Statistical properties and different methods of estimation of Gompertz distribution with application

Abstract

This article addresses the various properties and different methods of estimation of the unknown parameters of Gompertz distribution. Although, our main focus is on estimation from both frequentist and Bayesian point of view, yet, various mathematical and statistical properties of the Gompertz distribution (such as quantiles, moments, moment generating function, hazard rate, mean residual lifetime, mean past lifetime, stochasic ordering, stress-strength parameter, various entropies, Bonferroni and Lorenz curves and order statistics) are derived. We briefly describe different frequentist approaches, namely, maximum likelihood estimators, moments estimators, pseudo-moments estimators, modified moments estimators, L-moment estimators, percentile based estimators, least squares and weighted least squares estimators, maximum product of spacings estimators, minimum spacing absolute distance estimators, minimum spacing absolute-log distance estimator, Cramér-von-Mises estimators, Anderson-Darling and right-tail Anderson-Darling and compare them using extensive numerical simulations. Coverage probabilities for the frequentist methods are also obtained. Next we consider Bayes estimation under different types of loss function (symmetric and asymmetric loss functions) using gamma priors for both shape and scale parameters. Furthermore, the Bayes estimators and their respective posterior risks are computed and compared using MCMC algorithm. Finally, a real data set have been analyzed for illustrative purposes.

Subject Classification: (2010):

• 60E05
• 62F10

Keywords:

• Bayes estimator
• Maximum likelihood estimators
• Moment estimators
• Minimum distances estimators
• Failure rate function
• Mean residual life function