Bayesian Estimates Based On Record Values Under Weighted LINEX Loss Function

Mohammed Alhagyan,Fuad Al-Duais

doi:10.18187/pjsor.v16i1.2854

Abstract

In this paper, we developed linear exponential (LINEX) loss function by emerging weights to produce weighted linear exponential (WLINEX) loss function. Then we utilized WLINEX to derive scale parameter and reliability function of the Weibull distribution based on record values when the shape parameter is known. After, we estimated scale parameter and reliability function of Weibull distribution by using maximum likelihood (ML) estimation and by several Bayes estimations. The Bayes estimates were obtained with respect to symmetric loss function (squared error loss (SEL)), asymmetric loss function (LINEX) and asymmetric loss function (WLINEX). The ML and the different Bayes estimates are compared via a Monte Carlo simulation study. The result of simulation mentioned that the proposed WLINEX loss function is promising and can be used in real environment especially at the case of underestimate where it revealed better performance than LINEX loss function for estimating scale parameter.

Highlights

Record values arise naturally in many real life applications involving data relating to sport, weather and life testing studies
We developed linear exponential (LINEX) loss function by merging weights to produce weighted linear exponential (WLINEX) loss function
We utilized WLINEX to derive scale parameter and reliability function of the Weibull distribution based on record values when the shape parameter is known

Summary

Introduction

Record values arise naturally in many real life applications involving data relating to sport, weather and life testing studies. Many authors have studied record values and associated statistics, for example ; Nagaraja (1988), Ahsanullah (1995), Arnold, et al (1998), Ahsanullah (1980), Balakrishnan and Chan (1993), Shojaee et al (2012), Soliman et al (2006), Sultan (2008), Sultan, et al (2001), Asgharzadeh and Fallah (2010), Shawky and Badr (2012). We will derive Bayes estimator under weighted linear exponential (WLINEX) loss function to estimate scale parameter and reliability function of the Weibull distribution based on record values. Assuming that XU (1), XU (2), XU (3), XU (m) are the first m upper record values arising from a sequence { X i } of (iid) Weibull variables with density function (pdf). Suppose we observed the first m upper record values each of which has the WD whose pdf and cdf are respectively, given by (1) and (2).

LINEX Loss function

Weighted LINEX loss functions

Estimates Based on Squared Error Loss Function

Estimates Based on LINEX Loss Function