Empirical bayes shrinkage estimation of reliability in the exponential distribution

Chiou, Paul

doi:10.1080/03610929308831098

Cited by 14 publications

(10 citation statements)

References 17 publications

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“…{iu^(i)(l + x 2 h 2 ) x+0 · 5 } < oo, then the denominator of the integrand in (5.11) ~ 1. The proof is completed by noticing that 1 + (y + h^x) 2 where uh,y{x) = E"=i a,-(y) IZo ^(i -θ}ΐς{Η(χ -θ))μΗ{θ)άθ. Thus the family gN{e,z) generates also the set of Bayes estimators of f{9) ßN{y, ζ) = ΦΝ(ν, z)/fN(y, ζ).…”

Section: Proofsmentioning

confidence: 99%

“…Let us consider the cases a > 0 and a < 0 separately. If a < 0, then = o(q(f)), j = 1 ,...,n. We choose r(x) = q(-x), ϋ{ω) = v 0 (M" -1) 4 (4" -μ|") 4 μ| Ι '-1 βχρ(-ν / Μ^Ί Γ Τ)χ(|ω| e [1,2]). Substituting into (3.2), we get Ar(/ijv) 27~4s_1 exp(-4Wi^) ~ const.…”

Section: Corollary 3 Follows From the Fact That Ii(h) ~ H 2s I 2 (Hmentioning

confidence: 99%

“…After Robbins [22] formulated the empirical Bayes estimation problem many papers were devoted to the empirical Bayes method. Empirical Bayes estimators were constructed for many familiar probability distributions (see for instance, [1], [2], [5], [7], [8], [12], [13], [14], [21], [23], [24], [25], [26]). In some papers (see [10], [11], [14], [21], [23], [24]) the estimators are given along with the upper bounds for the risk function which leads to the following question: what is the best convergence rate of empirical Bayes estimators for a given family of conditional distributions?…”

Section: Introductionmentioning

confidence: 99%

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Empirical Bayes Estimation of a Location Parameter

Pensky¹

1997

Statistics &Amp; Risk Modeling

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Assume that in independent two-dimensional random vectors (Λι, θ\) ,..., (ΧΝ,ΘΝ), each Θϊ is distributed according to some unknown prior distribution density g, and that , given θ,, Xi has the conditional density function q(x -Θ,), i = 1,..., TV. In each pair the first component is observable, but the second is not. After the (JV+l)-th pair (X^r+i, is obtained, the objective is to construct the empirical Bayes estimator of a polynomial f(0w + 1) = with given coefficients bj. In the paper we derive the empirical Bayes estimator of b(9) without any parametric assumptions on g. The upper bound for the mean squared error is obtained. The lower bound for the mean squared error over the class of all possible empirical Bayes estimators is also derived. It is shown that the estimators constructed in the paper have the optimal or nearly optimal convergence rates. Examples for familiar families of conditional distributions are considered.

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Section: Proofsmentioning

confidence: 99%

Section: Corollary 3 Follows From the Fact That Ii(h) ~ H 2s I 2 (Hmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Empirical Bayes Estimation of a Location Parameter

Pensky¹

1997

Statistics &Amp; Risk Modeling

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“…Problems somewhat similar to those considered in this paper are investigated for special distributions such as exponential, gamma and Weibull by Chiou (1993), Lahiri and Park (1991). Li (1984) and Nakao and Liu (1990), among others.…”

Section: Empirical Bayes Approach When G Is Unknownmentioning

confidence: 99%

Empirical bayes estimation of reliability characteristics for an exponential family

Pensky

Singh

1999

Can J Statistics

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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.

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“…It is of interest to explore the potential generalization of the shrinkage estimator for rare events. There are quite many literatures that have been published to investigate the applications of the EB and the James–Stein type shrinkage approaches in various fields, including, but not limited to, signal processing, reliability engineering, and microarray data analysis …”

Section: Introductionmentioning

confidence: 99%

A Shrinkage Approach for Failure Rate Estimation of Rare Events

Xiao

Xie

2014

Qual. Reliab. Engng. Int.

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Systems have become more and more reliable due to technological advancement. For highly reliable systems, there are usually very few or even no failures during the testing and operation. On the other hand, given a short operating or testing time, the failure of the system is also rare even the failure rate is relatively high. The classical maximum likelihood estimation approach results in degenerated estimates zero when no failure occurs and hence meaningless. To overcome this problem, we investigate a shrinkage approach for the estimation of the failure rate of rare events from multiple heterogeneous systems provided that some of them have failures. The shrinkage estimator shrinks the MLE toward a predetermined data-dependent point in the parameter space and could be expressed as a weighted average of MLE and the data-dependent point. Examples are shown how the procedure can be implemented. Simulation studies show that our approach performs better than the other approaches in most cases.

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Empirical bayes shrinkage estimation of reliability in the exponential distribution

Cited by 14 publications

References 17 publications

Empirical Bayes Estimation of a Location Parameter

Empirical Bayes Estimation of a Location Parameter

Empirical bayes estimation of reliability characteristics for an exponential family

A Shrinkage Approach for Failure Rate Estimation of Rare Events

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