Statistical Inference of the Generalized Inverted Exponential Distribution under Joint Progressively Type-II Censoring
Statistical Inference of the Generalized Inverted Exponential Distribution under Joint...
Chen, Qiyue;Gui, Wenhao
2022-04-20 00:00:00
entropy Article Statistical Inference of the Generalized Inverted Exponential Distribution under Joint Progressively Type-II Censoring Qiyue Chen and Wenhao Gui * School of Mathematics and Statistics, Beijing Jiaotong University, Beijing 100044, China; 19271149@bjtu.edu.cn * Correspondence: whgui@bjtu.edu.cn Abstract: In this paper, we study the statistical inference of the generalized inverted exponential distribution with the same scale parameter and various shape parameters based on joint progressively type-II censored data. The expectation maximization (EM) algorithm is applied to calculate the maximum likelihood estimates (MLEs) of the parameters. We obtain the observed information matrix based on the missing value principle. Interval estimations are computed by the bootstrap method. We provide Bayesian inference for the informative prior and the non-informative prior. The importance sampling technique is performed to derive the Bayesian estimates and credible intervals under the squared error loss function and the linex loss function, respectively. Eventually, we conduct the Monte Carlo simulation and real data analysis. Moreover, we consider the parameters that have order restrictions and provide the maximum likelihood estimates and Bayesian inference. Keywords: generalized inverted exponential distribution; joint progressively type-II censoring scheme; EM algorithm; maximum likelihood estimation; bootstrap method; Bayesian inference; importance sampling; Monte Carlo simulation Citation: Chen, Q.; Gui, W. Statistical 1. Introduction Inference of the Generalized Inverted 1.1. Generalized Inverted Exponential Distribution Exponential Distribution under Joint Progressively Type-II Censoring. The generalized inverted exponential distribution (GIED) is a modification of the Entropy 2022, 24, 576. https:// inverse exponential distribution (IED). In this way, it can fit the lifetime data better. The doi.org/10.3390/e24050576 GIED was introduced by [1]. The distribution has a non-constant hazard rate function, which is unimodal and positively skewed. Due to these properties, the distribution can Academic Editor: Ali model different shapes of failure rates of aging criteria. Reference [2] proposed the method Mohammad-Djafari of the maximum product of spacings for the point estimation of the parameter of the GIED. Received: 24 March 2022 In [3], acceptance sampling plans were developed based on truncated lifetimes when the life Accepted: 15 April 2022 of an item follows a generalized inverted exponential distribution. Reference [4] performed Published: 20 April 2022 a Monte Carlo simulation for the GIED to analyze the performance of the estimations. Publisher’s Note: MDPI stays neutral Reference [5] studied the point estimation of the parameters of the GIED when the test with regard to jurisdictional claims in units are progressively type-II censored. Reference [6] generated samples from the GIED published maps and institutional affil- and computed the Bayes estimates. Reference [7] investigated the MLEs of the GIED when iations. the test units are progressively type-II censored. Reference [8] proposed a two-stage group acceptance sampling plan for the GIED under a truncated life experiment. Provided that X is a variable and it is subject to the GIED, the following is the form of the corresponding probability density function (pdf), the cumulative distribution function, Copyright: © 2022 by the authors. as well as hazard function. Here, l is the shape parameter and q is the scale parameter. Licensee MDPI, Basel, Switzerland. Besides, they are both positive. This article is an open access article distributed under the terms and q 1 l l x x f (x; q, l) = qle x 1 e , x > 0 (1) conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ F(x; q, l) = 1 1 e , x > 0 (2) 4.0/). Entropy 2022, 24, 576. https://doi.org/10.3390/e24050576 https://www.mdpi.com/journal/entropy Entropy 2022, 24, 576 2 of 20 l l x x h(x; q, l) = qle x 1 e , x > 0 (3) The plots of the pdf and hazard function are presented in Figures 1 and 2. =1, =1 0.9 =1, =2 =2, =1 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 2 4 6 8 10 Figure 1. pdf of GIED. 1.4 =1, =1 =1, =2 1.2 =2, =1 0.8 0.6 0.4 0.2 0 2 4 6 8 10 Figure 2. Hazard function of GIED. h f Entropy 2022, 24, 576 3 of 20 1.2. The Joint Progressive Type-II Censoring Scheme It is difficult to obtain the lifetime data of all the products given the cost and time in many practical situations. Hence, testing experiment censoring is of great importance. A great deal of work has been performed on a variety of censoring schemes. The experimental units cannot be withdrawn during the experiment under the type-I censoring scheme and type-II censoring scheme. Reference [9] described the progressive type-II censoring scheme, in which some units are allowed to be withdrawn during the test. Afterward, we describe the progressive type-II censoring briefly. Suppose n units are placed in a lifetime test. k is the effective sample size. It also represents the number of observed failures that satisfies k < n. R , , R stand for the number of units to be withdrawn for each failure time. Furthermore, 1 k they are non-negative integers and satisfy (R + 1) = n. At the first failure time, R i 1 i=1 units are withdrawn from the remaining n 1 surviving units randomly. When the second failure happens, we randomly withdraw R units from the remaining n 2 R surviving 2 1 units. Analogously, when the k-th failure happens, the remaining R surviving units are withdrawn randomly and the test ceases. Reference [10] provided an amount of work about the progressive censoring schemes. Reference [11] dealt with the Bayesian inference on step stress partially accelerated life tests using type-II progressive censored data in the presence of competing failure causes. Much research about progressive censoring schemes for one sample has been per- formed by many scholars. However, there is little research on two samples. Reference [12] first introduced the joint progressive censoring schemes for two samples. It is particularly beneficial to compare the life distribution of different products produced by two different assembly production lines on diverse equipment under the same environmental conditions. The joint progressively type-II censoring (JPC) scheme can be briefly described as follows. Suppose the samples are from two different lines, Line 1 and Line 2. The size of the samples of products in Line 1 is m and in Line 2 is n. Two samples are combined in the joint progressive censoring scheme, and they are placed on a lifetime test. Suppose N = m + n is the size of combined samples. R , , R stand for the number of units to be withdrawn in 1 k each failure time. Additionally, they are non-negative integers and satisfy (R + 1) = n. i=1 At the first point of failure w , R units are removed from the combined samples at random. 1 1 R units consist of s units from Line 1 and t units from Line 2. On the second failure 1 1 1 w , R units are withdrawn from the remaining m + n 2 R samples at random. R 2 2 1 2 units consist of s units from Line 1 and t units from Line 2. Analogously, at the k-th 2 2 failure time point w , the remaining R surviving units are withdrawn randomly and the k k test ceases. Let z , , z be random variables. If the i-th failure is from Line 1, let z = 1. 1 k i Otherwise, let z = 0. Suppose that the censored sample is ((w , z , s ), , (w , z , s )). i 1 1 1 k k k Here, we introduce k = z , which means the number of failures from Line 1. Similarly, 1 i i=1 k = (1 z ) = k k , which stands for the number of failures from Line 2. Figure 3 2 i 1 i=1 shows the scheme. Figure 3. JPC scheme. Reference [12] applied Bayesian estimation techniques to two exponential distribu- tions for the JPC scheme. Reference [13] considered the JPC scheme for more than two Entropy 2022, 24, 576 4 of 20 exponential populations and studied the statistical inference. Reference [14] investigated the conditional maximum likelihood estimations and the interval estimations of the Weibull distribution for the JPC scheme. Reference [15] discussed the point estimation and obtained the confidence intervals of two Weibull populations for the JPC scheme. Reference [16] obtained the Bayes estimation when data were sampled in the JPC scheme from a general class of distributions. Reference [17] studied the expected number of failures in the lifetime tests under the JPC scheme for various distributions. Besides, a new type-II progressive censoring scheme for two groups was introduced by [18]. Reference [19] extended the JPC scheme for multiple exponential populations and studied the statistical inference. Reference [20] studied the likelihood and Bayesian inference when data were sampled in the JPC scheme from the GIED. A few scholars have studied the statistical inference of the generalized inverted ex- ponential distribution when the test units are progressively type-II censored. However, no one has studied the statistical inference of the generalized inverted exponential dis- tribution under joint progressively type-II censoring. The research on this aspect is of great significance. In this article, we provide statistical inference and study the JPC scheme for two groups that follow the GIED with the same scale parameter. The expectation maximization algorithm is adopted to calculate the maximum likelihood estimates of the parameters. In light of the missing value principle, we derive the observed information matrix. We obtain the interval estimations by the bootstrap-p method based on the Fisher information matrix. We assume a Gamma prior for the shape and scale parameters. The Bayes estimates and credible intervals for the informative prior and the non-informative prior under the linex loss function and squared error loss function are calculated by adopting the importance sampling technique. The performances of various methods are compared through the Monte Carlo simulation. Besides, we conduct real data analysis. Moreover, in many practical cases, the experimenters may know that the lifetime of various populations is orderly. We investigate the problem that the parameters have order restrictions. We discuss the maximum likelihood estimation and Bayesian inference of the parameters. The rest of the article is arranged as follows. In Section 2, we obtain the likelihood function. In Section 3, we apply the EM algorithm to calculate the MLEs. In Section 4, we compute the observed information matrix based on the missing value principle. Next, we adopt the bootstrap method to obtain the confidence intervals. The Bayesian inference is presented in Section 5. In Section 6, a Monte Carlo simulation and real data analysis are shown. In Section 7, we derive the maximum likelihood estimation and Bayesian inference of the parameters that have order restrictions. 2. Likelihood Function Generate lifetime X , X , , X from the GIED with the progressive type-II 2:m:n 1:m:n i:m:n censoring scheme (R , , R ). The observed data are x = (x , x , , x ). The likelihood 1 k 1 2 m function is L(q, l) = C f (x )(1 F(x )) (4) Õ i i i=1 where x < x < < x . 1 2 m Substitute (2) and (1) for (4). Then, we can obtain the observed likelihood function: " # q 1 q m m m 1 l l m 2 i=1 x x x i i i L(q, l) = C(ql) e x 1 e 1 e (5) Õ i Õ i=1 i=1 Entropy 2022, 24, 576 5 of 20 Take the derivative of (5) to obtain the log-likelihood function. m m m l(q, l) µ m ln q + m ln l l + ln x + (q 1) ln(1 e ) å å i å i=1 i=1 i=1 + R ln 1 e (6) å i i=1 Suppose X , , X are m items from Line 1 that are i.i.d. GIED(q , l). Y , , Y 1 m 1 1 m are n items from Line 2 that are i.i.d. GIED(q , l). For a given joint progressive type-II censoring scheme (R , , R ), the observed data are ((w , z , s ), , (w , z , s )). Thus, 1 k 1 1 1 k k k the likelihood function without the normalizing constant is z q +(1 z )q 1 k 1 2 i i 1 l k k k 2 w w 1 2 i i L(q , q , ljdata) = q q l w e 1 e 1 2 1 2 Õ i i=1 q s q t 1 i 2 i l l w w i i 1 e 1 e (7) i=1 k k where k = å z , k = å (1 z ) = k k . 1 i 2 i 1 i=1 i=1 When k = 0, k = k, the likelihood function becomes 1 2 q 1 q s q t k k 2 1 i 2 i 1 l l l k 2 w w w w i i i i L(q , q , ljdata) = q l w e 1 e 1 e 1 e 2 2 1 Õ i Õ i=1 i=1 q s q s 1 i 1 i l l w w i i For s = 0, 1 e = 1. For s 6= 0 and a fixed l, 1 e is a strictly de- i i creasing function of q that decreases to 0. Thus, for k = 0, fixed q and l, L(q , q , ljdata) 1 1 2 1 2 is a strictly decreasing function of q . Therefore, there is no maximum likelihood estimate when k or k equals 0. Thus, we assume that k > 0 and k > 0. 1 2 1 2 The log-likelihood function is: l l w w i i ln L(q , q , ljdata) = k ln q + k ln q + k ln l + q s ln 1 e + q t ln 1 e 1 2 1 1 2 2 å 1 i 2 i i=1 + 2 ln w + (z q + (1 z )q 1) ln 1 e (8) 1 2 å i i i i=1 Then, we prove the MLEs of q , q for a given l are unique in the following. 1 2 Theorem 1. For a fixed l > 0, if k > 0 and k > 0, l (q , q ) = ln L(q , q , ljdata) is a 1 2 1 1 2 1 2 unimodal function of (q , q ). 1 2 Proof. Because the Hessian matrix of l (q , q ) is a negative definite matrix, l (q , q ) is a 2 2 1 1 1 1 concave function. Moreover, for fixed q (q ), when q (q ) tends to 0, l (q , q ) tends to ¥. 1 2 2 1 1 1 2 When q (q ) tends to ¥, l (q , q ) tends to ¥. 2 1 1 1 2 ˆ ˆ For a given l, the MLEs of q and q are q (l) and q (l). They can be written as: 1 2 1 2 k k 1 2 ˆ ˆ q (l) = and q (l) = (9) 1 2 M(l) N(l) Entropy 2022, 24, 576 6 of 20 where k k l l w w i i M(l) = z ln 1 e + s ln 1 e å i å i i=1 i=1 k k l l w w i i N(l) = (1 z ) ln 1 e + t ln 1 e (10) i i å å i=1 i=1 When l is unknown, the profile log-likelihood function of l is ˆ ˆ p (l) = ln L(q , q , ljdata) 1 1 2 = k ln M(l) k ln N(l) + k ln l + ln 1 e (11) 1 2 å i=1 Maximize (11) to obtain the MLEs of l. Then, we prove that the MLE of l exists and is unique in the following. Theorem 2. If k > 0 and k > 0, p (l) is a unimodal function of l. 1 1 Proof. The proof is given in Appendix A. Thus, we can obtain that for k > 0 and k > 0, the MLEs of (q , q , l) are unique 1 2 1 2 from Theorems 1 and 2. Next, take the partial derivatives of Equation (8) and let them equal 0. Therefore, we can calculate the MLEs of the parameters. However, the equation is nonlinear, which is cumbersome to compute the solution directly. Therefore, we propose to calculate the MLEs of (q , q , l) through the EM algorithm. 1 2 3. Expectation Maximization Algorithm The expectation maximization algorithm is an iterative optimization strategy based on the MLEs of the parameters. First, we introduce the potential data. The potential data can be interpreted as the data that do not have the missing variables. If we add extra variables, it becomes simpler to process. The potential data are the lifetime of the censored samples at each failure point time. It is assumed that at the i-th failure time point w , U is the lifetime of the j-th cen- i i j sored sample from Line 1, V 0 is the lifetime of the j -th censored sample from Line 2 for i j j = 1, , s , j = 1, , t , and i = 1, , k. The observed data are ((w , z , s ), , i i 1 1 1 (w , z , s )). The potential data are U = ((u , , u ), , (u , , u )) and k k k 11 1s k1 ks V = ((v , , v ), , (v , , v )). Therefore, the complete data are ((w , z , s ), , 11 1t 1 1 1 k1 kt 1 k (w , z , s ), U, V) = data , which are the combination of the observed data and the potential k k k data. The log-likelihood function based on the complete data is 0 1 s t k k i i @ A ln L(q , q , ljdata ) = m ln q + n ln q + (m + n) ln l 2 ln w + ln u + ln v 0 1 2 1 2 å i å i j å å i j i=1 j=1 i=1 j =1 0 1 k s t k s i i i l 1 1 1 i j @ A l + + + (q 1) ln 1 e (12) å å å å å w u v 0 i i j 0 i=1 j=1 i=1 j=1 i j j =1 k t k i j +(q 1) ln 1 e + (z q + (1 z )q 1) ln 1 e 2 i 1 i 2 å å å i=1 i=1 j =1 In the “E”-step, the pseudo log-likelihood function is given by Entropy 2022, 24, 576 7 of 20 l (q , q , l) = m ln q + n ln q + (m + n) ln l 2 [s E(ln U jU > w ) + t E(ln VjV > w )] s 1 2 1 2 å i i i i i i i i i=1 " # k k 1 1 1 2 ln w l + s E( ) + t E( jV > w ) i i i i i å å w u V i i j i i=1 i=1 k k l l U V i i +(q 1) s E ln(1 e jU > w ) + (q 1) t E ln(1 e jV > w ) 1 å i i i 2 å i i i i=1 i=1 + (z q + (1 z )q 1) ln 1 e (13) å i 1 i 2 i=1 The conditional pdfs of U and V 0 can be expressed, respectively, as i j i j f (u , q , l) G I ED i j 1 f (u jw ) = i j i U jW i j i 1 F (w , q , l) G I ED i 1 f (v 0 , q , l) G I ED 2 i j f (v 0jw ) = V jW i i i j 1 F (w , q , l) i j G I ED i 2 where i = 1, , k. The expectations associated with the functions of U can be written as follows: i j q 1 l l Z u u i j i j q lu ln u e (1 e ) 1 i j i j E(ln U jU > w ) = du , i j i j i i j i j (1 e ) q 1 l l 1 u u i j i j q lu e (1 e ) 1 i j E( jU > w ) = du , i j i i j l 1 i j i j (1 e ) q 1 l l l 1 Z u u u l i j i j i j q lu ln(1 e e (1 e ) 1 i j i j E ln(1 e )jU > w = du . i j i i j l 1 i j (1 e ) Similarly, we can obtain the expectations associated with the functions of V 0 . i j In the “M”-step, the estimates of q , q , and l can be calculated by maximizing the pseudo log-likelihood function with finite iterations. q and q with respect to l can be 1 2 calculated by taking partial derivatives of (13), as follows: q (l) = q = (14) 1 1 l l k k U w i i å s E ln(1 e jU > w ) + å z ln(1 e ) i i i i i=1 i=1 q (l) = q = (15) 2 2 l l k k V w i i t E ln(1 e jV > w ) + (1 z ) ln(1 e ) å å i i i i i=1 i=1 Use the equations above to rewrite (13) as the function only for l. Therefore, the three- dimensional optimization problem is transformed into a one-dimensional problem. At the (r) (r) (r 1) (r 1) (r) r-th iteration, (q , q , l ) denotes the estimate of (q , q , l). For fixed (q , q ), 1 2 1 2 1 2 (r) (r 1) (r 1) (r) (r) (r) maximize l (q , q , l) to obtain l . For fixed (q , q , l ), q , q can be s 1 2 1 2 1 2 obtained as follow: Entropy 2022, 24, 576 8 of 20 (r) q = (16) (q) k k U w i i s E ln(1 e jU > w ) + z ln(1 e ) å å i (r 1) (r 1) (r) i i i i=1 i=1 (q ,q ,l ) 1 2 (r) q = (17) (q) l l k k V w i i t E ln(1 e jV > w ) + (1 z ) ln(1 e ) å (r 1) (r 1) å i (r) i i i i=1 i=1 (q ,q ,l ) 1 2 (r) (r 1) (r) (r 1) We stop the iterations when jq q j 0.0001, jq q j 0.0001, 1 1 2 2 (r) (r 1) (r) (r) (r) jl l j 0.0001. Therefore, the MLEs of q , q , l are q , q , l . 2 2 1 1 4. Confidence Interval Estimation Our interest in this section is to obtain the observed information matrix based on the missing value principle. Then, we compute the interval estimations using the bootstrap-p method. 4.1. Observed Fisher Information Matrix Based on the idea of [21], we have 0 1 s t k k i i @ A I (q , q , l) = m I (q , q , l) + n I (q , q , l) I (q , q , l) + I (q , q , l) 1 2 1 1 2 2 1 2 å å u jw 1 2 å å v jw 1 2 i j i i i j i=1 j=1 i=1 j =1 and I represents the observed information matrix. 2 3 2 3 (1) (1) (1) (2) (2) (2) I I I I I I 11 12 13 11 12 13 6 7 6 7 (1) (1) (1) (2) (2) (2) I (q , q , l) = I I I , I (q , q , l) = I I I 1 1 2 4 5 2 1 2 4 5 21 22 23 21 22 23 (1) (1) (1) (2) (2) (2) I I I I I I 31 32 33 31 32 33 where ¶ ln f (x, q , l) G I ED 1 (1) (1) (1) (1) (1) (1) I = E( ), I = I = I = I = I = 0 11 12 21 22 23 32 ¶q 2 2 ¶ ln f (x, q , l) ¶ ln f (x, q , l) (1) (1) G I ED 1 (1) G I ED 1 I = I = E( ), I = E( ) 13 31 33 ¶q ¶l ¶l ¶ ln f (x, q , l) G I ED 2 (2) (2) (2) (2) (2) (2) I = I = I = I = I = 0, I = E( ) 11 12 13 21 31 22 ¶q 2 2 ¶ ln f (x, q , l) ¶ ln f (x, q , l) 2 2 (2) (2) G I ED (2) G I ED I = I = E( ), I = E( ). 23 32 33 ¶q ¶l ¶l The missing observed matrices are 2 3 2 3 (3) (3) (3) (4) (4) (4) I I I I I I 11 12 13 11 12 13 6 7 6 7 (3) (3) (3) (4) (4) (4) I (q , q , l) = , I (q , q , l) = 4 I I I 5 4 I I I 5 1 2 1 2 U jW 21 22 23 V jW 21 22 23 i j i 0 i i j (3) (3) (3) (4) (4) (4) I I I I I I 31 32 33 31 32 33 where ¶ ln f (u jw ) U jW i j i (3) i1 i (3) (3) (3) (3) (3) I = E( ), I = I = I = I = I = 0 11 12 21 22 23 32 ¶q 2 2 ¶ ln f (u jw ) ¶ ln f (u jw ) i j i i j i U jW U jW (3) (3) i1 i (3) i1 i I = I = E( ), I = E( ) 13 31 33 ¶q ¶l ¶l 1 Entropy 2022, 24, 576 9 of 20 ¶ ln f (v jw ) i j i V jW (4) (4) (4) (4) (4) (4) i1 i I = I = I = I = I = 0, I = E( ) 11 12 13 21 31 22 ¶q 2 2 ¶ ln f (v jw ) ¶ ln f (v jw ) i j i i j i V jW V jW (4) (4) i1 i (4) i1 i I = I = E( ), I = E( ). 23 32 33 ¶q ¶l ¶l All the expectations expressions are given in Appendix B. For every fixed (q , q , l), the 1 2 covariance matrix of the estimators is the inverse matrix of the observed information matrix. 4.2. Bootstrap-p Method The asymptotic confidence interval methods are based on the law of large numbers. In many practical cases, the sample size tends to be not enough. Thus, these methods have limitations about the small sample size. Reference [22] introduced the bootstrap method to construct the confidence interval. Therefore, we suggest the percentile bootstrap method to study the parametric bootstrap confidence intervals. The steps for estimating the confidence intervals are briefly summarized as follows: ˆ ˆ ˆ Step 1: Compute the MLEs of q , q and l from the joint progressively type-II censored (1) (2) samples. Step 2: Utilize the same censoring scheme and generate the joint progressively type-II bootstrap censored samples x , x , , x . 11 21 n1 (1) (1) (1) ˆ ˆ ˆ Step 3: Calculate new MLEs of q , q , and l, say q , q , and l , from the bootstrap 1 2 1 2 samples. Step 4: Repeat Step 2 and Step 3 until running B times to obtain a sequence of bootstrap estimates. (1) (2) (B) (1) (2) (B) (1) (2) (B) ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ Step 5: Sort (q , q , , q ), (q , q , , q ), and (l , l , , l ) in 1 1 1 2 2 2 1 2 B ascending order, respectively. ˆ ˆ ˆ ˆ ˆ ˆ Then, we obtain (q , q , , q ), (q , q , , q ), and 1(1) 1(2) 1(B) 2(1) 2(2) 2(B) ˆ ˆ ˆ (l , l , , l ). (1) (2) (B) Step 6: The 100(1 z)% bootstrap-p confidence intervals of q , q , and l are ˆ ˆ ˆ ˆ ˆ ˆ q , q , q , q and l , l (18) 1([B(z /2)]) 1([B(1 z /2)]) 2([B(z /2)]) 2([B(1 z /2)]) ([B(z /2)]) ([B(1 z /2)]) 5. Bayes Estimation Different from traditional statistics, Bayes estimation considers the prior information about life parameters. Thus, Bayesian estimation thinks about both the data provided and the prior probability to infer the interested parameters. It makes the inference of the interested parameters more objective and reasonable. 5.1. Bayes Estimation Suppose that the unknown parameters q , q , and l have Gamma prior distribu- 1 2 tions independently. a 1 b q 1 1 1 p (q ) = q e , q > 0; a , b > 0 (19) 1 1 1 1 1 1 G(a ) a 1 b q 2 2 2 p (q ) = q e , q > 0; a , b > 0 (20) 2 2 2 2 2 2 G(a ) c 1 dl p (l) = l e , l > 0; c, d > 0 (21) G(c) where a , a , b , b , c, d are the hyper-parameters that contain the prior information. 1 2 1 2 Entropy 2022, 24, 576 10 of 20 Thus, the joint prior possibility distribution can be written as a 1 a 1 c 1 b q b q dl 1 2 1 1 2 2 p (q , q , l) µ q q l e e e (22) 0 1 2 1 2 The joint posterior probability distribution is L(q , q , l, data) 1 2 p(q , q , l j data) = R R R 1 2 ¥ ¥ ¥ p (q , q , l)L(q , q , l j data)dq dq dl 0 1 2 1 2 1 2 0 0 0 p (q , q , l)L(q , q , l j data) 0 1 2 1 2 = R R R (23) ¥ ¥ ¥ p (q , q , l)L(q , q , l j data)dq dq dl 0 1 2 1 2 1 2 0 0 0 The denominator of p(q , q , l j data) is a function of the observed data. Thus, 1 2 L(q , q , l, data) and p(q , q , l j data) have a coefficient proportional relationship. There- 1 2 1 2 fore, the joint posterior probability distribution is p(q , q , l j data) µ L(q , q , l, data) = p (q , q , l)L(q , q , l j data) 1 2 1 2 0 1 2 1 2 l l k w k w i i b (z +s ) ln(1 e ) q b (1 z +t ) ln(1 e ) q å å 1 i i 1 2 i i 2 i=1 i=1 a +k 1 a +k 1 1 1 2 2 µ q e q e 1 2 k 1 d+ l k+c 1 i=1 w l e (24) (1 e ) i=1 We rewrite (24) as follows: p(q , q , l j data) µ p(q j l, data) p(q j l, data) p(l j data) d(q , q , l) 1 2 1 2 1 2 where p(q j l, data) Ga a + k , b (z + s ) ln(1 e ) 1 1 1 1 å i i i=1 p(q j l, data) Ga a + k , b (1 z + t ) ln(1 e ) 2 2 2 2 å i i i=1 p(l j data) Ga c + k, d + i=1 " # (a +k ) 1 1 d(q , q , l) µ b (z + s ) ln(1 e ) 1 2 1 i i i i=1 Õ (1 e ) i=1 " # (a +k ) 2 2 b (1 z + t ) ln(1 e ) 2 å i i i=1 5.2. Loss Functions In Bayesian statistics, the Bayesian estimation of a function f(q , q , l) is derived on a 1 2 prescribed loss function. Thus, it is critical to select the loss function: Squared error loss function (SEL) The SEL function is given by L (w, w ˆ ) = (w ˆ w) (25) s Entropy 2022, 24, 576 11 of 20 Here, w ˆ is an estimate of w. The corresponding Bayes estimate w ˆ of w can be obtained from w ˆ = E[w j x] (26) Thus, f(q , q , l) represents the Bayes estimation of f(q , q , l) under the SEL func- 1 2 1 2 tion, that is, Z Z Z ¥ ¥ ¥ f(q , q , l) = f(q , q , l)p(q , q , l j data)dq dq dl (27) 1 2 1 2 1 2 1 2 0 0 0 Linex loss function (LL) The LL function is the most universally used asymmetric loss function. The asym- metric loss function is considered more comprehensive in many respects. The linex loss function is given below: h(w ˆ