P. V. Rao

Berry-esseen bound for the kaplan-meier estimator

The Berry-Esseen bound for U-statistics, established by Helmers and Van Zwet (1982), is combined with the Breslow-Crowley (1974) bounds for the difference between the empirical cumulative hazard and the Kaplan-Meier cumulative hazard estimators of the survival function to derive a Berry-Esseen bound for the Kaplan-Meier estimator. We show that there exists an absolute quantity K such that the absolute difference between the standardized distribution function of Kaplan-Meier estimator at a fixed time point t and the standard normal cumulative distribution function is bounded above by where S(·) is the survival function and σ1is defined in Lemma 1.

An estimator of relative potency

The Kolmogorov-Sxnirnov statistic is used as a basis for defining nonparaim trie point and interval estimators for the relative potency of a test preparation with respect to a standard preparation These estimators compare favorably with those proposed by Sen (1963) if the dosage distribuiton has heavy mass in either of its tails.

Tests of symmetry based on watson statistics

A family of statistics, having the form of the goodness-of-fit and two-sample statistics of Watson ‘1961, 1962», is defined for testing the symmetry of a continuous distribution about a known mediam. The common asymptotic null distribution of these statistics is derived. One statistic in the family is shown to possess desirable invariance and power properties. Tables of exact critical values for sample sizes 3 < n < 26 are given for this statistic

A modified tiao-guttman rule for multiple outliers

A modification of the rule suggested by Tiao and Guttman (1967) is considered for the estimation of the mean of a normal population using data with an unspecified number of outliers. The properties of the modified rule are investigated by considering its premium and protection. For the case where the number of outliers, i, is specified, it is shown that the Tiao-Guttman tables for m 2 are adequate for all values of m.

Improved Estimation of Survival Functions in the New-Better-Than-Used Class

In this article, we consider the problem of nonparametric estimation of life distributions that satisfy the New-Better-Than-Used (NBU) property. Focus is on estimators that possess the NBU property. We propose several possible estimators and compare their properties with the properties of two previously known estimators. We conclude that two of the estimators introduced in this article perform better than the previously known estimators—especially if the sample sizes are not large.

On minimum distance estimation of location based on the kolmogorov statistic

The asymptotic distribution is derived for the minimum distance estimator of a location parameter based on the Kolmogorov goodness of fit statistic. The distribution is expressed in terms of the distribution of a functional of a Brownian bridge. An upper bound is obtained for the length of the confidence interval based on the Kolmogorov statistic. A simulation study with sample sizes 10 and 20 compares the length of the interval based on the Kolmogorov statistic to the length of the interval based on the maximum likelihood estimator. Another simulation shows the effect of model misspecification on the coverage probabilities of the interval based on the Kolmogorov statistic.

Efficacies of some rank tests for censored bivariate data

Counting process representations are used to study the efficacies of two classes of censored paired data statistics one proposed by Woolson and Lachenbruch (1980) and the other by Popovich and Rao (1985). Expressions for efficacies are derived using contiguous sequences of parametric alternatives. The results are used to determine the form of the statistic with maximum efficacy in each class. Relative efficiencies are evaluated under a log-linear model assuming a variety of values for the correlation coefficient between the log of survival times and for the probability of double censoring. The relative efficiency comparisons indicate that the statistics proposed by Popovich and Rao perform as well as those proposed by Woolson and Lachenbruch. When the probability of double censoring is high. the former statistics have a slight advantage over the other.

On the estimation of a survival function when new is better than used of a specified age

Hollander, Park and Proschan [1] introduced the NBU-{t 0} class of survival functions for modelling lifetime distributions for which the conditional probability of surviving an additional time x, given survival up to time t 0, is less than or equal to the unconditional probability of surviving longer than time x. Reneau and Samaniego [2] proposed a method of estimating an NBU-{t 0} survival function. In this article, we propose a modification of the Reneau-Samaniego estimator with a view to improve its optimally properties. Based on some theoretical investigations and a simulation study, we conclude that the modified estimator has several desirable properties including almost sure consistency, improved goodness-of-fit and reduced mean square error.

Optimal Weights for a Class of Rank Tests for Censored Bivariate Data

The problem of testing equality of survival distributions on the basis of paired censored survival data has received considerable attention in literature. Some of the important statistics used for such purposes can be expressed as linear combinations of two statistics, one based on uncensored pairs and the other based on the censored pairs. Raychaudhuri and Rao (Nonparametric Statistics, 1996, 6, 1–11) investigated properties of two classes of such statistics and derived expressions for the optimal coefficients (weights) for the linear combination that will maximize efficacy within each class. As the optimal weights depend upon the form of the underlying survival and censoring distributions, statistics with optimal weights can only be used with estimated weights. This article presents a method of estimating optimal weights on the basis of an assumed model that specifies the distribution of the difference between the observed survival times conditional on the censoring pattern. The model, in addition to dispensing with the usual assumption that the survival and censoring variables are independent, also permits a graphical check of its lack of fit on the basis of observed data. The performance of statistics with the estimated weights is evaluated by using two simulation studies – one with data generated under the assumed model and the other assuming independence of the survival and censoring times. Simulation results show that the optimal statistics with estimated weights have good power properties in all cases considered, and that they compare well with other commonly used tests for paired censored survival data. An advantage of the tests with optimal weights is that, unlike their competitors, these tests have demonstrated performance characteristics in some cases where the assumption of independent censoring may not be justified.

On the use of the scholz-sievers method for dlstribution-free inference about the difference in slopes of two regression lines

A simple adaptation of a distribution-free method due to Scholz (1978) and Sievers (1978) for inference in a single regression setting is proposed for inference about the difference in slopes of two regression lines. We assume that the data are obtained from a designed experiment with common regression constants. A comparison of the proposed method to its competitors-one due to Hollander and the other due to Rao and Gore-indicates superiority of the new method.