Subhash C. Kochar

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.

Stochastic comparisons of parallel systems of heterogeneous exponential components

Abstract Let X1, …, Xn be independent exponential random variables with Xi having hazard rate λi, i = 1, …, n. Let λ = (λ1, …, λn). Let Y1, …, Yn be a random sample of size n from an exponential distribution with common hazard rate λ = Σ n i = 1 λ i n . The purpose of this paper is to study stochastic comparisons between the largest order statistics Xn:n and Yn:n from these two samples. It is proved that the hazard rate of Xn:n is smaller than that of Yn:n. This gives a convenient upper bound on the hazard rate of Xn:n in terms of that of Yn:n. It is also proved that Yn:n is smaller than Xn:n according to dispersive ordering. While it is known that the survival function of Xn:n is Schur convex in λ, Boland, El-Neweihi and Proschan [J. Appl. Prohab. 31 (1994) 180–192] have shown that for n > 2, the hazard rate of Xn:n is not Schur concave. It is shown here that, however, the reversed hazard rate of Xn:n is Schur convex in λ.

On tail-ordering and comparison of failure rates

In this paper, we have studied some implications between tail-ordering (also known as dispersive ordering) and failure rate ordering (also called TP2 ordering) of two probability distribution functions. Based on independent random samples from these distributions, a class of distribution-free tests has been proposed for testing the null hypothesis that the two life distributions are identical against the alternative that one failure rate is uniformly smaller than the other. The tests have good efficiencies as compared to their competitors.

Stochastic comparisons of parallel systems when components have proportional hazard rates

Let X1, …, Xn be independent random variables with Xi having survival function Fλi, i = 1, …, n, and let Y1, …,Yn be a random sample with common population survival distribution Fλ, where λ=Σi=1 nλi/n. Let Xn:n and Yn:n denote the lifetimes of the parallel systems consisting of these components, respectively. It is shown that Xn:n is greater than Yn:n in terms of likelihood ratio order. It is also proved that the sample range Xn:n X1:n is larger than Yn:n Y1:n according to reverse hazard rate ordering. These two results strengthen and generalize the results in Dykstra, Kochar, and Rojo [6] and Kochar and Rojo [11], respectively.

Some tests for comparing cumulative incidence functions and cause-specific hazard rates

Abstract We consider the competing risks problem with the available data in the form of times and causes of failure. In many practical situations (e.g., in reliability testing) it is important to know whether two risks are equal or whether one is “more serious” than the other. We propose some distribution-free tests for comparing cumulative incidence functions and cause-specific hazard rates against ordered alternatives without making any assumptions on the nature of dependence between the risks. Both the censored and the uncensored cases are studied. The performance of the proposed tests is assessed in a simulation study. As an illustration, we compare the risks of two types of cancer mortality (thymic lymphoma and reticulum cell carcinoma) in a strain of laboratory mice.

The Total Time on Test Transform and the Excess Wealth Stochastic Orders of Distributions

For nonnegative random variables X and Y we write X ≤TTT Y if ∫0 F -1(p)(1-F(x))dx ≤ ∫0 G -1(p)(1-G(x))dx all p ∈ (0,1), where F and G denote the distribution functions of X and Y respectively. The purpose of this article is to study some properties of this new stochastic order. New properties of the excess wealth (or right-spread) order, and of other related stochastic orders, are also obtained. Applications in the statistical theory of reliability and in economics are included.

Inference for Likelihood Ratio Ordering in the Two-Sample Problem

Abstract We obtain the maximum likelihood estimators of two multinomial probability vectors under the constraint that they are likelihood ratio ordered. We extend this estimation approach to the case of two univariate distributions and show strong consistency of the estimators. We also derive and study the asymptotic distribution of the likelihood ratio statistic for testing the equality of two discrete probability distributions against the alternative that one distribution is greater than the other in the likelihood ratio ordering sense. Finally, we examine a data set pertaining to average daily insulin dose from the Boston Collaborative Drug Surveillance Program and compare our testing procedure to testing procedures for other stochastic orderings.

Partial orderings of life distributions with respect to their aging properties

New partial orderings of life distributions are given. The concepts of decreasing mean residual life, new better than used in expectation, harmonic new better than used in expectation, new better than used in failure rate, and new better than used in failure rate average are generalized, so as to compare the aging properties of two arbitrary life distributions.

Stochastic orderings between distributions and their sample spacings – II

Let X1:n[less-than-or-equals, slant]X2:n[less-than-or-equals, slant]...[less-than-or-equals, slant]Xn:n denote the order statistics of a random sample X1,X2,...,Xn from a probability distribution with distribution function F. Similarly, let Y1:n[less-than-or-equals, slant]Y2:n[less-than-or-equals, slant]...[less-than-or-equals, slant]Yn:n denote the order statistics of an independent random sample Y1,Y2,...,Yn from G. The corresponding spacings are defined by Ui:n[reverse not equivalent]Xi:n-Xi-1:n and Vi:n[reverse not equivalent]Yi:n-Yi-1:n, for i=1,2,...,n, where X0:n=Y0:n[reverse not equivalent]0. It is proved that if X is smaller than Y in the hazard rate order sense and if either F or G is a DFR (decreasing failure rate) distribution, then the vector of Ui:ns is stochastically smaller than the vector of Vi:ns. If instead, we assume that X is smaller than Y in the likelihood ratio order and if either F or G is DFR, then Ui:n is smaller than Vi:n in the hazard rate sense for 1[less-than-or-equals, slant]i[less-than-or-equals, slant]n. Finally, if we make a stronger assumption on the shapes of the distributions that either X or Y has log-convex density, then the random vector of Ui:ns is smaller than the corresponding random vector of Vi:ns in the sense of multivariate likelihood ratio ordering.

A general composition theorem and its applications to certain partial orderings of distributions

A composition theorem for functions obeying certain positive ordering is proved. The novelty of the present version is that unlike earlier results which assume both components of the composition to be distributions or survival functions, one of the components is allowed to be negative and unbounded. The theorem is applied to yield very simple proof of characterizations for failure rate orderings of distributions given recently by Caperaa (1988). We also use this composition theorem to give a characterization of two distributions with ordered mean residual life functions.