Permissible noninformative priors for the accelerated life test model with censored data

Fode Zhang,Yimin Shi

doi:10.1186/s40064-016-2004-0

Abstract

In this paper, the Jeffreys priors for the step-stress partially accelerated life test with Type-II adaptive progressive hybrid censoring scheme data are considered. Given a density function family satisfied certain regularity conditions, the Fisher information matrix and Jeffreys priors are obtained. Taking the Weibull distribution as an example, the Jeffreys priors, posterior analysis and its permissibility are discussed. The results, which present that how the accelerated stress levels, censored size, hybrid censoring time and stress change time etc. affect the Jeffreys priors, are obtained. In addition, a theorem which shows there exists a relationship between single observation and multi observations for permissible priors is proved. Finally, using Metroplis with in Gibbs sampling algorithm, these factors are confirmed by computing the frequentist coverage probabilities.

Highlights

Noninformative priors, which make the Bayesian analysis is a distinct field in some sense, have received a lot of attention in the past decades
We investigate the Jeffreys priors for SSPALT with Type-II APHCS, this censoring scheme will be illustrated
Theorem 4 Let Y n be the failure times observed from Weibull (θ, β), the Jeffreys prior based on the SSPALT with Type-II APHCS is given by π1J (θ, β) ∝ θ −1 |ψ1(β)|, where ψ1(β) > 0 is a constraint which may not be satisfied in practice

Summary

Introduction

Noninformative priors, which make the Bayesian analysis is a distinct field in some sense, have received a lot of attention in the past decades (see references Berger et al 2009, 2014; Guan et al 2013; Jeffreys 1946). Theorem 1 Let Y n be a sample of independent life data, with likelihood function L(θ; ym) given in (4), we have the following results: (a) Jhj = −Eθ ∂h2jL (θ ) | θ = − HD(θh, θj) + HC (θh, θj) , where HD(θh, θj) = E D1 (θ )P{δ1i = 1 | θ } + E D2 (θ )P{δ1i = 0 | θ }, HC (θh, θj) = E C1 (θ )P{δ1i = 1 | θ } + E C2 (θ )P δ2ci = 1 | θ .

Results

Conclusion