Objective and subjective prior distributions for the Gompertz distribution

Abstract

This paper takes into account the estimation for the unknown parameters of the Gompertz distribution from the frequentist and Bayesian view points by using both objective and subjective prior distributions. We first derive non-informative priors using formal rules, such as Jefreys prior and maximal data information prior (MDIP), based on Fisher information and entropy, respectively. We also propose a prior distribution that incorporate the expert's knowledge about the issue under study. In this regard, we assume two independent gamma distributions for the parameters of the Gompertz distribution and it is employed for an elicitation process based on the predictive prior distribution by using Laplace approximation for integrals. We suppose that an expert can summarize his/her knowledge about the reliability of an item through statements of percentiles. We also present a set of priors proposed by Singpurwala assuming a truncated normal prior distribution for the median of distribution and a gamma prior for the scale parameter. Next, we investigate the effects of these priors in the posterior estimates of the parameters of the Gompertz distribution. The Bayes estimates are computed using Markov Chain Monte Carlo (MCMC) algorithm. An extensive numerical simulation is carried out to evaluate the performance of the maximum likelihood estimates and Bayes estimates based on bias, mean-squared error and coverage probabilities. Finally, a real data set have been analyzed for illustrative purposes.

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“…Therefore, the Jeffreys prior leads to an improper posterior distribution and cannot be used in a Bayesian analysis. Another objective prior known as the maximum data information prior could be considered [16,[24][25][26]; however, such a prior is not invariant under one-to-one transformations, which limits its use. Additionally, the main aim was to consider objective priors.…”

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“…The posterior density for the latter is improper unless the prior is truncated; we follow Northrop & Attalides (2016) and restrict the range to ξ ≥ −1. Moala & Dey (2018) discuss the validity of the prior for the Gompertz distribution, π(σ, β) ∝ σ −1 exp{exp(1/β) ∞ 1/β exp(−t)/tdt}, for a different parametrization. In these low-dimensional settings we can use the ratio-of-uniforms method (Wakefield et al, 1991;Northrop, 2021) to sample from the posterior distribution.…”

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“…Dey et al [5] provided various mathematical and statistical properties and compared different estimation methods from both frequentist and Bayesian point of view. Moala and Dey [6] provided Bayesian analysis methods under the objective and subjective priors including Jeffreys prior, maximal data information prior, Singpurwalla's prior, and elicited prior.…”