi.org/10.2307/2684400 doi: 10.2307/2684400
![]() |
[115] | P. Flach, J. Hernández-Orallo, C. Ferri, A coherent interpretation of AUC as a measure of aggregated classification performance, In: Proceedings of the 28th International Conference on Machine Learning, 2011. |
[116] |
J. A. Hanley, B. J. McNeil, The meaning and use of the area under a receiver operating characteristic (ROC) curve, Radiology, 143 (1982), 29–36. http://dx.doi.org/10.1148/radiology.143.1.7063747 doi: 10.1148/radiology.143.1.7063747
![]() |
[117] | B. Efron, The jackknife, the bootstrap, and other resampling plans, In: Society of Industrial and Applied Mathematics CBMS-NSF Monographs 38, 1982. http://dx.doi.org/10.1137/1.9781611970319 |
[118] |
I. A. Boboc, M. C. Dinică, An Algorithm for testing the efficient market hypothesis, PloS One, 8 (2013), e78177. https://doi.org/10.1371/journal.pone.0078177 doi: 10.1371/journal.pone.0078177
![]() |
[119] |
M. A. Sánchez-Granero, K. A. Balladares, J. P. Ramos-Requena, J. E. Trinidad-Segovia, Testing the efficient market hypothesis in Latin American stock markets, Physica A, 540 (2020), 123082. https://doi.org/10.1016/j.physa.2019.123082 doi: 10.1016/j.physa.2019.123082
![]() |
[120] |
E. M. Sent, Rationality and bounded rationality: You can't have one without the other, Eur. J.Hist. Econ. Thou., 25 (2018), 1370–1386. http://dx.doi.org/10.1080/09672567.2018.1523206 doi: 10.1080/09672567.2018.1523206
![]() |
[121] |
M. Hahn, R. Futrell, R. Levy, E. Gibson, A resource-rational model of human processing of recursive linguistic structure, P. Natl. Acad. Sci., 119 (2022), e2122602119. http://dx.doi.org/10.1073/pnas.2122602119 doi: 10.1073/pnas.2122602119
![]() |
[122] |
M. Szczepański, M. Pawlicki, R. Kozik, M. Choraś, New explainability method for BERT-based model in fake news detection, Sci. Rep., 11 (2021), 23705. http://dx.doi.org/10.1038/s41598-021-03100-6 doi: 10.1038/s41598-021-03100-6
![]() |
[123] |
G. Pennycook, Z. Epstein, M. Mosleh, A. A. Arechar, D. Eckles, D. G. Rand, Shifting attention to accuracy can reduce misinformation online, Nature, 592 (2021) 590–595. http://dx.doi.org/10.1038/s41586-021-03344-2 doi: 10.1038/s41586-021-03344-2
![]() |
β | μ1 | VarZ | CV | N | M | |
θ=0.4 | 0.3 | 0.0874 | 0.0234 | 1.7508 | 4.3227 | 30.106 |
0.6 | 0.2287 | 0.0351 | 0.8190 | 1.5817 | 3.3285 | |
0.9 | 0.3488 | 0.0369 | 0.5506 | 0.8949 | 0.7335 | |
1.2 | 0.4412 | 0.034 | 0.4179 | 0.5665 | 0.0143 | |
θ=0.6 | 0.3 | 0.3375 | 0.3492 | 1.7508 | 4.3227 | 30.106 |
0.6 | 0.4495 | 0.1355 | 0.8190 | 1.5817 | 3.3285 | |
0.9 | 0.5473 | 0.0908 | 0.5506 | 0.8949 | 0.7335 | |
1.2 | 0.6186 | 0.0668 | 0.4179 | 0.5665 | 0.0143 | |
θ=0.8 | 0.3 | 0.8806 | 2.3770 | 1.7508 | 4.3227 | 30.106 |
0.6 | 0.7260 | 0.3535 | 0.8190 | 1.5817 | 3.3285 | |
0.9 | 0.7534 | 0.1721 | 0.5506 | 0.8949 | 0.7335 | |
1.2 | 0.7862 | 0.1080 | 0.4179 | 0.5665 | 0.0143 | |
θ=1.2 | 0.3 | 3.4022 | 35.479 | 1.7508 | 4.3227 | 30.106 |
0.6 | 1.4270 | 1.3659 | 0.8190 | 1.5817 | 3.3285 | |
0.9 | 1.1822 | 0.4238 | 0.5506 | 0.8949 | 0.7335 | |
1.2 | 1.1022 | 0.2122 | 0.4179 | 0.5665 | 0.0143 | |
θ=1.5 | 0.3 | 7.1581 | 157.05 | 1.7508 | 4.3227 | 30.106 |
0.6 | 2.0699 | 2.8737 | 0.8190 | 1.5817 | 3.3285 | |
0.9 | 1.5149 | 0.6958 | 0.5506 | 0.8949 | 0.7335 | |
1.2 | 1.3274 | 0.3078 | 0.4179 | 0.5665 | 0.0143 |
m | MLE | BSE | BLI | BGE | |||||
Mean | MSE | Mean | MSE | Mean | MSE | Mean | MSE | ||
30 | θ | 0.4919 | 0.0039 | 0.4502 | 0.0035 | 0.4505 | 0.0037 | 0.4496 | 0.0039 |
β | 0.7802 | 0.0115 | 0.7496 | 0.0008 | 0.7497 | 0.0101 | 0.7494 | 0.0103 | |
60 | θ | 0.4928 | 0.0018 | 0.5093 | 0.0010 | 0.5093 | 0.0013 | 0.5085 | 0.0015 |
β | 0.7681 | 0.0051 | 0.7395 | 0.0007 | 0.7396 | 0.0009 | 0.7393 | 0.0101 | |
80 | θ | 0.5007 | 0.0011 | 0.4888 | 0.0006 | 0.4888 | 0.0008 | 0.4886 | 0.0010 |
β | 0.7586 | 0.0034 | 0.7882 | 0.0005 | 0.7883 | 0.0008 | 0.7880 | 0.0009 | |
100 | θ | 0.4943 | 0.0010 | 0.5091 | 0.0004 | 0.5093 | 0.0005 | 0.5087 | 0.0008 |
β | 0.7591 | 0.0024 | 0.7577 | 0.0004 | 0.7581 | 0.0006 | 0.7573 | 0.0008 |
m | MLE | BSE | BLI | BGE | |||||
Mean | MSE | Mean | MSE | Mean | MSE | Mean | MSE | ||
30 | θ | 0.7998 | 0.0055 | 0.8643 | 0.0039 | 0.8645 | 0.0041 | 0.4816 | 0.0043 |
β | 1.2981 | 0.0309 | 1.3373 | 0.0121 | 1.3384 | 0.0123 | 1.3364 | 0.0125 | |
60 | θ | 0.8001 | 0.0024 | 0.8358 | 0.0015 | 0.8359 | 0.0017 | 0.8357 | 0.0019 |
β | 1.2646 | 0.0148 | 1.1742 | 0.0075 | 1.1746 | 0.0078 | 1.1738 | 0.0079 | |
80 | θ | 0.7993 | 0.0023 | 0.7856 | 0.0014 | 0.7859 | 0.0016 | 0.7852 | 0.0018 |
β | 1.2641 | 0.0075 | 1.2241 | 0.0017 | 1.2244 | 0.0020 | 1.2239 | 0.0021 | |
100 | θ | 0.7948 | 0.0014 | 0.8046 | 0.0007 | 0.8048 | 0.0009 | 0.8044 | 0.0011 |
β | 1.2673 | 0.0072 | 1.2320 | 0.0011 | 1.2322 | 0.0013 | 1.2319 | 0.0015 |
m | MLE | BSE | BLI | BGE | ||||||
Mean | MSE | Mean | MSE | Mean | MSE | Mean | MSE | |||
30 | θ | 1.2240 | 0.0188 | 1.2206 | 0.0051 | 1.2217 | 0.0053 | 1.2196 | 0.0055 | |
β | 1.5799 | 0.0559 | 1.6200 | 0.0245 | 1.6225 | 0.0248 | 1.6184 | 0.0249 | ||
60 | θ | 1.2049 | 0.0065 | 1.2310 | 0.0032 | 1.2316 | 0.0034 | 1.2306 | 0.0036 | |
β | 1.5556 | 0.0215 | 1.5530 | 0.0058 | 1.5538 | 0.0061 | 1.5526 | 0.0062 | ||
80 | θ | 1.2035 | 0.0040 | 1.2276 | 0.0023 | 1.2280 | 0.0025 | 1.2272 | 0.0028 | |
β | 1.5524 | 0.0159 | 1.4679 | 0.0032 | 1.4685 | 0.0034 | 1.4676 | 0.0035 | ||
100 | θ | 1.2139 | 0.0041 | 1.2160 | 0.0019 | 1.2165 | 0.0022 | 1.2157 | 0.0024 | |
β | 1.5158 | 0.0113 | 1.4818 | 0.0022 | 1.4822 | 0.0023 | 1.4815 | 0.0025 |
8 | 6 | 8 | 16 | 23 | 20 | 28 | 40 |
43 | 50 | 54 | 32 | 52 | 29 | 33 | 42 |
41 | 52 | 56 | 79 | 155 | 84 | 95 | 111 |
136 | 161 | 204 | 241 |
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 |
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 |
Zone | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 |
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 |
Data | Q1 | Q2 | μ′1 | Q3 | CV | N | M |
1 | 28.75 | 46.50 | 67.82 | 27.60 | 54.57 | 1.31 | 0.83 |
2 | 6.55 | 7.72 | 7.67 | 9.35 | 0.35 | -0.18 | -0.96 |
3 | 3.445 | 4.475 | 4.521 | 5.39 | 0.35 | 0.3589 | -0.0846 |
Data | Model | ˆθ | ˆβ | KS | P-value | LL | A | B | C | D |
EF-IW | 10.609 | 0.5686 | 0.1425 | 0.6197 | -144.601 | 293.202 | 295.866 | 294.016 | 293.682 | |
IW | 8.5601 | 0.6934 | 0.2135 | 0.1557 | -152.624 | 309.269 | 311.934 | 300.958 | 300.624 | |
1 | EF-W | 224.821 | 0.7022 | 0.1961 | 0.2316 | -147.318 | 298.637 | 301.301 | 299.451 | 299.117 |
EF-E | 0.0047 | 0.3475 | 0.0023 | -149.955 | 301.911 | 303.243 | 302.318 | 302.064 | ||
PBX | 0.0771 | 0.5983 | 0.1575 | 0.4901 | -145.356 | 294.713 | 297.378 | 295.603 | 295.269 | |
GE | 0.0188 | 1.4887 | 0.1495 | 0.5582 | -145.041 | 294.082 | 296.746 | 294.897 | 294.562 | |
EF-IW | 169.011 | 2.4633 | 0.1114 | 0.3500 | -132.473 | 268.947 | 273.444 | 270.734 | 269.126 | |
IW | 2168.36 | 4.0579 | 0.1697 | 0.0354 | -146.885 | 297.771 | 302.268 | 299.557 | 297.950 | |
2 | EF-W | 10.415 | 3.6140 | 0.1468 | 0.0978 | -132.736 | 269.472 | 273.969 | 271.259 | 269.651 |
EF-E | 0.0709 | 0.4251 | <0.0001 | -179.247 | 360.494 | 362.743 | 361.387 | 360.553 | ||
PBX | 0.0407 | 1.5378 | 0.2102 | 0.0041 | -146.950 | 297.936 | 302.433 | 299.723 | 298.115 | |
GE | 0.6226 | 69.883 | 0.1399 | 0.1290 | -139.298 | 282.596 | 287.093 | 284.382 | 282.775 | |
EF-IW | 11.193 | 1.5481 | 0.0756 | 0.6168 | -177.432 | 358.864 | 364.074 | 360.973 | 358.988 | |
IW | 25.075 | 2.5284 | 0.1395 | 0.0408 | -198.317 | 400.634 | 405.845 | 402.743 | 400.758 | |
3 | EF-W | 1.9167 | 7.5766 | 0.1301 | 0.0676 | -185.737 | 375.475 | 380.686 | 377.584 | 375.599 |
EF-E | 0.1131 | 0.3139 | <0.0001 | -210.347 | 422.694 | 425.300 | 423.749 | 422.735 | ||
PBX | 0.0639 | 1.7021 | 0.0909 | 0.3791 | -177.673 | 359.347 | 364.557 | 361.455 | 359.470 | |
GE | 0.7435 | 16.358 | 0.1060 | 0.2107 | -180.239 | 364.478 | 69.688 | 366.587 | 364.602 |
Data | Par | Bayes | ||
BSE | BLI | BGE | ||
1 | θ | 10.468 | 10.470 | 10.468 |
β | 0.5765 | 0.5766 | 0.5764 | |
2 | θ | 168.989 | 168.990 | 168.989 |
β | 2.457 | 2.457 | 2.457 | |
3 | θ | 11.692 | 11.689 | 11.694 |
β | 1.5635 | 1.5637 | 1.5634 |
β | μ1 | VarZ | CV | N | M | |
θ=0.4 | 0.3 | 0.0874 | 0.0234 | 1.7508 | 4.3227 | 30.106 |
0.6 | 0.2287 | 0.0351 | 0.8190 | 1.5817 | 3.3285 | |
0.9 | 0.3488 | 0.0369 | 0.5506 | 0.8949 | 0.7335 | |
1.2 | 0.4412 | 0.034 | 0.4179 | 0.5665 | 0.0143 | |
θ=0.6 | 0.3 | 0.3375 | 0.3492 | 1.7508 | 4.3227 | 30.106 |
0.6 | 0.4495 | 0.1355 | 0.8190 | 1.5817 | 3.3285 | |
0.9 | 0.5473 | 0.0908 | 0.5506 | 0.8949 | 0.7335 | |
1.2 | 0.6186 | 0.0668 | 0.4179 | 0.5665 | 0.0143 | |
θ=0.8 | 0.3 | 0.8806 | 2.3770 | 1.7508 | 4.3227 | 30.106 |
0.6 | 0.7260 | 0.3535 | 0.8190 | 1.5817 | 3.3285 | |
0.9 | 0.7534 | 0.1721 | 0.5506 | 0.8949 | 0.7335 | |
1.2 | 0.7862 | 0.1080 | 0.4179 | 0.5665 | 0.0143 | |
θ=1.2 | 0.3 | 3.4022 | 35.479 | 1.7508 | 4.3227 | 30.106 |
0.6 | 1.4270 | 1.3659 | 0.8190 | 1.5817 | 3.3285 | |
0.9 | 1.1822 | 0.4238 | 0.5506 | 0.8949 | 0.7335 | |
1.2 | 1.1022 | 0.2122 | 0.4179 | 0.5665 | 0.0143 | |
θ=1.5 | 0.3 | 7.1581 | 157.05 | 1.7508 | 4.3227 | 30.106 |
0.6 | 2.0699 | 2.8737 | 0.8190 | 1.5817 | 3.3285 | |
0.9 | 1.5149 | 0.6958 | 0.5506 | 0.8949 | 0.7335 | |
1.2 | 1.3274 | 0.3078 | 0.4179 | 0.5665 | 0.0143 |
m | MLE | BSE | BLI | BGE | |||||
Mean | MSE | Mean | MSE | Mean | MSE | Mean | MSE | ||
30 | θ | 0.4919 | 0.0039 | 0.4502 | 0.0035 | 0.4505 | 0.0037 | 0.4496 | 0.0039 |
β | 0.7802 | 0.0115 | 0.7496 | 0.0008 | 0.7497 | 0.0101 | 0.7494 | 0.0103 | |
60 | θ | 0.4928 | 0.0018 | 0.5093 | 0.0010 | 0.5093 | 0.0013 | 0.5085 | 0.0015 |
β | 0.7681 | 0.0051 | 0.7395 | 0.0007 | 0.7396 | 0.0009 | 0.7393 | 0.0101 | |
80 | θ | 0.5007 | 0.0011 | 0.4888 | 0.0006 | 0.4888 | 0.0008 | 0.4886 | 0.0010 |
β | 0.7586 | 0.0034 | 0.7882 | 0.0005 | 0.7883 | 0.0008 | 0.7880 | 0.0009 | |
100 | θ | 0.4943 | 0.0010 | 0.5091 | 0.0004 | 0.5093 | 0.0005 | 0.5087 | 0.0008 |
β | 0.7591 | 0.0024 | 0.7577 | 0.0004 | 0.7581 | 0.0006 | 0.7573 | 0.0008 |
m | MLE | BSE | BLI | BGE | |||||
Mean | MSE | Mean | MSE | Mean | MSE | Mean | MSE | ||
30 | θ | 0.7998 | 0.0055 | 0.8643 | 0.0039 | 0.8645 | 0.0041 | 0.4816 | 0.0043 |
β | 1.2981 | 0.0309 | 1.3373 | 0.0121 | 1.3384 | 0.0123 | 1.3364 | 0.0125 | |
60 | θ | 0.8001 | 0.0024 | 0.8358 | 0.0015 | 0.8359 | 0.0017 | 0.8357 | 0.0019 |
β | 1.2646 | 0.0148 | 1.1742 | 0.0075 | 1.1746 | 0.0078 | 1.1738 | 0.0079 | |
80 | θ | 0.7993 | 0.0023 | 0.7856 | 0.0014 | 0.7859 | 0.0016 | 0.7852 | 0.0018 |
β | 1.2641 | 0.0075 | 1.2241 | 0.0017 | 1.2244 | 0.0020 | 1.2239 | 0.0021 | |
100 | θ | 0.7948 | 0.0014 | 0.8046 | 0.0007 | 0.8048 | 0.0009 | 0.8044 | 0.0011 |
β | 1.2673 | 0.0072 | 1.2320 | 0.0011 | 1.2322 | 0.0013 | 1.2319 | 0.0015 |
m | MLE | BSE | BLI | BGE | ||||||
Mean | MSE | Mean | MSE | Mean | MSE | Mean | MSE | |||
30 | θ | 1.2240 | 0.0188 | 1.2206 | 0.0051 | 1.2217 | 0.0053 | 1.2196 | 0.0055 | |
β | 1.5799 | 0.0559 | 1.6200 | 0.0245 | 1.6225 | 0.0248 | 1.6184 | 0.0249 | ||
60 | θ | 1.2049 | 0.0065 | 1.2310 | 0.0032 | 1.2316 | 0.0034 | 1.2306 | 0.0036 | |
β | 1.5556 | 0.0215 | 1.5530 | 0.0058 | 1.5538 | 0.0061 | 1.5526 | 0.0062 | ||
80 | θ | 1.2035 | 0.0040 | 1.2276 | 0.0023 | 1.2280 | 0.0025 | 1.2272 | 0.0028 | |
β | 1.5524 | 0.0159 | 1.4679 | 0.0032 | 1.4685 | 0.0034 | 1.4676 | 0.0035 | ||
100 | θ | 1.2139 | 0.0041 | 1.2160 | 0.0019 | 1.2165 | 0.0022 | 1.2157 | 0.0024 | |
β | 1.5158 | 0.0113 | 1.4818 | 0.0022 | 1.4822 | 0.0023 | 1.4815 | 0.0025 |
8 | 6 | 8 | 16 | 23 | 20 | 28 | 40 |
43 | 50 | 54 | 32 | 52 | 29 | 33 | 42 |
41 | 52 | 56 | 79 | 155 | 84 | 95 | 111 |
136 | 161 | 204 | 241 |
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 |
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 |
Zone | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021 | 2022 |
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 |
Data | Q1 | Q2 | μ′1 | Q3 | CV | N | M |
1 | 28.75 | 46.50 | 67.82 | 27.60 | 54.57 | 1.31 | 0.83 |
2 | 6.55 | 7.72 | 7.67 | 9.35 | 0.35 | -0.18 | -0.96 |
3 | 3.445 | 4.475 | 4.521 | 5.39 | 0.35 | 0.3589 | -0.0846 |
Data | Model | ˆθ | ˆβ | KS | P-value | LL | A | B | C | D |
EF-IW | 10.609 | 0.5686 | 0.1425 | 0.6197 | -144.601 | 293.202 | 295.866 | 294.016 | 293.682 | |
IW | 8.5601 | 0.6934 | 0.2135 | 0.1557 | -152.624 | 309.269 | 311.934 | 300.958 | 300.624 | |
1 | EF-W | 224.821 | 0.7022 | 0.1961 | 0.2316 | -147.318 | 298.637 | 301.301 | 299.451 | 299.117 |
EF-E | 0.0047 | 0.3475 | 0.0023 | -149.955 | 301.911 | 303.243 | 302.318 | 302.064 | ||
PBX | 0.0771 | 0.5983 | 0.1575 | 0.4901 | -145.356 | 294.713 | 297.378 | 295.603 | 295.269 | |
GE | 0.0188 | 1.4887 | 0.1495 | 0.5582 | -145.041 | 294.082 | 296.746 | 294.897 | 294.562 | |
EF-IW | 169.011 | 2.4633 | 0.1114 | 0.3500 | -132.473 | 268.947 | 273.444 | 270.734 | 269.126 | |
IW | 2168.36 | 4.0579 | 0.1697 | 0.0354 | -146.885 | 297.771 | 302.268 | 299.557 | 297.950 | |
2 | EF-W | 10.415 | 3.6140 | 0.1468 | 0.0978 | -132.736 | 269.472 | 273.969 | 271.259 | 269.651 |
EF-E | 0.0709 | 0.4251 | <0.0001 | -179.247 | 360.494 | 362.743 | 361.387 | 360.553 | ||
PBX | 0.0407 | 1.5378 | 0.2102 | 0.0041 | -146.950 | 297.936 | 302.433 | 299.723 | 298.115 | |
GE | 0.6226 | 69.883 | 0.1399 | 0.1290 | -139.298 | 282.596 | 287.093 | 284.382 | 282.775 | |
EF-IW | 11.193 | 1.5481 | 0.0756 | 0.6168 | -177.432 | 358.864 | 364.074 | 360.973 | 358.988 | |
IW | 25.075 | 2.5284 | 0.1395 | 0.0408 | -198.317 | 400.634 | 405.845 | 402.743 | 400.758 | |
3 | EF-W | 1.9167 | 7.5766 | 0.1301 | 0.0676 | -185.737 | 375.475 | 380.686 | 377.584 | 375.599 |
EF-E | 0.1131 | 0.3139 | <0.0001 | -210.347 | 422.694 | 425.300 | 423.749 | 422.735 | ||
PBX | 0.0639 | 1.7021 | 0.0909 | 0.3791 | -177.673 | 359.347 | 364.557 | 361.455 | 359.470 | |
GE | 0.7435 | 16.358 | 0.1060 | 0.2107 | -180.239 | 364.478 | 69.688 | 366.587 | 364.602 |
Data | Par | Bayes | ||
BSE | BLI | BGE | ||
1 | θ | 10.468 | 10.470 | 10.468 |
β | 0.5765 | 0.5766 | 0.5764 | |
2 | θ | 168.989 | 168.990 | 168.989 |
β | 2.457 | 2.457 | 2.457 | |
3 | θ | 11.692 | 11.689 | 11.694 |
β | 1.5635 | 1.5637 | 1.5634 |