Robust Assessing the Lifetime Performance of Products with Inverse Gaussian Distribution in Bayesian and Classical Setup

Abdullah Ali H Ahmadini,M H Tahir,Christophe Chesneau,Sohail Akhtar,Amara Javed,Shokrya S Alshqaq,Farrukh Jamal,Waleed Almutiry,Sanaa Al-Marzouki,Mohammed Elgarhy,Amer Al-Omari

doi:10.1155/2021/4582958

Abstract

The inverse Gaussian (Wald) distribution belongs to the two-parameter family of continuous distributions having a range from 0 to ∞ and is considered as a potential candidate to model diffusion processes and lifetime datasets. Bayesian analysis is a modern inferential technique in which we estimate the parameters of the posterior distribution obtained by formally combining a prior distribution with an observed data distribution. In this article, we have attempted to perform the Bayesian and classical analyses of the Wald distribution and compare the results. Jeffreys' and uniform priors are used as noninformative priors, while the exponential distribution is used as an informative prior. The analysis comprises finding joint posterior distributions, the posterior means, predictive distributions, and credible intervals. To illustrate the entire estimation procedure, we have used real and simulated datasets, and the results thus obtained are discussed and compared. We have used the Bayesian specialized Open BUGS software to perform Markov Chain Monte Carlo (MCMC) simulations using a real dataset.

Highlights

The inverse Gaussian distribution (IGD), known as the Wald distribution, belongs to the two-parameter family of continuous distributions with support 0 to ∞ [1]. e concept of Brownian motion is applicable in describing the inherent process of many phenomena, in the natural and physical sciences. e time in which a Brownian motion with a positive drift reaches a fixed value is distributed as an IGD
We have considered the squared error loss function (SELF), which describes the posterior means as the Bayes estimates. at is
E results reveal that the values of the Akaike information criterion (AIC) and Bayes information criterion (BIC) computed by Bayesian estimates are the smallest as compared with those produced by frequentist estimates

Summary

Introduction

The inverse Gaussian distribution (IGD), known as the Wald distribution, belongs to the two-parameter family of continuous distributions with support 0 to ∞ [1]. e concept of Brownian motion is applicable in describing the inherent process of many phenomena, in the natural and physical sciences. e time in which a Brownian motion with a positive drift reaches a fixed value is distributed as an IGD. E author in [4] considered estimating the inverse Gaussian (μ, λ) model in a Bayesian framework. E authors in [14] investigated Bayesian estimation for the parameters of the IGD distribution. E author in [5] obtained some Bayesian results for the inverse Gaussian family of distributions with a noninformative reference prior as well as the natural conjugate prior. E authors in [6] derived Bayesian results for the IGD by using a proper prior under reparameterization with reference to the distribution mean and of the inverse of the squared variation coefficient, for obtaining Bayes estimates as well as of their inverses.

Numerical Example

Bayesian Analysis

Predictive Inference

Estimation methods ML UMVUE Bayes Lindley

Simulation Study