Francisco J. Samaniego

On Closure of the IFR Class Under Formation of Coherent Systems

A representation is derived for the failure rate of an arbitrary s-coherent system when the lifetimes of its components are s-independently distributed according to a common absolutely continuous distribution F. The system failure rate is written explicitly as a function of F and its failure rate. The representation is used in several examples, including an example showing that the closure theorem for k-out-of-n systems in i.i.d. IFR components proven by Barlow & Proschan cannot be extended to all s-coherent systems. The class of s-coherent systems for which such closure obtains is characterized.

System Signatures and their Applications in Engineering Reliability

Background on Coherent Systems.- System Signatures.- Signature-Based Closure, Preservation and Characterization Theorems.- Further Signature-Based Analysis of System Lifetimes.- Applications of Signatures to Network Reliability.- Applications of Signatures in Reliability Economics.- Summary and Discussion.

The “signature” of a coherent system and its application to comparisons among systems

Various methods and criteria for comparing coherent systems are discussed. Theoretical results are derived for comparing systems of a given order when components are assumed to have independent and identically distributed lifetimes. All comparisons rely on the representation of a systems lifetime distribution as a function of the systems “signature,” that is, as a function of the vector p= (p1, … , pn), where pi is the probability that the system fails upon the occurrence of the ith component failure. Sufficient conditions are provided for the lifetime of one system to be larger than that of another system in three different senses: stochastic ordering, hazard rate ordering, and likelihood ratio ordering. Further, a new preservation theorem for hazard rate ordering is established. In the final section, the notion of system signature is used to examine a recently published conjecture regarding componentwise and systemwise redundancy.

Nonparametric Maximum Likelihood Estimation Based on Ranked Set Samples

Abstract A ranked set sample consists entirely of independently distributed order statistics and can occur naturally in many experimental settings, including problems in reliability. When each ranked set from which an order statistic is drawn is of the same size, and when the statistic of each fixed order is sampled the same number of times, the ranked set sample is said to be balanced. Stokes and Sager have shown that the edf F n of a balanced ranked set sample from the cdf F is an unbiased estimator of F and is more precise than the edf of a simple random sample of the same size. The nonparametric maximum likelihood estimator (MLE) F of F is studied in this article. Its existence and uniqueness is demonstrated, and a general numerical procedure is presented and is shown to converge to F. If the ranked set sample is balanced, it is shown that the EM algorithm, with F n as a seed, converges to the unique solution (F) of the problems self-consistency equations; the consistency of every iterate of the EM a...

Estimating the Reliability of Systems Subject to Imperfect Repair

Abstract This study of statistical inference for repairable systems focuses on the development of estimation procedures for the life distribution F of a new system based on data on system lifetimes between consecutive repairs. The Brown—Proschan imperfect-repair model postulates that at failure the system is repaired to a condition as good as new with probability p, and is otherwise repaired to the condition just prior to failure. In treating issues of statistical inference for this model, the article first points out the lack of identifiability of the pair (p, F) as an index of the distribution of interfailure times T 1, T 2, …. It is then shown that data pairs (Ti, Zi ) (i = 1, 2, …) render the parameter pair (p, F) identifiable, where Zi is a Bernoulli variable that records the mode of repair (perfect or imperfect) following the ith failure. Under the assumption that data of the form {(Ti, Zi )} are drawn via inverse sampling until the occurrence of the mth perfect repair, the problem of estimating the...

The Practice of Bayesian Analysis

The purpose of this book is twofold. First, it has been written to advertise the advantages a Bayesian analysis can bring. New statistical and decision models can be tailored to the unique beliefs, values and needs of the user, and the implications of the data she collects can be analysed with reference to this underlying structure. The second purpose is to provide an analyst wanting to apply the Bayesian methodology with a resource of examples and techniques, so that she may be better informed about the scope of opportunities and the pitfalls inherent in such an analysis.

On the Efficacy of Bayesian Inference for Nonidentifiable Models

Abstract Although classical statistical methods are inapplicable in point estimation problems involving nonidentifiable parameters, a Bayesian analysis using proper priors can produce a closed form, interpretable point estimate in such problems. The question of whether, and when, the Bayesian approach produces worthwhile answers is investigated. In contrast to the preposterior analysis of this question offered by Kadane, we examine the question conditionally, given the information provided by the experiment. An important initial insight on the matter is that posterior estimates of a nonidentifiable parameter can actually be inferior to the prior (no-data) estimate of that parameter, even as the sample size grows to infinity. In general, our goal is to characterize, within the space of prior distributions, classes of priors that lead to posterior estimates that are superior, in some reasonable sense, to ones prior estimate. This goal is shown to be feasible through a detailed examination of a particular t...

Toward a Reconciliation of the Bayesian and Frequentist Approaches to Point Estimation

Abstract The Bayesian and frequentist approaches to point estimation are reviewed. The status of the debate regarding the use of one approach over the other is discussed, and its inconclusive character is noted. A criterion for comparing Bayesian and frequentist estimators within a given experimental framework is proposed. The competition between a Bayesian and a frequentist is viewed as a contest with the following components: a random observable, a true prior distribution unknown to both statisticians, an operational prior used by the Bayesian, a fixed frequentist rule used by the frequentist, and a fixed loss criterion. This competition is studied in the context of exponential families, conjugate priors, and squared error loss. The class of operational priors that yield Bayes estimators superior to the “best” frequentist estimator is characterized. The implications of the existence of a threshold separating the space of operational priors into good and bad priors are explored, and their relevance in ar...

The Signature of a Coherent System and Its Applications in Reliability

We describe and interpret the interesting characteristic of a system called its “signature”. There is a connection between a system’s signature and other well-known reliability concepts, and the signature is a useful tool in making compar-isons between systems. For example, it is often possible to rate one system as better than another by a quick glance at their respective signatures, and various stochastic comparisons between systems may be established by comparing their signatures. Many of the theoretical aspects of signatures discussed give rise to interesting practical applications involving in particular consecutive K-out-of-n systems, redundancy enhancements to a system, and majority systems.

Nonparametric Estimation of a Distribution Subject to a Stochastic Precedence Constraint

For any two random variables X and Y with distributions F and G defined on [0,∞), X is said to stochastically precede Y if P(X≤Y) ≥ 1/2. For independent X and Y, stochastic precedence (denoted by X≤spY) is equivalent to E[G(X–)] ≤ 1/2. The applicability of stochastic precedence in various statistical contexts, including reliability modeling, tests for distributional equality versus various alternatives, and the relative performance of comparable tolerance bounds, is discussed. The problem of estimating the underlying distribution(s) of experimental data under the assumption that they obey a stochastic precedence (sp) constraint is treated in detail. Two estimation approaches, one based on data shrinkage and the other involving data translation, are used to construct estimators that conform to the sp constraint, and each is shown to lead to a root n-consistent estimator of the underlying distribution. The asymptotic behavior of each of the estimators is fully characterized. Conditions are given under which each estimator is asymptotically equivalent to the corresponding empirical distribution function or, in the case of right censoring, the Kaplan–Meier estimator. In the complementary cases, evidence is presented, both analytically and via simulation, demonstrating that the new estimators tend to outperform the empirical distribution function when sample sizes are sufficiently large.