Naftali A. Langberg

Small-Sample Results for the Kaplan-Meier Estimator

Abstract Whereas much is known about the asymptotic properties of the Kaplan-Meier (1958) estimator (KME) of a survival function, exact results for small samples have been difficult to obtain. In this article, we obtain an exact expression for the αth moment (α > 0) of the KME under a model of proportional hazards. This enables us, under proportional hazards, to (a) study the bias of the KME, and (b) compare the exact variance of the KME to its asymptotic variance.

Multivariate Phase-Type Distributions

A univariate random variable is said to be of phase type if it can be represented as the time until absorption in a finite state absorbing Markov chain. Univariate phase type random variables are useful because they arise from processes that are often encountered in applications, they have densities that can be written in a closed form, they possess some useful closure properties, and they can approximate any nonnegative random variable. This paper introduces and discusses several extensions to the multivariate case. It shows that the multivariate random variables possess many of the properties of univariate phase type distributions and derives explicit formulas for various probabilistic quantities of interest. Some examples are included.

REPAIR REPLACEMENT POLICIES

In this paper we introduce the concept of repair replacement. Repair replacement is a maintenance policy in which items are preventively maintained when a certain time has elapsed since their last repair. This differs from age replacement where a certain amount of time has elapsed since the last replacement. If the last repair was a complete repair, repair replacement is essentially the same as age replacement. It is in the case of minimal repair that these two policies differ. We make comparison between various types of policies in order to determine when and under which condition one type of policy is better than another.

Tests for Monotone Mean Residual Life Using Randomly Censored Data.

At any age the mean residual life function gives the expected remaining life at that age. Reliabilists and biometricians have found it useful to categorize failure distributions by the monotonicity properties of the mean residual life function. Hollander and Proschan (1975, Biometrika 62, 585-593) have derived tests of the null hypothesis that the underlying failure distribution is exponential, versus the alternative that it has a monotone mean residual life function. These tests are based on a complete sample. Often, however, data are incomplete because of withdrawals from the study and because of survivors at the time the data are analyzed. In this paper we generalize the Hollander-Proschan tests to accommodate randomly censored data. The efficiency loss due to the presence of censoring is also investigated.

MAINTENANCE COMPARISONS: BLOCK POLICIES

Complete repair and minimal repair models with a block maintenance policy are considered. Each of these models gives rise to a counting process, and these processes are compared stochastically. This contrasts with most previous work on maintenance policies where only univariate marginal comparisons were made. Also a more general block schedule is considered than is customary.

A New-Method For Estimating Life Distributions From Incomplete Data

Abstract : We construct a new estimator for a continuous life distribution from incomplete data, the Piecewise Exponential Estimator (PEXE). We show that the PEXE is strongly consistent under a mild restriction on the distribution of the censoring random variables (possible non-identical and non-continuous). Then we consider the Product Limit Estimator (PLE), introduced by Kaplan and Meier (1958). We prove the strong consistency of the PLE under a mild regularity condition on the distributions of the censoring random variables. This result extends previous ones obtained by various researchers. Finally we compare the new PEXE and traditional PLE. (Author)

COMPARISONS OF REPLACEMENT POLICIES

For independent random lifelengths of the units in use stochastic comparisons of the number of failures and removal in [0, s] under age and block replacement policies are performed. A new concept of NBU (NWU) in sequence is introduced.

DIFFERENTIAL ATTRITION Estimating the Effect of Crossovers on the Evaluation of a Medical Technology

The differential attrition of persons from comparison groups severely restricts the inferences that can be made from results of evaluative research. This problem is particularly troublesome in the evaluation of medical technologies, such as coronary artery bypass graft surgery, since a substantial percentage of medical or control patients cross over to the surgical group. A procedure using worst case assumptions is developed that allows researchers to estimate the maximum effect of differential attrition, and therefore enhance the quality of their inferences. The article first illustrates theprocedure, then concludes with a discussion of the generality of the estimation procedure to other instances in which differential attrition is a problem, and points out the limitations of the approach.

PRESENTATION OF PHASE-TYPE DISTRIBUTIONS AS PROPER MIXTURES

It is shown that any phase-type distribution can be represented as a proper mixture of two distinct phase-type distributions. Using different terms, it is shown that the class of phase-type distributions does not include any extreme ones. A similar result holds for the subclass of upper-triangular phase-type distributions.

A mixture of exponential and ifr gamma distributions having an upsidedown bathtub-shaped failure rate

We consider a mixture of one exponential distribution and one gamma distribution with increasing failure rate. For the right choice of parameters, it is shown that its failure rate has an upsidedown bathtub shape failure rate. We also consider a mixture of a family of exponentials and a family of gamma distributions and obtain a similar result.