Ashok K. Singh

Bayes estimation of hazard and acceleration in accelerated testing

In accelerated life testing, the time transformation function theta (t) is often unknown, even if that function is assumed to be linear. If theta (t) is known, data in the accelerated condition can be adjusted to provide information about the failure time distribution in the use condition. If theta (t) is unknown, the usual estimation procedures require data from the use condition as well as data from the acceleration condition. In this work it is assumed that the uncertainty about theta can be modeled by a prior distribution, chosen from the truncated Pareto family of distributions, and that the uncertainty in lambda , the failure rate, can be modeled by a prior distribution from the gamma family. Under these assumptions, the posterior distributions and their first two moments are provided for both lambda and theta . Thus, this complete Bayes approach to accelerated life testing with the assumed model allows the adjustment of data taken in the accelerated condition to provide the user with the important estimates in the use condition. The results are illustrated by examples. >

Bayes estimation of the linear hazard-rate model

In life testing, the failure-time distributions are often specified by choosing an appropriate hazard-rate function. The class of life-time distribution characterized by a linear hazard-rate includes the one-parameter exponential and Rayleigh distributions. Usually the parameters of the linear hazard-rate model are estimated by the method of least squares. This work is concerned with Bayes estimation of the two-parameters from a type-2 censored sample. Monte Carlo simulation is used to compare the Bayes risk of the regression estimator with the minimum Bayes risk. Discrete mixtures of decreasing failure rate distributions are known to have decreasing failure rates. The authors prove that the result holds for continuous mixtures as well. >

On rules based on sample medians for selection of the largest location parameter

The problem of selection of a subset containing the largest of several location parameters is considered, and a Gupta-type selection rule based on sample medians is investigated for normal and double exponential populations. Numerical comparisons between rules based on medians and means of small samples are made for normal and contaminated normal populations, assuming the popula-tion means to be equally spaced. It appears that the rule based on sample means loses its superiority over the rule based on sample medians in case the samples are heavily contaminated. The asymptotic relative efficiency (ARE) of the medians procedure relative to the means procedure is also computed, assuming the normal means to be in a slippage configuration. The means proce-dure is found to be superior to the median procedure in the sense of ARE. As in the small sample case, the situation is reversed if the normal populations are highly contaminate.

Empirical Bayesian estimation of mean life from an accelerated life test

Abstract In accelerated life testing, the time to failure of an item is observed under a high stress level, and the estimates are needed for the normal stress levels. This paper is concerned with empirical Bayes estimation of the hazard rate constant for exponentially distributed life times. We have assumed that the effect of acceleration is to scale up the hazard rate, and that the hazard rate has the natural conjugate prior with a known mean and unknown variance. We have considered two smooth empirical Bayes estimates of the hazard rate, and computed their Bayes risks by simulation.

Empirical-Bayes Estimation of Mean Life for a Censored Sample, Constant Hazard Rate Model

The estimation of mean life based on a censored sample from a population with a constant failure rate is considered in an empirical Bayes situation. Bayes risks of empirical Bayes estimates for various numbers of past samples are compared with the Bayes risk of the sample mean, and the minimum Bayes risk. A Monte Carlo simulation shows that the Bayes risk of the empirical Bayes estimator is very close to the minimum Bayes risk for as few as 5 past estimates of the mean life. Morever, it is better than the uniformly minimum variance unbiased estimator (the sample mean) in terms of the Bayes risk. Effects of misclassification of the prior are also considered.

Testing multiple slippages

A class of invariant Bayes rules is derived for testing homogeneity of k (≥2) different populations against (kt) slippage alternatives that some (unknown) subset of size t of the given populations has parameter larger than the remaining k-t, where t is a given integer between 1 and k-1. For a similar problem in nonparametric situations, locally best tests based on ranks are derived.

The EM Algorithm and Related Statistical Models

Chapters 5 and 6 discuss how to extend the CE method for various optimization problems. Chapter 5 introduces a fully automated version of the CE algorithm (FACE) where all the parameters of the CE algorithm are automatically tuned. Chapter 6 considers the problem of optimization for noisy objective functions. Chapter 7 discusses applications of the CE method to some combinatorial optimization problems such as pairwise sequence alignment problems in bioinformatics, scheduling problems in operation research, and clique problem in graph theory. Finally, in Chapter 8, applications of the CE method to some problems in machine learning are presented. Various Matlab implementations of CE algorithms are given in the Appendix. Since the CE method is a young and developing field, there is no book available in this area where the two authors are the pioneers. Therefore, it is quite a unique book and it may become a classic reference in the CE method literature.

Life Estimation after Testing for Early Failures

Life estimation based on ordered observations from the exponential distribution is considered for the case when several early failures may be present. The method proposed here requires a preliminary statistical test to decide whether the suspected observations are indeed early failures. The role of the suspected observations (early failures) in the subsequent estimation problem is based on the result of this test. The proposed estimator has, under certain conditions, smaller mean square error than that of the minimum variance unbiased estimator, and its bias remains small.

Estimation of series system reliability for exponentially distributed component life times

Abstract The reliability of a series system with P components which have exponential life times is estimated using Type II censored samples. The series system reliability is a function of μ( λ )= ∑ j=1 P λ j , where λj is the hazard rate constant for the jth component, j=1,2,…P. An estimator of μ( λ ) which dominates the MLE in terms of the risk is derived. This improved estimator of μ( λ ) is used for estimating the series system reliability. Monte Carlo simulation is used to estimate the risks of the proposed estimators, and comparisons with the ML estimators are made.

A Manager's Guide to the Design and Conduct of Clinical Trials

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