M. M. Siddiqui

Some Criteria for Aging

Abstract The concept of “aging,” or progressive shortening of an entitys residual lifetime, is discussed in terms of the entitys survival time distribution. Quantities defined to describe the aging phenomenon include the “specific aging factor,” “hazard rate,” “hazard rate average,” and “mean residual lifetime.” A set of seven criteria for aging is established, based on these quantities, and a chain of implications among the criteria is developed. The hazard rate average and mean residual lifetime are noted as being particularly useful for empirical studies. An application of these two quantities is illustrated for a set of empirical survival time data.

Robust Estimation of Location

Abstract The problem of estimating a location parameter from a random sample when the form of distribution is unknown or there is contamination of the target distribution is attacked by deriving estimators which are efficient over a class of two or more forms (“pencils”) of continuous symmetric unimodal distributions. The pencils considered are the normal, double exponential, Cauchy, parabolic, triangular, and rectangular (a limiting case). The estimators considered are special symmetrical linear combinations of order statistics: trimmed means, Winsorized means, “linearly weighted” means, and a combination of the median and two other order statistics. These are also compared asymptotically with a Hodges-Lehmann estimator. The theory required for deriving asymptotic variances is outlined. Efficiences are tabulated for sample sizes of 4 or 5, 8 or 9, 16 or 17, and ∞. Asymptotic efficiences of at least 0.82 relative to the best estimator for any single pencil are achieved by using the best trimmed mean or li...

Simple Regression Methods for Survival Time Studies

Abstract Given a set of grouped survival data, least squares estimates are proposed for the parameters of four survival distributions that can be fit: exponential, linear hazard, Gompertz and Weibull. Sample estimates of the hazard function are utilized in the least squares procedures and a method is given for selecting a distribution for further investigation based on the likelihood under the four survival models. A Monte Carlo study demonstrated that the least squares estimates are nearly as efficient as maximum likelihood when the sample size is 50 or more. The methods are applied to survival data for 112 patients with plasma cell myeloma.

ASYMPTOTICALLY ROBUST ESTIMATORS OF LOCATION

Abstract The robustness properties of four estimators of location are studied with respect to eight distribution types. For each type, the probability density function is symmetric about the median and the range of variate is infinite. For the entire class of distributions, the estimator with the highest guaranteed efficiency is the mean of the middle fifty percent of the sample. This study supplements the paper by Crow and Siddiqui (1967).

Residual lifetime distribution and its applications

Abstract Let T be a continuous positive random variable representing the lifetime of an entity. This entity could be a human being, an animal or a plant, or a component of a mechanical or electrical system. For nonliving objects the lifetime is defined as the total amount of time for which the entity carries out its function satisfactorily. The concept of aging involves the adverse effects of age such as increased probability of failure due to wear. In this paper, we consider certain characteristics of the residual lifetime distribution at age t , such as the mean, median, and variance, as describing aging. Gamma and Weibull families of distributions are studied from this point of view. Explicit asymptotic expressions for the mean, variance and the percentiles of corresponding residual lifetime distributions are found. Finally these families of distributions are fitted to four sets of actual data, two of which are entirely new. The results can be used in discriminating different shape parameters.

Asymptotic Joint Distribution of Linear Systematic Statistics from Multivariate Distributions

Abstract The asymptotic joint distribution of an arbitrary number of linear systematic statistics (that is, linear combinations of order statistics), when observations are made on a random vector, is shown to be normal under fairly general conditions. The linear systematic statistics may correspond to the same or to different components of the vector. Formulas for evaluating the parameters of the asymptotic normal distribution are derived. As an illustration, these are applied to the case of trimmed means when the distribution sampled is bivariate normal.

A Combinatorial Test for Independence of Dichotomous Responses

Abstract Dealt with in this paper is the analysis of phenomena involving simultaneous responses among members of small groups during a finite number of discrete time intervals. An exact combinatorial test obtained by conditioning on the sufficient statistics under a null hypothesis is proposed. A specific application of this test is given for the analysis of asthma attack data of asthmatic individuals.

Record Values in Markov Sequences

Publisher Summary This chapter discusses the upper record values of the sample sequences of a stochastic process. If {Xn, n ≥ 1} is a Markov sequence possessing transition densities, the record sequence obtained from it is also a Markov sequence and has transition densities. The possible defectiveness of the record sequence makes a straightforward analysis of the record process, through the transition densities, very troublesome. The chapter describes two transition-density-like functions, which together can give complete information about the record process.

When a- and d-oftimality conflict

Let . be an m≠1 vector of unknown parameters. We consider a situation where, within a class of two unbiased estimators is D-optimal and is A-optimal. For example, such a situation arises in reliability studies (Srivastava, 1986). We present some results on comparing the variances of the unbiased estimators of certain interesting linear functions .

The consistency of a matching test

Abstract Let X ∗ 1 ⩽X ∗ 2 ⩽⋯⩽X ∗ n be the order statistics of a random sample from a distribution on [0, 1]. Let Ak, the kth match, be the event that X ∗ k ϵ( (k−1) n k n ], and let Sn be the total number of matches. The consistency of Sn for testing uniform df, U, against df G≠U is investigated, and it is shown that Sn is consistent if the intersection of G with U has Lebesgue measure zero. It is also consistent against a sequence of alternatives approaching U at a rate less faster than n -1 2 .