Recently, extropy has gained interest by the academic researchers. This work explores the features of parametric and non-parametric estimators based on upper record values under the a two-parameter Weibull distribution. We apply the Markov Chain Monte Carlo (MCMC) method to provide a Bayesian estimator. A considerable number of theoretical properties of the procedures are determined

Zouaoui, Nour El Houda

English

UFV Journals (Universidade Federal de Viçosa)

The Journal of Engineering and Exact Sciences – jCEC, Vol. 09 N. 12 (2023) journal homepage: https://periodicos.ufv.br/jcec  eISSN: 2527-1075 ISSN: 2446-9416 1 Extropy estimation of Weibull distribution under upper records Article Info: Article history: Received 2023-11-05 / Accepted 2023-12-12 / Available online 2023-12-12 doi:  10.18540/jcecvl9iss12pp19490   Nour El Houda Zouaoui  ORCID: https://orcid.org/0000-0003-3224-8273  Laboratory of Applied Mathematics, Mathematics Department, Biskra University, Algeria E-mail: nourelhouda.zouaoui@univ-biskra.dz, nor9180327@ju.edu.jo  Abstract  Recently, extropy has gained interest by the academic researchers. This work explores the features of parametric and non-parametric estimators based on upper record values under the a two-parameter Weibull distribution. We apply the Markov Chain Monte Carlo (MCMC) method to provide a Bayesian estimator. A considerable number of theoretical properties of the procedures are determined. Keywords: Extropy. Upper record. Bayesian estimation  1. Introduction  In reliability theory, the concept of uncertainty plays a pivotal role in our understanding of data and decision-making processes. Uncertainty, often quantified by entropy, see Shannon (1984), measures the degree to which outcomes are unpredictable or unknown. In the other hand, an alternative measurement for uncertainty was suggested by Lad et al. (2015) named extropy. offering a different perspective on the information contained within a system. For a continuous random variable (rv) 𝑋 with with probability density function (pdf) 𝑓 (𝑥) and cumulative distribution function (cdf) 𝐹(𝑥),  the extropy is defined by  𝐽(𝑋) = −12∫ 𝑓2(𝑥)𝑑𝑥                                                                                                                    (1)    These measurements are highly regarded in the framework of order and record statistics. In this article, we adopt the concept of record values which is important in a variety of practical applications, ranging from reliability engineering to environmental research. Understanding the behavior and characteristics of these records can provide useful information about the underlying distribution and its properties. readers may refer to works such as Arnold et al. (1998). Let 𝑋1, … , 𝑋𝑛 be a sequence of independent and identically distributed (iid) rv with pdf 𝑓 (𝑥) and cdf  𝐹(𝑥)  . We say an upper record value is an observation 𝑋𝑗 that outperforms all prior observations in a series of random variables  𝑋𝑗 > 𝑋𝑖, Ɐ 𝑖 < 𝑗 (Ahsanullah, 1995). Assume 𝐷𝑛 = 𝑈1, … , 𝑈𝑛  be the first n upper record values. The joint density function for 𝐷𝑛 is as follows:  𝑓(𝑢1, … , 𝑢𝑛) = 𝑓(𝑢𝑛)∏𝑓(𝑢𝑖)1−𝑓(𝑢𝑖)𝑛−1𝑖=1                                                                                         (2)  The estimation of extropy has recently attracted attention from several researchers. We may refer to Qiu et al. (2017), who provided some estimators for extropy with applications in testing uniformity. On the other hand, the problem of estimating the extropy based on complete sample has been considered recently by some authors, see for example, Qiu et al. (2018). Characterization results are also given. Additionally, Zouaoui et al. (2022) investigated the Evaluation of the The Journal of Engineering and Exact Sciences – jCEC 2 uncertainty increments for the records and they also provided related characterization results. For more statistical inference see Jose et al. (2019) and Ahmed et al. (2023). This paper aims to explore the extropy of record values within the context of the Weibull distribution. We seek to gain deeper insights into the estimation of extropy for the Weibull distribution where it is noted for its adaptability and application, it is commonly used to describe life-cycle data and dependability problems, see Baratpour et al. (2007), Chacko et al. (2021) and Murthy et al. (2004).  The pdf of Weibull distribution is defined respectively as follows:  𝑓(𝑥) = 𝑏𝜆𝑥𝜆−1𝑒−𝑏𝑥𝜆                                                                                                                   (3)  Understanding the extropy associated with Weibull-distributed records can enhance our ability to model and predict phenomena in various fields, from engineering to environmental studies.  2. Maximum likelihood estimation The MLE method can be used to explore the range of possible distributions and parameters. it seeks to identify model parameter values that maximize the probability function over the parameter space. The maximum likelihood method is a widely used statistical inference technique that may be used to a variety of distributions and models. The Fisher information matrix (FIM) can be used to calculate confidence intervals (CIs) due to its asymptotic features.  1.1 Likelihood Equations    Let 𝐷𝑛 = 𝑈1, … , 𝑈𝑛 be  the first n upper record values from Weibull distribution 𝑊(𝑏, 𝜆). from (1), the likelihood function, say, 𝐿(𝑏, 𝜆; 𝑢), can be presented as  𝐿(𝑏, 𝜆; 𝑢) = 𝑒−𝑏𝑢𝑛𝜆∏ 𝑏𝜆 𝑢𝑖𝜆−1𝑛𝑖=1                            (4)  The log-likelihood function is given by:  𝑙(𝑏, 𝜆; 𝑢) = −𝑏𝑢𝑛𝜆 + ∑log 𝑏 + log  𝜆 + (𝜆 − 1) log 𝑢𝑖𝑛𝑖=1                  =  −𝑏𝑢𝑛𝜆 + 𝑛 log 𝑏 + 𝑛 log  𝜆 + ∑ (𝜆 − 1) log 𝑢𝑖𝑛𝑖=1                                                           (5)  The partial derivatives of 𝑙(𝑏, 𝜆; 𝑢) for 𝑏, 𝜆 is derived respectively as  𝜕𝑙(𝑏,𝜆;𝑢)𝜕𝑏= −𝑢𝑛𝜆 +𝑛𝑏                                                                                                                           (6) 𝜕𝑙(𝑏,𝜆;𝑢)𝜕𝜆= −𝑏𝜆𝑢𝑛𝜆−1 +𝑛𝜆+ ∑ log 𝑢𝑖𝑛𝑖=1                                                                                              (7)  Now, to get the  MLEs of 𝑏  and 𝜆 we set the equations (6) and (7) to zero. Therefore, the MLE of 𝑏 is given by    ?̂?𝑀𝐿 =𝑛𝑢𝑛?̂?                                                                                                                                         (8)  For ?̂?𝑀𝐿 cannot be derived with an explicit form. Therefore, we need to solve the nonlinear equation (7) numerically. One of the most used methods is the Newton–Raphson (N–R) method. Using the invariant property, the MLE of 𝐽𝑋 is :  The Journal of Engineering and Exact Sciences – jCEC 3 𝐽𝑋 = −?̂?𝑀𝐿−2?̂?𝑀𝐿+3 ?̂?𝑀𝐿22?̂?𝑀𝐿2 𝛤(2?̂?𝑀𝐿 − 1)                                                                                         (9)  1.2 Asymptotic Confidence Intervals for MLEs For more accuracy of MLEs, we use the asymptotic variance of MLE to determine the ACIs of 𝑏  and 𝜆. Let 𝐼(𝛷) be the fisher information matrix where 𝛷 = (𝑏, 𝜆 ), the FIM can be given as follows   𝐼(𝛷) = [−𝜕2𝑙(𝑏,𝜆;𝑢)𝜕𝑏2−𝜕2𝑙(𝑏,𝜆;𝑢)𝜕𝑏𝜕𝜆−𝜕2𝑙(𝑏,𝜆;𝑢)𝜕𝑏𝜕𝜆−𝜕2𝑙(𝑏,𝜆;𝑢)𝜕𝜆2]                                                                                                (10)   Thus,   𝐼(𝛷) = [𝑛𝑏2𝜆𝑢𝑛𝜆−1𝜆𝑢𝑛𝜆−1 𝑏𝑢𝑛𝜆−1 + 𝑏𝜆(𝜆 − 1)𝑢𝑛𝜆−2 +𝑛𝜆2]                                                                         (11)                             To find the 𝑉𝑎𝑟(?̂?𝑀𝐿) and 𝑉𝑎𝑟(?̂?𝑀𝐿), we should calculate the inverse of FIM of the MLEs under the asymptotic property. Thus:  𝐼(?̂?) = [𝑛?̂?𝑀𝐿2 ?̂?𝑀𝐿𝑢𝑛?̂?𝑀𝐿−1?̂?𝑀𝐿𝑢𝑛?̂?𝑀𝐿−1 ?̂?𝑀𝐿𝑢𝑛?̂?𝑀𝐿−1 + ?̂?𝑀𝐿?̂?𝑀𝐿(?̂?𝑀𝐿 − 1)𝑢𝑛?̂?𝑀𝐿−2 +𝑛?̂?𝑀𝐿2]                                    (12)  Where, ?̂?  is the estimate of 𝛷. Thus   𝐼(?̂?)−1=1det (𝐼(?̂?))[    ?̂?𝑀𝐿𝑢𝑛?̂?𝑀𝐿−1 + ?̂?𝑀𝐿?̂?𝑀𝐿(?̂?𝑀𝐿 − 1)𝑢𝑛?̂?𝑀𝐿−2 +𝑛?̂?𝑀𝐿2 −?̂?𝑀𝐿𝑢𝑛?̂?𝑀𝐿−1−?̂?𝑀𝐿𝑢𝑛?̂?𝑀𝐿−1𝑛?̂?𝑀𝐿2]                = [𝑉𝑎𝑟(?̂?𝑀𝐿) 𝐶𝑂𝑉(?̂?𝑀𝐿 , ?̂?𝑀𝐿)𝐶𝑂𝑉(?̂?𝑀𝐿 , ?̂?𝑀𝐿) 𝑉𝑎𝑟(?̂?𝑀𝐿)]                                                                            (13)  Where  det (𝐼(?̂?)) =𝑛?̂?𝑀𝐿(𝑢𝑛?̂?𝑀𝐿−1 + ?̂?𝑀𝐿(?̂?𝑀𝐿 − 1)𝑢𝑛?̂?𝑀𝐿−2) + (𝑛?̂?𝑀𝐿?̂?𝑀𝐿)2 − (?̂?𝑀𝐿𝑢𝑛?̂?𝑀𝐿−1)2          (14)  The (1 −  𝜀)100% confidence intervals for ?̂?𝑀𝐿, ?̂?𝑀𝐿 are given as ?̂?𝑀𝐿 ± 𝑍𝜀2√𝑉𝑎𝑟(?̂?𝑀𝐿)  , ?̂?𝑀𝐿 ± 𝑍𝜀2√𝑉𝑎𝑟(?̂?𝑀𝐿)                                                                (15)  respectively, where 𝑍𝜀2 is 𝑍𝜀2100% the lower percentile of standard normal distribution.  𝐽𝑋 = −?̂?𝑀𝐿−2?̂?𝑀𝐿+3 ?̂?𝑀𝐿22?̂?𝑀𝐿2 𝛤(2?̂?𝑀𝐿 − 1)                                                                          (16)  The Journal of Engineering and Exact Sciences – jCEC 4 To attain the 100(1 − 𝑍𝜀)% two-sided asymptotic approximation CIs for 𝐽𝑋. The delta method can be used to approximate the variances of extropy. Let:  𝐷𝐽𝑋 = (𝜕𝐽𝑋𝜕𝑏𝜕𝐽𝑋𝜕𝜆)𝑏=?̂?𝑀𝐿,   𝜆=?̂?𝑀𝐿                                                                                                 (17)  Then we find the estimated variance of extropy as follows  𝑉𝑎𝑟(𝐽?̂?) = 𝐷𝐽𝑋  𝐼(?̂?)−1𝐷𝐽𝑋𝑇                                                                                                      (18)  Therefore  [𝐽?̂? − 𝑍𝜀2√𝑉𝑎𝑟(𝐽?̂?) , 𝐽?̂? + 𝑍𝜀2√𝑉𝑎𝑟(𝐽?̂?)]                                                                                      (19)  3. Bayes inference   In this section, we concentrate on the main objective which is the Bayesian estimation to estimate the parameters 𝑏 and 𝜆 and also 𝐽𝑋. For this method we use the squared error and LINEX loss functions, it can be defined respectively as follows:  𝐿1(𝛷, ?̂?) = (𝛷 − ?̂?)2, 𝐿2(𝛷, ?̂?) = e(𝜂(𝛷−?̂?)) − 𝜂(𝛷 − ?̂?) − 1                                               (20)  Let’s choose the prior distribution of 𝑏 and 𝜆. We propose the parameters independently follow a Gamma distribution (𝑏~ 𝐺𝑎𝑚𝑚𝑎(α, 𝛽) and 𝜆~ 𝐺𝑎𝑚𝑚𝑎(γ, 𝜏), where α, 𝛽, γ, 𝑎𝑛𝑑 𝜏 are positive real constants). Thus, the joint prior distribution  𝑃(𝑏, 𝜆) 𝛼  𝜆γ−1𝑏α−1𝑒−𝜆𝜏−𝑏𝛽;  α, 𝛽, γ, 𝜏 > 0                                                                                     (21)  Hence, The joint posterior distribution  𝑃∗(𝑏, 𝜆|𝑢) =   𝐿(𝑏,𝜆;𝑢)𝑃(𝑏,𝜆)∬ 𝐿(𝑏,𝜆;𝑢)𝑃(𝑏,𝜆) 𝑑𝑏 𝑑𝜆                      =𝜆γ−1𝑏α−1𝑒−𝜆𝜏−𝑏𝛽−𝑏𝑢𝑛𝜆∏ 𝑏𝜆 𝑢𝑖𝜆−1𝑛𝑖=1∬𝑒−𝑏𝑢𝑛𝜆∏ 𝑏𝜆 𝑢𝑖𝜆−1𝑛𝑖=1 𝜆γ−1𝑏α−1𝑒−𝜆𝜏−𝑏𝛽 𝑑𝑏 𝑑𝜆                    (22)  The joint posterior density can be written as:  𝑃∗(𝑏, 𝜆|𝑢) 𝛼  𝜆2𝑛+γ−1𝑏2𝑛+α−1 ∏ 𝑢𝑖𝜆−1 𝑛𝑖=1 𝑒−𝜆(𝜏−𝑢𝑛𝑏)−𝑏(𝛽−∑ log𝑢𝑖𝑛𝑖=1 −𝑢𝑛𝜆)                                        (23)   2.1 Markov Chain Monte Carlo  The Bayes estimates for determining the posterior mean for the parameters are difficult to get unless numerical approximation methods are used.  There are numerous approximation approaches in the literature for dealing with this type of situation. We consider the (MCMC) approximation approach and the Gibbs sampling algorithm which are popular Bayesian estimating techniques that rely on marginal posterior distributions for sampling. Readers may refer to Pradhan et al. (2011), Chib et al. (1995) and Al-Labadi et al. (2020).  The full conditional posterior distributions for 𝑏 and 𝜆 s are as follows:  The Journal of Engineering and Exact Sciences – jCEC 5 𝑃∗(𝜆|𝑏, 𝑢) 𝛼  𝜆2𝑛+γ−1 ∏ 𝑢𝑖𝜆−1 𝑛𝑖=1 𝑒−𝜆(𝜏−𝑢𝑛𝑏)+𝑏𝑢𝑛𝜆                                                                          (24)  𝑃∗(𝑏|𝜆, 𝑢) 𝛼  𝑏2𝑛+α−1𝑒𝜆𝑢𝑛𝑏−𝑏(𝛽−∑ log𝑢𝑖𝑛𝑖=1 −𝑢𝑛𝜆)                                                                               (25)  Thus, we must use the Metropolis–Hastings (M–H) algorithm to generate the unknown parameters because the densities in Equations (24,25) cannot be written as known densities. As a result, it is impossible to generate 𝑏 and 𝜆 directly from these densities using conventional methods; for more information. The M–H algorithm aims to minimize rejection rates as much as possible. To find the (BEs) and generate credible intervals for the required parameters, the M–H algorithm relies on selecting the normal distribution. The Gibbs technique, which can be summarized as the following algorithm, is as follows:  Step 1: Put the  ML estimators of 𝑏 and 𝜆 as initial values 𝑏0 and 𝜆0. Step 2: Let 𝑇 = 1, … . , 𝑁 be the observations generated from the conditional posterior distributions for 𝑏 and 𝜆 (24) and (25) respectively.   Step 3: Repeat Steps 2 𝑀 times to obtain MCMC samples (𝑏1, 𝜆1),… , (𝑏𝑀, 𝜆𝑀) where 𝑀 The total amount of cycles needed.). Step 4: The Bayes estimator of extropy given in (9) under SE and LINEX are presented as follow 𝐽𝑆?̂? = 1𝑀−𝑚∑ −𝑏𝑡−2𝜆𝑡+3 (𝜆𝑡)22(𝜆𝑡)2𝛤(2𝜆𝑡 − 1)𝑀𝑡=𝑚+1                                                                           (26)  𝐽𝐿?̂? =−1𝜂 𝑙𝑜𝑔 [1𝑀−𝑚∑ 𝑒𝜂𝑏𝑡−2𝜆𝑡+3 (𝜆𝑡)22(𝜆𝑡)2𝛤(2𝜆𝑡−1)𝑀𝑡=𝑚+1 ]                                                                     (27)  where m is the first iterations as burn in period.  4. Simulation A simulation study was carried out to assess the performance of the estimating techniques created in Sections 2 and 3. First, we get the extropy of k-records for unknown parameters 𝑏, 𝜆  at equation (9) by calculating the MLE ?̂?𝑀𝐿 and ?̂?𝑀𝐿 which we were able to obtain the bias of MLEs for various values k . Now, assuming the model parameters α = 2, 𝛽 = 2 and γ = 2, 𝜏 = 2,  500 observations. Based on these data sets, the maximum likelihood estimates (MLEs) and Bayes estimates of the parameters were obtained. For Bayesian estimation, we generated 10,000 realizations of the Markov chains using the Gibbs and Metropolis–Hastings algorithms. The results of the simulation study are summarized in Table 1. We observe that the bias of all estimators decrease as the sample size n increases and the bias of bayes estimation under the SE loss function is smaller than the MLE.    The Journal of Engineering and Exact Sciences – jCEC 6  Table 1 – The bias of MLE and BE for Weibull distribution k Shape Scale Extropy MLE Bayes  SEL LX: h=1 |Bias| |Bias| |Bias| 6 1.5 2 0.078998995 0.006430693 0.006030694 0.012340693 7 1.5 2 0.072187881 0.013241806 0.012141809 0.019671806 8 1.5 2 0.077499109 0.007930578 0.007820571 0.009830578 9 1.5 2 0.067900459 0.017529228 0.017419221 0.018929228 6 1.5 2.5 0.078998995 0.047277892 0.04417789 0.051277892 7 1.5 2.5 0.072187881 0.054089006 0.051289009 0.062289006 8 1.5 2.5 0.077499109 0.048777778 0.042377772 0.057877778 9 1.5 2.5 0.06790046 0.058376427 0.054276421 0.061276427 6 1.6 2 0.118035454 0.016182274 0.011682279 0.019682274 7 1.6 2 0.106579012 0.027638716 0.022238717 0.027638716 8 1.6 2 0.115357326 0.018860402 0.013160403 0.018860402 9 1.6 2 0.09954352 0.034674208 0.031674201 0.034674208 6 1.6 2.5 0.118035454 0.093583058 0.091783051 0.093583058 7 1.6 2.5 0.106579012 0.1050395 0.102234394 0.196620395 8 1.6 2.5 0.115357325 0.096261187 0.091161189 0.126261187 9 1.6 2.5 0.09954352 0.112074992 0.100074993 0.187874992  References Ahmed, E. 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Extropy estimation of Weibull distribution under upper records

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Extropy estimation of Weibull distribution under upper records

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