Bayesian Analysis of Record Statistic from the Inverse Weibull Distribution under Balanced Loss Function

Abstract

The main contribution of this work is to develop a linear exponential loss function (LINEX) to estimate the scale parameter and reliability function of the inverse Weibull distribution (IWD) based on lower record values. We do this by merging a weight into LINEX to produce a new loss function called weighted linear exponential loss function (WLINEX). We then use WLINEX to derive the scale parameter and reliability function of the IWD. Subsequently, we discuss the balanced loss functions for three different types of loss function, which include squared error (SE), LINEX, and WLINEX. The majority of previous scholars determined the weighted balanced coefficients without mathematical justification. One of the main contributions of this work is to utilize nonlinear programming to obtain the optimal values of the weighted coefficients for balanced squared error (BSE), balanced linear exponential (BLINEX), and balanced weighted linear exponential (BWLINEX) loss functions. Furthermore, to examine the performance of the proposed methods—WLINEX and BWLINEX—we conduct a Monte Carlo simulation. The comparison is between the proposed methods and other methods including maximum likelihood estimation, SE loss function, LINEX, BSE, and BLINEX. The results of simulation show that the proposed models BWLINEX and WLINEX in this work have the best performance in estimating scale parameter and reliability, respectively, according to the smallest values of mean SE. This result means that the proposed approach is promising and can be applied in a real environment.

On Entropy Estimation of Inverse Weibull Distribution under Improved Adaptive Progressively Type-II Censoring with Applications

“…The IW distribution was considered by Keller [14] to model failures of mechanical components subject to degradation. Afterward, many researchers studied the IW distribution, including, but not limited to, Calabria and Pulcini [15], Jiang et al [16], Mahmoud et al [17], Sultan [18], Kundu and Howlader [19], Hassan et al [20], Kumar and Kumar [21], and Al-Duais [22]. The random variable Y follows the two-parameter IW distribution, denoted by IW(θ, λ), if the corresponding PDF and CDF, are given by: g(y; θ, λ) = λθy −(θ+1) e −λy −θ , y ≥ 0, θ, λ > 0, (4) and G(y; θ, λ) = e −λy −θ , y ≥ 0, θ, λ > 0, (5) respectively, where θ > 0 is the shape parameter and λ > 0 is the rate parameter.…”

Bayesian Estimations of Shannon Entropy and Rényi Entropy of Inverse Weibull Distribution

“…Lemma 1. Suppose that T is the historical data information about the entropy function H. Then, under the SE loss function (25), the Bayesian estimator Ĥ1 for any prior distribution is shown in Equation ( 26):…”

“…Mohammad and Sana [24] obtained the Bayes estimators and ML estimators for the unknown parameters of the IWD under lower record values. Faud [25] developed a linear exponential loss function, and estimated the parameter and reliability of the IWD based on lower record values under this loss function. Li and Hao [26] considered the estimation of a stress-strength model when stress and strength are two independent IWDs with different parameters.…”

Bayesian Estimation of Different Scale Parameters Using a LINEX Loss Function

“…For example, it offered simple but appealing large enough sample approximations to optimal timings, dubbed asymptotically pointwise optimal (APO) rules, and demonstrated that APO rules were asymptotically optimal (AO) under a second scenario. Many publications have examined the APO rule and how it might be used to address those other challenges [ 10 ]. Discrete-time events are the focus of the studies in these publications.…”

“…The LINEX loss function was officially created, and its properties were investigated further. It is a handy asymmetrical nonlinear function that increases dramatically on one end of zero and gradually on the other [ 15 ]. Much research has looked into estimating issues with LINEX loss function [ 16 ].…”