Shoutir Kishore Chatterjee

Nonparametric Testing Under Progressive Censoring

Summary Progressive censoring schemes (allowing a continuous monitoring of experimentation until a terminal decision is reached) are often adopted in clinical trials and life testing problems. In this paper, a general class of rank order tests for progressive censoring is proposed. Along with a basic martinga:le property and a Brownian motion approximation for a related rank order process, asymptotic distribution theory of the proposed statistics is developed. Asymptotic performance characteristics of the proposed tests (in the light of Bahadur efficiency and the stochastic smallness of the stopping variables) are studied.

Non-Parametric Tests for the Bivariate Two-Sample Location Problem

Summary In this paper, the Wilcoxon-Mann-Whitney rank-sum test and Moods median test for the univariate two-sample location problem have been extended to the case of two variables, following a conditional approach. Various properties of the proposed tests have been studied and their asymptotic powers compared.

Inverse Sampling Based on General Scores for Non-Parametric Two-Sample Problems

The problem considered in this paper is that of testing the identity of two univariate distribution functions F1(x) and F9(x) when there is a sample of fixed size from F1 (x) and the observations (Y) from F2(x) are drawn sequentially. Two score functions 4> 1(u), 0 < u < 1, l= I, 2, which reflect departures from the null hypothesis in specific ways are taken. Writing F~. (x) for a certain uniformly consistent estimate of F1 (x) based on the first sample, sums of the type S,. 01 = £ 4> 1 (Ft. (Y;)) are calculated for successive observations ;-1 on Y. Observation is stopped as soon as S,. ( 11 reaches a pre-fixed upper bound. The hypothesis is rejected or accepted on the basis of the terminal value of Sn <•>. Different large sample procedures based on this approach are formulated and examined.

On the Power Superiority of Certain Bivariate Location Tests Against Restricted Alternatives

FoR testing H0 : TJ = 0 against H1 : TJ > 0 on the basis of a random sample from N (TJ, 1), the one-tailed test based on the sample mean is uniformly most powerful, and hence more powerful than the two-tailed test. Similarly in the non-parametric formulation for the univariate location problem against one-sided alternatives standard one-tailed tests are asymptotically more powerful than the corresponding two-tailed tests. In the bivariate case, for testing H0 : TJ 1 = 0, TJ 2 = 0 against the restricted alternatives H1 : TJ1 > 0, TJ 2 > 0, (TJ1 , TJ2) t= (0, 0) for a bivariate normal population with means TJ1 , TJ 2 , no uniformly most powerful test is available. However, a likelihood ratio test has been derived [see Kudo (1963), Nuesch (1966)] for the problem. Although numerical studies indicate that this likelihood ratio test would be more powerful in the first quadrant than the standard test for H0 against all alternatives, the result has not so far been established theoretically. For a corresponding (two-sample) nonparametric problem, tests against restricted alternatives have been derived by an extended Union-Intersection principle in Chatterjee and De (1972). Here again, a gap has to be filled by proving that these tests would be asymptotically more powerful in the first quadrant than the corresponding unrestricted tests. The object of this paper is to prove a result which fills in these gaps. Let p(x1 , x 2/TJ 1 , TJ 2, p) denote the density function .of a bivariate normal distribution with mean vector (TJ~> TJ2), variance I and correlation coefficient p.

On Stochastic Ordering and a General Class of Poverty Indexes

Although measures of diversity and inequality have been extensively proposed in socio-economic and health perspectives, violation of a subtle monotonicity criterion (under stochastic ordering) diminishes their rationality and utility in the context of poverty and other ecomomic indexes. A modification is proposed here to develop certain stochastic ordering monotonicity preserving diversity measures that overcome this drawback. Such measures are shown to be invariant under increasing transformation, and thereby appropriate for (partially) ordered categorical response data models. AMS{2000) Subject Classification: 62H17, 62-07.

Asymptotically Minimal Multivariate Tolerance Sets

Multivariate β-content tolerance sets with asymptotic confidence level γ ( 0 < γ ⩽ 1 ) are defined and the requirement of asymptotic minimality for such sets is formulated. A method of construction of asymptotically minimal tolerance sets baFed on uniformly consistent density estimates ia developed. Application of the method in certain parametric situations based on parameter estimates and certain nonparametric situations based on window estimates of the density is considered.

Estimation of variance components in an unbalanced one-way classification

Abstract In the unbalanced one-way random effects model the weighted least squares approach with estimated weights is used to develop a relatively simple estimator of variance components. As the number of classes increases, the proposed estimator is seen not only to be best asymptotically normal but also to be asymptotically equivalent to the maximum likelihood estimator.

Rank procedures for some two-population multivariate extended classification problems

Given independent samples from three multivariate populations with cumulative distribution functions F(1)(x), F(2)(x), and F(0)(x) = [theta]F(1)(x) + (1 - [theta])F(2)(x), where 0

An Alternative Approach to the Anova Problem

When the hypothesis of equality of means of several univariate homoscedastic normal populations from which independent samples are available, is rejected on the basis of the ANOV A F-test, the populations have to be divided into disjoint groups such that within each group the populations are all same. For any such division a natural index of correctness of the grouping is defined and a procedure is proposed such that it leads to the choice of a division for which the true index does not fall below a stipulated threshold at a given level of confidence. The procedure is illustrated through some numerical examples.

Detailed Statistical Inference-multiple Decision Problem

The procedure for detailed statistical inference developed by the authors in an earlier paper (1992) for the two-decision case, is extended here to the case of several decisions. Alongwith the rule for choosing the decision, the problem of stating data dependent measures of confidence in terms of betting odds, is considered. The extension involves generalization oftlie coneept of legitimacy of betting odds introduced in the earlier paper and the choice of a suitable utility function for bets. The actual solution is worked out in the case of logarithmic utility. A rather intricate mathematical result requires to be established to prove the existence of an optimum rule in this case. Application of the procedure is illustrated through some numerical examples.