Bayesian inference of dynamic cumulative residual entropy from Pareto II distribution with application to COVID-19

: Dynamic cumulative residual entropy is a recent measure of uncertainty which plays a substantial role in reliability and survival studies. This article comes up with Bayesian estimation of the dynamic cumulative residual entropy of Pareto II distribution in case of non-informative and informative priors. The Bayesian estimator and the corresponding credible interval are obtained under squared error, linear exponential (LINEX) and precautionary loss functions. The Metropolis-Hastings algorithm is employed to generate Markov chain Monte Carlo samples from the posterior distribution. A simulation study is done to implement and compare the accuracy of considered estimates in terms of their relative absolute bias, estimated risk and the width of credible intervals. Regarding the outputs of simulation study, Bayesian estimate of dynamic cumulative residual entropy under LINEX loss function is preferable than the other estimates in most of situations. Further, the estimated risks of dynamic cumulative residual entropy decrease as the value of estimated entropy decreases. Eventually, inferential procedure developed in this paper is illustrated via a real data .


Introduction
The primary measure of the uncertainty contained in random variable X is the Shannon entropy [1]. It plays an undeniable essential role in the field of probability and statistics, financial analysis, engineering, and information theory. Now, there are considerable literatures assigned to the applications, generalizations and properties of Shannon's measure of entropy. Özç am [2] studied an econometric procedure which revises and updates the technical production coefficients of latest Turkish input/output table, as new information about sectorial productions become available. Genç ay and Gradojevic [3] provided a comparative analysis of stock market dynamics of the 1987 and 2008 financial crises and discussed the extent to which risk management measures based on entropy which can be successful in predicting aggregate market expectations. Rashidi et al. [4] discussed and simulated the heat transfer flow (using entropy generation) in solar still. Zhang et al. [5] discussed a network entropy method to measure connectivity uncertainty of functional connectivity graphs of the brain sequences. The predictability of Brazilian agricultural commodity prices during the period after food crisis using information theory has been studied by De Araujo et al. [6]. Different types of entropy measures have been discussed by many researchers (see for examples Shakhatreh et al. [7] and Klein and Doll [8]).
Let X be a non-negative random variable, the Shannon entropy, say H(X), of the probability density function density (PDF) is defined by In recent times, inference problems associated with entropy measures are of interest to several researches. An estimator of the entropy from the generalized half-logistic distribution using upper record value was obtained by Seo et al. [9]. The Bayesian estimators of entropy from Weibull distribution based on generalized progressive hybrid censoring scheme were studied by Cho et al. [10]. Estimation of entropy from generalized exponential distribution via record values was discussed by Chacko and Asha [11]. The maximum likelihood (ML) estimator of Shannon entropy from inverse Weibull distribution was obtained by Hassan and Zaky [12] using multiple censored data. Bayesian estimator of entropy for Lomax distribution was provided by Hassan and Zaky [13] via upper record values. In recent years, measurement of uncertainty for probability distributions became more interested. The entropy for residual lifetime X t = (X −t | X>t) was defined by Ebrahimi [14] as a dynamic form of uncertainty called the residual entropy at time t and defined as is the survival function. More recently, Rao et al. [15] proposed an alternative measure of uncertainty known as the cumulative residual entropy (CRE), denoted by ( )

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The CRE has the benefits: (i) it possess consistent definitions for the continuous and discrete domains, (ii) it forever non-negative and (iii) it is straightforwardly determined from sample data and these computations asymptotically converge to the true values.
Another measure of uncertainty deals with residual lifetime function is dynamic cumulative residual entropy (DCRE) which can be attractive in many fields like reliability and survival analysis. The entropy for residual lifetime X t as a dynamic form of uncertainty was defined by Asadi and Zohrev and [16] as follows It can be noted that, at t = 0, the DCRE tends to CRE. The Bayesian estimators of DCRE for the Pareto distribution were studied by Renjini et al. [17] using type II right censored data. Renjini et al. [18] considered the DCRE Bayesian estimators for Pareto distribution via upper record values. The ML and Bayesian estimate of the entropy of inverse Weibull distribution based on generalized progressive hybrid censoring scheme were discussed by Lee [19]. Renjini et al. [20] provided Bayesian estimator of DCRE for Pareto distribution from complete data.
Pareto II (Lomax) distribution was originally developed by Lomax [21] to model business failure data. This distribution has extensive applications in many fields such as income and wealth inequality, firm size and queuing problems, computer science, risk analysis and economics, actuarial science and reliability, the reader can refer to [22][23][24][25][26][27][28][29][30]. The cumulative distribution function (CDF) and the PDF of Pareto II distribution with shape parameter  and scale parameter  are defined by ( 1) and, The DCRE for Pareto II distribution can be obtained by substituting (3) in (1) as follows: Using integration by parts, the DCRE of Pareto II distribution will be as follows: This is the required expression of the DCRE for Pareto II distribution which it is a function of  and .

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From the previous literatures, it can see that the Pareto II distribution take attention from theoretical and statisticians basically due to its use in multiple areas. In addition to, the DCRE has found nice interpretations and applications in the fields of reliability and survival analysis. Recently, statistical inference for the DCRE for lifetime distributions attracted appreciable attention. This motivates us to propose the estimation of the DCRE for Pareto II distribution in view of Bayesian procedure. The Bayesian estimator is obtained using non-informative prior (NIP) and informative prior (IP). The considered loss functions are squared error (SE), LINEX and precautionary (PRE). Markov Chain Monte Carlo (MCMC) technique is utilized due to the complicated forms of DCRE Bayesian estimator. Application to COVID 19 data in Egypt appeared that these data contain more information which is useful in mathematical and statistical purposes.
The form of the article is as follows. The next section presents Bayesian estimator of DCRE for Pareto II distribution under NIP for the considered loss functions. Section 3 gives Bayesian estimators of DCRE for Pareto II distribution using proposed loss functions under IP. Section 4 provides simulation issue and application to real data. The paper ends with summary of this work.

Bayesian estimation of DCRE under NIP
are obtained as posterior mean as follows:  (11) and, Integrals (7)- (12) do not have a closed form, therefore Metropolis-Hastings (M-H) and random-walk Metropolis algorithms are employed to generate MCMC samples from posterior density functions (6). After getting MCMC samples from the posterior distribution, we can find the Bayes estimate for the parameters. The M-H algorithm is described as follows: Step 1: Let g(.) be the density of Pareto II distribution.
Step 2: Initialize a starting value x 0 and determine the number of samples N.
Step 3: For i = 2 to N set x = x i-1 .
Step 5: If Step 6: Set i = i + 1 and return to step 2 and repeat the previous steps N times.
Hence ; ) PRE Xt  are obtained, by using Equation (5). Furthermore, BCI is a useful summary of the posterior distribution which reflects its variation that is used to quantify the statistical uncertainty. The Bayesian analogy of a confidence interval is called a credible interval. A credible interval of entropy is the probability that a real value of entropy will fall between an upper and lower bounds of a probability distribution. Therefore, using the same algorithm introduced by Chen and Shao [31], we obtain an approximate highest posterior density interval for ( ; ). Xt 

Bayesian estimation of DCRE under IP
In this section, we obtain the Bayesian estimator of DCRE under SE, LINEX and PRE loss functions by considering the prior of parameters  and  has a gamma distribution. Additionally, the BCI estimators are constructed. Following [32], assuming that the prior of  and  denoted by 3 ()  and 4 ( ),  has a gamma distribution with parameters (a 1 , b 1 ) and (a 2 , b 2 ) respectively.
where a j and b j , j = 1,2 are known and non-negative, so, the joint posterior for parameters, denoted by * So, the marginal posterior PDF of  and  are given respectively by:  PRE are obtained as follows: Similarly, the Bayesian estimators of  under, SE, LINEX and PRE loss functions are obtained. As mentioned in previous section, MCMC technique is used to approximate the integral Equations (13). The M-H algorithm will be implemented to compute the BE as well as BCI width under proposed loss functions. Hence, the BE of ( ; ) Xt  under different loss functions is obtained based on Equation (5).

Simulation and application
Here, performance of the different estimates is examined and a real data set is provided to illustrate the theoretical results.  Tables 1-6 give simulation results of the DCRE in case of the NIP. Also, numerical outcomes are represented in Figures 1-4. So, we conclude the following about the behavior of the DCRE estimates.  Table 1).     Table 2).

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The ER of DCRE under LINEX at  = 2 gets the smallest values at n = 10, 30 and 70. For n = 50, 70 and 100, the width of BCI for DCRE under SE is the shortest compared to the width of BCI in case of PRE and LINEX loss functions (see Table 3). The ERs for DCRE estimates get the smallest values at ( ; ) 2.2222. Xt   for all n (see for example Figures 3 and 4).  Table 4).

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The ER of DCRE under SE takes the smallest values for all n. The width of BCI of DCRE under LINEX at  = 2 is the shortest compared to the width of BCI in case of PRE and SE loss functions for all n except at n = 100 (see Table 5).

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The ER of DCRE under PRE takes the smallest values for all n except at n = 100. The width of BCI for DCRE under LINEX at  = −2 is the shortest compared to the width of BCI in case of PRE and SE loss functions for all n except at n = 10 (see Table 6).         Tables 7-12 give simulation results of DCRE under IP. Also, numerical outcomes are illustrated through Figures 5-8. So, we conclude the following observations about the behavior of the entropy estimates.    Table 7). The ER of DCRE under SE gets the smallest values for most n. The width of BCI for DCRE under PRE is the shortest compared to the width of BCI in case of LINEX and SE loss functions for all n (see Table 8).  Table 9). The ERs for DCRE estimates get    Table 10). for all values of n (see Table 11). From

4.4.Application to real data
Here, the real data sets can be used to illustrate the method proposed in previous sections. The validity of the fitted model has been checked by Abd-El-Monsef et al. [33]. Regarding this data, the Bayesian estimates of DCRE under SE, LINEX and PRE loss functions are obtained and listed in Table 13. The estimated entropy of total deaths is very low;this indicates that these data containing more information that can be useful in mathematical and statistical purposes. So,it's better to study the entropy forthe detailed information about the daily death in Egypt for COVID-19 in future researches.

Data 2:
The real data set was obtained from a meteorological study by Simpson [34] which represent the radar-evaluated rainfalls from single Florida cumulus clouds (from 1968 to 1970) from 52 south Florida cumulus clouds, 26 seeded clouds, and 26 control clouds. The validity of the fitted model has been checked by [32]. The Kolmogorov-Smirnov goodness of fit test is employed for real data and its p value indicates that the Pareto II distribution fits the data. The data are recorded as follows 129. 6   As anticipated, from this example that the estimates of DCRE are increasing function on time as the time t increases under gamma and uniform priors for the proposed loss functions. Based on ER, the Bayes estimate of DCRE, under LINEX loss function is suitable than the other estimates.

Summary and conclusion
The Bayesian estimation of dynamic cumulative residual entropy is considered for Pareto II distribution. The Bayesian estimators of DCRE for Pareto II model are obtained in case of non-informative and informative priorsfor symmetric and asymmetric loss functions. The MCMC procedure is employed to compute the Bayes estimates and the BCIs. The behavior of DCRE estimates for Pareto II distribution is evaluated through their relative absolute bias, estimated risk and the width of credible intervals. Application to real data and simulation issues are provided.
According to outcomes of study we conclude that, under NIP, for small true values of DCRE the width of BCIs for estimated values of DCRE under LINEX loss function is smaller than the corresponding based on SE and PRE loss functions for large n at t = 0.5 and 1.5. For large true values of DCRE, at t = 0.5, the width of BCIs for estimated values of DCRE under SE loss function is smaller than the corresponding other loss functions for large n, but the width of BCIs for estimated values of DCRE under LINEX loss function is smaller than the corresponding based on the other loss functions for large sample size at t = 1.5.
Under IP, for small true values of DCRE, the width of BCIs of DCRE under LINEX loss function is smaller than the corresponding based on SE and PRE loss functions for large sample size at t = 0.5 and 1.5. For large true values of DCRE, at t = 0.5, the width of BCIs for estimated values of entropy under LINEX loss function is smaller than the corresponding estimated values based on SE and PRE loss functions for selected large n, but the width of BCIs for estimated values of DCRE under SE loss function is smaller than the corresponding estimated values based on SE and PRE loss functions for large n at t = 1.5.
Generally, the Bayesian estimate of DCRE approaches the true value as n increases. The DCRE values and ERs are directly proportional, that is; if the real value of DCRE decreases, the ERs decrease. As the time increases, the Bayesian estimate of DCRE increases. As n increases, the ER and the width of BCIs decrease. Bayesian estimates under LINEX loss function are more suitable than other loss functions for different types of prior functions in most of situations.