Myles Hollander

Nonparametric Statistical Methods.

This Second Edition of Myles Hollander and Douglas A. Wolfes successful Nonparametric Statistical Methods meets the needs of a new generation of users, with completely up-to-date coverage of this important statistical area. Like its predecessor, the revised edition, along with its companion ftp site, aims to equip readers with the conceptual and technical skills necessary to select and apply the appropriate procedures for a given situation. An extensive array of examples drawn from actual experiments illustrates clearly how to use nonparametric approaches to handle one- or two-sample location and dispersion problems, dichotomous data, and one-way and two-way layout problems. An ideal text for an upper-level undergraduate or first-year graduate course, Nonparametric Statistical Methods, Second Edition is also an invaluable source for professionals who want to keep abreast of the latest developments within this dynamic branch of modern statistics.

Nonparametric Estimation With Recurrent Event Data

The problem of nonparametric estimation for the distribution function governing the time to occurrence of a recurrent event in the presence of censoring is considered. We derive Nelson–Aalen and Kaplan–Meier-type estimators for the distribution function, and establish their respective finite-sample and asymptotic properties. We allow for random observation periods for each subject under study and explicitly account for the informative sum-quota nature of the data accrual scheme. These allowances complicate technical matters considerably and, in particular, invalidate the direct use of martingale methods. Consistency and weak convergence of our estimators are obtained by extending an approach due to Sellke, who considered a single renewal process (i.e., recurrent events on a single subject) observed over an infinite time period. A useful feature of the present analysis is that strong parallels are drawn to the usual “single-event” setting, providing a natural route toward developing extensions that involve covariates (e.g., weighted log-rank tests, Cox-type regression, and frailty models). Another advantage is that we obtain explicit, closed-form expressions for the asymptotic variances for these estimators. This enables, for instance, the characterization of the efficiency loss that results from employing only the first, possibly right-censored, observation per subject. An interesting feature of these results is the prominent role of the renewal function. Finally, we discuss the case of correlated interoccurrence times, propose an estimator in the case where the within-unit interoccurrence times follow a gamma frailty model, and compare the performance of our estimators to an estimator recently proposed by Wang and Chang.

Testing to Determine the Underlying Distribution Using Randomly Censored Data.

Abstract : For right-censored data, a goodness-of-fit procedure is developed for testing whether the underlying distribution is a specified functions G. The test statistic C is the one-sample limit of Efrons (1967) two-sample statistic W. The test based on C is compared with recently proposed competitors due to Koziol and Green (1976) and Hyde (1977). The comparisons are on the basis of applicability, the extent to which the censoring distribution can affect the inference, and power. It is shown that in certain situations the C test compares favourably with the tests of Koziol-Green and Hyde.

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.

Likelihood Ratio-Based Confidence Bands for Survival Functions

Abstract Thomas and Grunkemeier introduced a nonparametric likelihood ratio approach to confidence interval estimation of survival probabilities based on right-censored data. We construct simultaneous confidence bands using this approach. The boundaries of the bands are contained within [0, 1]. A procedure essentially equivalent to a bias correction is developed. The resulting increase in coverage accuracy is illustrated by an example and a simulation study. We look at various versions of log-likelihood ratio-based confidence bands and compare them to the Hall-Wellner band and Nairs equal precision band. We also construct likelihood ratio-based bands for cumulative hazard functions.

27 Nonparametric concepts and methods in reliability

Publisher Summary This chapter reveals the nonparametric analysis in reliability and several classes of life distributions. Classes of life distributions are based on the notions of aging afford nonparametric statisticians an opportunity to consider problems of a character somewhat different from the usual. The chapter summarizes the definitions, physical interpretations, useful geometric characterizations, probabilistic properties of and logical relationships, and implication .among these classes of life distributions and formulates a variety of classes of life distributions based on notions of aging. The reliability operations considered are: (1) formation of coherent systems, (2) addition of independent life lengths (convolution of life distributions), (3) selection of a life length observation from one of a set of distributions (mixture of distributions), and (4) subjecting a device to shocks. The chapter also surveys nonparametric inference for these classes along with the discussion on the total time- on-test plots and empirical mean residual life functions.

Models for Recurrent Events in Reliability and Survival Analysis

Existing models forrecurrent phenomena occurring in public health, biomedicine, reliability, engineering, economics, and sociology are reviewed. A new and general class of models for recurrent events is proposed. This class simultaneously takes into account intervention effects, effects of accumulating event occurrences, and effects of concomitant variables. It subsumes as special cases existing models for recurrent phenomena. The statistical identifiability issue for the proposed class of models is addressed.

Dynamic reliability models with conditional proportional hazards

A dynamic approach to the stochastic modelling of reliability systems is further explored. This modelling approach is particularly appropriate for load-sharing, software reliability, and multivariate failure-time models, where component failure characteristics are affected by their degree of use, amount of load, or extent of stresses experienced. This approach incorporates the intuitive notion that when a set of components in a coherent system fail at a certain time, there is a ‘jump’ from one structure function to another which governs the residual lifetimes of the remaining functioning components, and since the component lifetimes are intrinsically affected by the structure function which they constitute, then at such a failure time there should also be a jump in the stochastic structure of the lifetimes of the remaining components. For such dynamically-modelled systems, the stochastic characteristics of their jump times are studied. These properties of the jump times allow us to obtain the properties of the lifetime of the system. In particular, for a Markov dynamic model, specific expressions for the exact distribution functions of the jump times are obtained for a general coherent system, a parallel system, and a series-parallel system. We derive a new family of distribution functions which describes the distributions of the jump times for a dynamically-modelled system.

A Class of Life Distributions for Aging

Abstract We introduce a new better than used of age t 0 (NBU-t 0) class of life distributions, where the survival probability at age 0 is greater than or equal to the conditional survival probability at specified age t 0 > 0. The dual class of new worse than used of age t 0 (NWU-t 0) life distributions is defined by reversing the direction of inequality. Preservation and nonpreservation properties of the two classes under various reliability operations are presented. We also show how to test, using a random sample, whether the NBU-t 0 class obtains.

A Chi-Squared Goodness-of-Fit Test for Randomly Censored Data

Abstract In this article, procedures analogous to Karl Pearsons well-known chi-squared goodness-of-fit test for a simple null hypothesis are developed under the random censorship model. It is shown that one straightforward analog of Pearsons statistic is diminished in applicability due to the form of its limiting distribution. This leads to the development of an asymptotically exact test based on a Wald-type statistic with a chi-squared limiting null distribution. This test is compared and contrasted theoretically and via a simulation with Akritas’ test with respect to significance levels, asymptotic local powers, and finite sample powers. The general conclusions from the simulation study are that the proposed test usually achieves the desired significance levels when the probability of observing a censored or an uncensored value in the last interval is not small, whereas Akritas’ test tends to be a bit anticonservative. On the other hand, Akritas’ test is more powerful than the proposed test in a model...