Emad El-Neweihi

APPLICATIONS OF THE HAZARD RATE ORDERING IN RELIABILITY AND ORDER STATISTICS

The hazard rate ordering is an ordering for random variables which compares lifetimes with respect to their hazard rate functions. It is stronger than the usual stochastic order for random variables, yet is weaker than the likelihood ratio ordering. The hazard rate ordering is particularly useful in reliability theory and survival analysis, owing to the importance of the hazard rate function in these areas. In this paper earlier work on the hazard rate ordering is reviewed, and extensive new results related to coherent systems are derived. Initially we fix the form of a coherent structure and investigate the effect on the hazard rate function of the system when we switch the lifetimes of its components from the vector (T, - .., T,) to the vector (T1,. ? ., T,), where the hazard rate functions of the two vectors are assumed to be comparable in some sense. Although the hazard rate ordering is closed under the formation of series systems, we see that this is not the case for parallel systems even when the system is a two-component parallel system with exponentially distributed lifetimes. A positive result shows that for two-component parallel systems with proportional hazards (A1r(t), A2r(t)), the more diverse (A1, 2) is in the sense of majorization the stronger is the system in the hazard rate ordering. Unfortunately even this result does not extend to parallel systems with more than two components, demonstrating again the delicate nature of the hazard rate ordering. The principal result of the paper concerns the hazard rate ordering for the lifetime of a k-out-of-n system. It is shown that if TkIn is the lifetime of a k-out-of-n system, then Tkl n is greater than Tk +ln in the hazard rate ordering for any k. This has an interesting interpretation in the language of order statistics. For independent (not necessarily identically distributed) lifetimes T1, - * *, Tn, we let Tk:n represent the kth order statistic (in increasing order). Then it is shown that Tk +1: is greater than Tk:n in the hazard rate ordering for all k = 1, - - -, n - 1. The result does not, however, extend to the stronger likelihood ratio order.

Stochastic order for redundancy allocations in series and parallel systems

The problem of where to allocate a redundant component in a system in order to optimize the lifetime of a system is an important problem in reliability theory which also poses many interesting questions in mathematical statistics. We consider both active redundancy and standby redundancy, and investigate the problem of where to allocate a spare in a system in order to stochastically optimize the lifetime of the resulting system. Extensive results are obtained in particular for series and parallel systems.

Measures of component importance in reliability theory

Abstract In this paper we survey the literature concerning the topic of measures of importance for components in binary coherent systems. The diverse measures discussed fall into one of the following three broad categories: structural measures, time dependent measures and time independent measures. Comparisons are made between various measures, and a framework for possible new measures is suggested. More research is needed in assessing the quantitative and qualitative properties of each of these measures as well as their interrelationships, and it is suggested that the most appropriate measure for a given situation is often highly dependent on the type of improvement for a system that is being envisaged.

Redundancy importance and allocation of spares in coherent systems

Abstract We study the model in which a set of spares is available for redundancy in a coherent system. In some circumstances, parallel (or active) redundancy is used to improve the reliability of the system, while in others series redundancy is used to improve a different measure of utility. Hence we define the two concepts of parallel and series redundancy importance for components in a coherent system relative to an available set of redundant spares. These measures of importance are compared with the structural importance and reliability importance of a component due to Birnbaum. Various results for the optimal allocation of rebundant spares are given, with particular reference to k out of n systems and modules of coherent systems.

Clusure of the NBUE and DMRL classes under formation of parallel systems

The class of new better than used in expectation life distributions is shown to be closed under the formation of parallel systems with independent and identically distributed components. The class of differentiable life distributions with decreasing mean residual life is also proved to have the same closure property.

Characterizations of Geometric Distribution and Discrete IFR (DFR) Distributions Using Order Statistics.

Let X be a discrete random variable the set of possible values (finite or infinite) of which can be arranged as an increasing sequence of real numbers a1<a2<a3<…. In particular, ai could be equal to i for all i. Let X1n≦X2n≦⋯≦Xnn denote the order statistics in a random sample of size n drawn from the distribution of X, where n is a fixed integer ≧2. Then, we show that for some arbitrary fixed k(2≦k≦n), independence of the event {Xkn=X1n} and X1n is equivalent to X being either degenerate or geometric. We also show that the montonicity in i of P{Xkn = X1n | X1n = ai} is equivalent to X having the IFR (DFR) property. Let ai = i and G(i) = P(X≧i), i = 1, 2, …. We prove that the independence of {X2n − X1n ∈B} and X1n for all i is equivalent to X being geometric, where B = {m} (B = {m,m+1,…}), provided G(i) = qi−1, 1≦i≦m+2 (1≦i≦m+1), where 0<q<1.

Stochastic Ordering and a Class of Multivariate New Better Than Used Distributions.

A new characterization for the univariate class of new better than used ‘NBU’ distributions in terms of stochastic ordering is introduced. A multivariate version of this characterization is then used to define a multivariate class of NBU distributions. Basic properties of this class are derived. Comparisons and relationships of this new class with earlier classes are developed. Two multivariate new worse than used (NWU) classes of life distributions are defined and compared and their basic properties are studied.

Reliability estimation based on ranked set sampling

In this paper we consider the problem of estimating the reliability of an exponential component based on a Ranked Set Sample (RSS) of size n. Given the first r observations of that sample, 1≤r≤n, we construct an unbiased estimator for this reliability and we show that these n unbiased estimators are the only ones in a certain class of estimators. The variances of some of these estimators are compared. By viewing the observations of the RSS of size n as the lifetimes of n independent k-out-of-n systems, 1≤k≤n, we are able to utilize known properties of these systems in conjunction with the powerful tools of majorization and Schur functions to derive our results.

21 Optimal allocation of multistate components

Publisher Summary This chapter presents some results in the optimal allocation of multistate components to k series systems so that some performance characteristic like expected number of systems functioning at level α or higher, the probability that at least one of the systems functions at level α or higher, is maximized. The chapter reviews a general optimal allocation result for multistate systems, and describes some detail reliability models in which these results can be used. The basic mathematical tools used are majorization and Schur functions.

A Multivariate New Better than Used Class Derived from a Shock Model

We introduce a new class of multivariate new better than used (MNBU) life distributions based on a shock model similar to that yielding the Marshall-Olkin multivariate exponential distribution. Let T1, …, TM be independent new better than used (NBU) life lengths. Let F(t1, …, tn) be the joint survival function of minI∈A1 T1, I = 1, …, n, where A1, …, An are nonempty subsets of {1, …, M} and Ur=1n A1 = {1, …, M} F(t1, …, tn) is said to be a MNBU survival function. Basic properties of MNBU survival functions are derived. Comparisons and relationships of this new class of MNBU survival functions are developed with earlier classes.