Empirical bayes estimation of reliability characteristics for an exponential family

Abstract

This paper considers empirical Bayes (EB) squared-error-loss estimations of mean lifetime, variance and reliability function for failure-time distributions belonging to an exponential family, which includes gamma and Weibull distributions as special cases. EB estimators are proposed when the prior distribution of the lifetime parameter is completely unknown but has a compact (known or unknown) support. Asymptotic optimality and rates of convergence of these estimators are investigated. The rates established here under the compact support restriction are better than the polynomial rates of convergence obtained previously. RESUMELes auteurs abordent I'estimation baytsienne empirique, sous fonction de perte quadratique, de la duke de vie moyenne, de la variance et de la fiabilitt de distributions de temps de panne appartenant ti une famille exponentielle, comme c'est le cas pour les lois gamma ou de Weibull par exemple. 11s proposent des estimateurs de Bayes empiriques pour les situations ou la loi a priori du parambtre de durte de vie est complttement inconnue, sauf pour le fait qu'elle posstde un support compact (prCcisC ou non). 11s Ctudient I'optimalitC asymptotique et les taux de convergence de tels estimateurs. Les taux qu'ils obtiennent en postulant la compacitt du support sont meilleurs que les taux de convergence polynomiaux dtja connus.

The Log Gamma Distribution in the Reliability Analysis of Highly Reliable Systems

“…In carrying out a Bayesian probability analysis, one regards the unknown reliability R as a random variable with a prior p.d.f., usually based on past history, followed by the application of Bayes' theorem (see, e.g., Keeping (1962); Lindley (1965); Mendel (1992); Pensky and Singh (1999)). …”

Empirical Bayes Estimation of Reliability in a Positive Exponential Family

“…Nakao and Liu (1990) studied empirical Bayes interval estimation for the reliability of a bivariate exponential model. Pensky and Singh (1999) studied empirical Bayes estimation for the reliability in a positive exponential family. Srivastava et al (2004) investigated Bayesian estimation of reliability for generalized Burr lifetime distribution under time censoring.…”

“…When = 1 and = 1, f y reduces to an exponential density, and to the gamma or Weibull densities when = 1 or = 1, respectively. For the probability density f y of (1.1), the reliability function at a given time t 0 is: Pensky and Singh (1999) have initiated the research problem of estimating the reliability function S t 0 using the empirical Bayes approach. The interest of this research problem is: the usual kernel functions, such as those used in van Houwelingen (1976), Singh (1979), and Karunamuni (1996), among many others, fail to provide consistent estimators for some functionals, which are involved in the concerned empirical Bayes estimation problem.…”

“…The interest of this research problem is: the usual kernel functions, such as those used in van Houwelingen (1976), Singh (1979), and Karunamuni (1996), among many others, fail to provide consistent estimators for some functionals, which are involved in the concerned empirical Bayes estimation problem. Pensky and Singh (1999) proposed an empirical Bayes estimator PS n for S t 0 , claiming PS n to be asymptotically optimal with a rate O ln +1 n/n without providing proof. However, the claimed result is based on an erroneous statement.…”

On Empirical Bayes Procedures for Selecting Good Populations in Positive Exponential Family

“…Since Robbins (1956Robbins ( , 1964 introduced the empirical Bayes approach, it has become a powerful tool in decision-making. Empirical Bayes approach in statistical inferences has been used recently by Singh and Wei (1992), van Houwelingen and Stijnen (1993), Pensky (1998), Pensky and Singh (1999), and Liang (2000aLiang ( , 2000b. The regret Bayes risk is defined as r n = R(G,6 n ) -R(G,5 G ), which is used to measure the performance of empirical Bayes rule 6 n .…”