Hari Mukerjee

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.

Inferences Under a Stochastic Ordering Constraint: The k-Sample Case

If X1 and X2 are random variables with distribution functions F1 and F2, then X1 is said to be stochastically larger than X2 if F1 ≤F2. Statistical inferences under stochastic ordering for the two-sample case has a long and rich history. In this article we consider the k-sample case; that is, we have k populations with distribution functions F1, F2, … , Fk,k ≥ 2, and we assume that F1 ≤ F2 ≤ ˙˙˙ ≤ Fk. For k = 2, the nonparametric maximum likelihood estimators of F1 and F2 under this order restriction have been known for a long time; their asymptotic distributions have been derived only recently. These results have very complicated forms and are hard to deal with when making statistical inferences. We provide simple estimators when k ≥ 2. These are strongly uniformly consistent, and their asymptotic distributions have simple forms. If and are the empirical and our restricted estimators of Fi, then we show that, asymptotically, for all x and all u > 0, with strict inequality in some cases. This clearly shows a uniform improvement of the restricted estimator over the unrestricted one. We consider simultaneous confidence bands and a test of hypothesis of homogeneity against the stochastic ordering of the k distributions. The results have also been extended to the case of censored observations. Examples of application to real life data are provided.

Estimation of Survival Functions under Uniform Stochastic Ordering

Abstract If S and T are survival functions for two life distributions, then S is said to be uniformly stochastically smaller than T, denoted by S ≪ T, if θ(x) ≡ S(x)/T(x) is nonincreasing in x on {x: T(x) > 0}. This ordering is transitive. Uniform stochastic ordering (USO) has found important applications in nonparametric accelerated life testing, among other areas. It has been shown that the nonparametric maximum likelihood estimator (NPMLE) of S under USO when T is known is inconsistent. Dykstra, Kochar Robertson derived the restricted NPMLEs of several unknown survival functions linearly ordered by USO. This article shows that these too are inconsistent in general. Rojo and Samaniego gave excellent ad hoc estimators of S and T when the other is known. Based on their idea for the one-sample problem, they gave two ad hoc estimators (one of them only implied) of S and T when they are both unknown. These are consistent, but they lack some desirable properties. This article introduces a one-parameter famil...

Order-Restricted Inferences in Linear Regression

Abstract Regression analysis constitutes a large portion of the statistical repertoire in applications. In cases where such analysis is used for exploratory purposes with no previous knowledge of the structure, one would not wish to impose any constraints on the problem. But in many applications we are interested in curve fitting with a simple parametric model to describe the structure of a system with some prior knowledge about the structure. An important example of this occurs when the experimenter has a strong belief that the regression function changes monotonically with some or all of the predictor variables in a region of interest. The analyses needed for statistical inferences under such constraints are nonstandard. Considering the present body of knowledge developed for unconstrained regression, it will be an enormous task to derive the analogs of even a small fraction of this for the restricted case. In this article we initiate the study with simple linear regression on a single variable. The est...

Estimation of Cumulative Incidence Functions in Competing Risks Studies Under an Order Restriction

Abstract In the competing risks problem an important role is played by the cumulative incidence function (CIF), whose value at time t is the probability of failure by time t for a particular type of failure in the presence of other risks. Its estimation and asymptotic distribution theory have been studied by many. In some cases there are reasons to believe that the CIFs due to two types of failure are order restricted. Several procedures have appeared in the literature for testing for such orders. In this paper we initiate the study of estimation of two CIFs subject to a type of stochastic ordering, both when there are just two causes of failure and when there are more than two causes of failure, treating those other than the two of interest as a censoring mechanism. We do not assume independence of the two types of failure of interest; however, these are assumed to be independent of the other causes in the censored case. Weak convergence results for the estimators have been derived. It is shown that when the order restriction is strict, the asymptotic distributions are the same as those for the empirical estimators without the order restriction. Thus we get the restricted estimators “free of charge”, at least in the asymptotic sense. When the two CIFs are equal, the asymptotic MSE is reduced by using the order restriction. For finite sample sizes simulations seem to indicate that the restricted estimators have uniformly smaller MSEs than the unrestricted ones in all cases.

Estimation of two ordered mean residual life functions

Abstract If X is a life distribution with finite mean then its mean residual life function (MRLF) is defined by M ( x )= E [ X − x | X > x ]. It has been found to be a very intuitive way of describing the aging process. Suppose that M 1 and M 2 are two MRLFs, e.g., those corresponding to the control and the experimental groups in a clinical trial. It may be reasonable to assume that the remaining life expectancy for the experimental group is higher than that of the control group at all times in the future, i.e., M 1 ( x )⩽ M 2 ( x ) for all x . Randomness of data will frequently show reversals of this order restriction in the empirical observations. In this paper we propose estimators of M 1 and M 2 subject to this order restriction. They are shown to be strongly uniformly consistent and asymptotically unbiased. We have also developed the weak convergence theory for these estimators. Simulations seem to indicate that, even when M 1 = M 2 , both of the restricted estimators improve on the empirical (unrestricted) estimators in terms of mean squared error, uniformly at all quantiles, and for a variety of distributions.

Restricted estimation of the cumulative incidence functions corresponding to competing risks

In the competing risks problem, an important role is played by the cumulative incidence function (CIF), whose value at time t is the probability of failure by time t from a particular type of failure in the presence of other risks. In some cases there are reasons to believe that the CIFs due to various types of failure are linearly ordered. El Barmi et al. (3) studied the estimation and inference procedures under this ordering when there are only two causes of failure. In this paper we extend the results to the case of k CIFs, where k ≥ 3. Although the analyses are more challenging, we show that most of the results in the 2-sample case carry over to this k-sample case.

Consistent estimation of survival functions under uniform stochastic ordering; the k -sample case

Let S 1 , S 2 , ? , S k be survival functions of life distributions. They are said to be uniformly stochastically ordered, S 1 ? u s o S 2 ? u s o ? ? u s o S k , if S i / S i + 1 is a survival function for 1 ? i ? k - 1 . The nonparametric maximum likelihood estimators of the survival functions subject to this ordering constraint are known to be inconsistent in general. Consistent estimators were developed only for the case of k = 2 . In this paper we provide consistent estimators in the k -sample case, with and without censoring. In proving consistency, we needed to develop a new algorithm for isotonic regression that may be of independent interest.

Order restricted inference for comparing the cumulative incidence of a competing risk over several populations

There is a substantial literature on testing for the equality of the cumulative incidence functions associated with one specific cause in a competing risks setting across several populations against specific or all alternatives. In this paper we propose an asymptotically distribution-free test when the alternative is that the incidence functions are linearly ordered, but not equal. The motivation stems from the fact that in many examples such a linear ordering seems reasonable intuitively and is borne out generally from empirical observations. These tests are more powerful when the ordering is justified. We also provide estimators of the incidence functions under this ordering constraint, derive their asymptotic properties for statistical inference purposes, and show improvements over the unrestricted estimators when the order restriction holds.

A strong law of large numbers for nonparametric regression

Suppose (i1,n, ..., in,n) is permutation of (1, ..., n) for each positive integer n such that the order of the indices {1, h., n - 1} in the permutation corresponding to n - 1 is preserved. If {Zn} is a sequence of mean-zero random variables and {kn} is a sequence of positive integers with kn --> [infinity] and kn/n --> 0, we prove max1 0 a.s. under a first moment-type assumption on {Zn} and appropriate conditions on the permutations and the growth rate of {kn}. The result is applied to prove strong consistency of nonparametric estimators of regression functions with heavy-tailed error distributions using the k-nearest neighbor and the unikform kernel methods under similar moment assumptions on the conditional distributions of the regressed variable.