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.

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

In the present work, we introduce a partially sequential test for monitoring the location parameter of a population at phase II when a standard sample of prefixed size from phase I population is available. We consider a typical pairwise dependence between the samples from phase I and phase II. The entire problem is motivated from the statistical issues related to environmental monitoring. The proposed test is primarily based on ranks and hence it is closely nonparametric in nature. We use an inverse sampling scheme for collecting samples in phase II. Asymptotic distribution of the stopping variable is derived under suitable assumptions. We carry out detailed simulation studies to examine the power performance of the proposed test. Some numerical findings obtained through Monte Carlo studies are summarized graphically. The key advantage of the proposed method over traditional fixed-sample-size-based tests is its quicker detection property with high precision. This helps us to save precious phase I...

A Near-Nonparametric Partially Sequential Test for Monitoring Phase II Location Under Pairwise Dependence Between Two Phases

The object of the present investigation is to provide suitable nonparametric tests for the identity of two univariate populations when there is a random sample of fixed size from one of the populations and the observations from the second population are drawn sequentially by an inverse binomial sampling. A progressive censoring test procedure corresponding to the inverse binomial sample is proposed. Different large sample results of the proposed tests are formulated and examined.

Progressive censoring under inverse sampling for nonparametric two-sample problems

The present article provides two partially sequential nonparametric tests (see Wolfe, 1977a) for testing the identity of two location parameters without making any assumptions on the shapes of the distributions of the underlying populations. The proposed tests use 0–1 scores while defining the stopping variable, and are based on the partial sequential placement statistics introduced by Orban and Wolfe (1980). Some asymptotic results related to the proposed procedures are studied. Competitive tests based on fixed sample sizes are also found. Further various exact and asymptotic performance measures of the tests are investigated. Recommended by Edit Gombay

Partial Sequential Tests for Generalized Behrens-Fisher Problem

In this article, we consider the problem of testing the equality of the medians of two continuous populations without the assumption that the populations have the same shape. This is called the generalised Behrens-Fisher problem. We propose a partial sequential test procedure using placement statistics (see Orban, J. and Wolfe, D.A. (1980). Distribution-free partially sequential placement procedures. Commun. Statist- Theor., 9, 883–904) for 0–1 scores. The proposed test procedure is based on a progressively censored scheme (see Sen, P.K. (1978). Nonparametric repeated significance tests. In: Krishnaiah, P.R. (Ed.), Development in Statistics, Vol. 1, pp. 227–264 and also Chatterjee, S.K. and Sen, P.K. (1973). Nonparametric testing under progressive censoring. Cal. Statist. Assoc. Bull., 22, 13–50), where we have a random sample of fixed size from one population and, from the other population, the observations are drawn sequentially up to a pre-fixed target sample size n (usually large). Some asymptotic res...

A progressive censored test for the generalised Behrens-Fisher problem

Io (3.10) (p 50, first line) of the paper, it was stated that hr.mEv+ in ~ the one-sided case <-win the two-sided case). But as with r tending to ao ,m also tends to infinity, the conditional argument given in support of this was not correct. However, the conclusion (3.10), and hence, Theorem 3.1 holds, This can be seen as follows. It may be verified that {S n .m ; l <;n < r} where S n .m is given by ( 3.4 ), is a martingale sequence uader 8 0 • Applying Theorem 2 of Brown (1971, p 60), it can be easily shown that, under H 0 , for fixed t,

Uttam Bandyopadhyay

Papers

Semi-Sequential Nonparametric tests for two Populations with one Sample size fixed: