Inverse Sampling Based on General Scores for Non-Parametric Two-Sample Problems
TL;DR: In this paper, 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, is investigated.
Abstract: 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.
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TL;DR: In this paper, the authors developed two nonparametric partially sequential tests for detecting possible presence of linear trend among the incoming series of observations, assuming that a sample of fixed size is available a priori from some unknown univariate continuous population and there is no sign of trend among these historical observations.
8 citations
TL;DR: In this article, a simple two-stage monitoring rule for detecting small disorders in a two-sample location problem is proposed, which is based on ranks and hence is nonparametric in nature.
Abstract: We propose a simple two-stage monitoring rule for detecting small disorders in a two-sample location problem. The proposed rule is based on ranks and hence is nonparametric in nature. In the first stage, we use a sequential monitoring scheme to decide the necessity of employing a location test at some point of time. If there is urgency, we simply use a two-sample Wilcoxon rank sum test in the second stage. This leads to a semi sequential one-shot monitoring procedure. We study some asymptotic performance of the proposed rule. We also present some numerical findings obtained through Monte Carlo studies. The proposed rule meets the challenge of controlling type I error rate in sequential monitoring of an incoming series of observations.
7 citations
TL;DR: Two nonparametric tests for the identity of some unknown univariate continuous distribution functions against monotone or unidirectional trend in location are developed.
Abstract: Recently, Mukherjee and Bandyopadhyay (J Stat Plan Inference, 2011, doi:
10.1016/j.jspi.2011.02.017
) introduced some partially sequential tests for detecting liner trend among the incoming series of observations when a training sample is available a-priori. Their work is very useful in econometric or environmental monitoring under certain situations. The present work is intended for generalization of their tests for any monotone trend. We develop two nonparametric tests for the identity of some unknown univariate continuous distribution functions against monotone or unidirectional trend in location. One of these two tests is based on usual ranks and the other is based on sequential ranks. These are typical nonparametric tests for monitoring structural changes. Performance of the two tests are compared using asymptotic studies as well as through some numerical results based on Monte-Carlo simulations. An illustration is offered using a real data on monthly production of certain beverage.
7 citations
TL;DR: A new nonparametric simultaneous test for dual alternatives based on contrasts that helps to detect three underlying pattern much more efficiently and is illustrated through a motivating example related to highly sensitive issue of arsenic contamination in ground water.
Abstract: In this paper, we propose a new nonparametric simultaneous test for dual alternatives Simultaneous tests for dual alternatives are used for pattern detection of arsenic contamination level in ground water We consider two possible patterns, namely, monotone shift and an umbrella-type location alternative, as the dual alternatives Pattern recognition problems of this nature are addressed in Bandyopadhyay et al [5], stretching the idea of multiple hypotheses tests as in Benjamini and Hochberg [6] In the present context, we develop an alternative approach based on contrasts that helps us to detect three underlying pattern much more efficiently We illustrate the new methodology through a motivating example related to highly sensitive issue of arsenic contamination in ground water We provide some Monte-Carlo studies related to the proposed technique and give a comparative study between different detection procedures We also obtain some related asymptotic results
7 citations
Cites background from "Inverse Sampling Based on General S..."
...Readers may be refereed to Wolfe [26], Orban and Wolfe [19,20], Costello and Wolfe [12], Chatterjee and Bandyopadhyay [10], Bandyopadhyay and Mukherjee [2], Bandyopadhyay et al....
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...Readers may be refereed to Wolfe [26], Orban and Wolfe [19,20], Costello and Wolfe [12], Chatterjee and Bandyopadhyay [10], Bandyopadhyay and Mukherjee [2], Bandyopadhyay et al. [3–5] and Mukherjee [16,17], among others, for further details....
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TL;DR: In this paper, 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 is introduced.
Abstract: 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...
6 citations
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TL;DR: In this paper, a class of test procedures for the two-sample setting when one of the samples has a fixed sample size, and the second is obtained by an inverse binomial sequential sampling scheme is proposed.
Abstract: A class of test procedures is proposed for the two-sample setting when one of the samples has a fixed sample size, and the second is obtained by an inverse binomial sequential sampling scheme. These procedures allow either a classical approach or a completely nonparametric attack to a wide variety of two-sample problems, Properties of the tests are discussed, Including limiting distributions as the fixed sample size tends to infinity, and the capability of obtaining asymptotic power restrictions against specified alternatives of interest. Criteria for selecting a particular procedure are discussed.
43 citations
Additional excerpts
..., m +I are known to have (see Wilks (1962) Chapter 7) EF(C,)-= (m + o-l, CovF (C,, Ck)= ((m + 1)o,kf(m + 1) 2 (m +2))-( (m + 1)2(m + 2) )....
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TL;DR: In this article, it was shown that the conditional argument given in support of this was not correct, 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,
Abstract: 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,
7 citations