International Journal of Science and Research (IJSR) ISSN (Online): 2319-7064 Index Copernicus Value (2013): 6.14 | Impact Factor (2015): 6.391 Volume 5 Issue 10, October 2016 www.ijsr.net Licensed Under Creative Commons Attribution CC BY Comparison between Two-Sample Adaptive Tests and Traditional Tests in Location Problem Chikhla Jun Gogoi 1 , Dr. Bipin Gogoi 2 1 PhD Research Scholar, Department of Statistics, Dibrugarh University, Assam, India 2 Professor, Department of Statistics, Dibrugarh University, Assam, India Abstract: The present paper considers power comparison between two sample Adaptive tests and traditional non parametric tests in the field of location problem. In the proposed approach it is shown by Monte- Carlo simulation study, that the Adaptive tests are superior to the other existing competitors in terms of both robustness of size and power. Keywords: Adaptive tests, Gastwirth test, Wilcoxon test, HFR test, long tailed test, t Test , Monte Carlo simulation 1. Introduction One of the fundamental problems of statistics, often encountered in applications, is the two-sample location problem. In the two-sample location problem the application of the t-test depends on very restrictive assumptions such as normality and equal variances of the two random variables X 1 and X 2 . If the assumptions of the t-test are not satisfied it is more appropriate to apply a robust version of the t-test, like the Welch test or the trimmed t-test, or a nonparametric test, like the Wilcoxon. But usually we have no information about the underlying distribution of the data. Therefore, an adaptive test should be applied. It would be desirable, therefore to use data itself to determine the nature of F(.), and on the basis of that information, we could choose an appropriate set of scores. We would then use that same data to perform the test. Such two-stage analyses are termed as adaptive test. In the past seven decades many important distribution free tests for differences in location between samples had been developed. In the mid 1940s the Wilcoxon -Mann- Whitney test was introduced for testing differences in location between two samples and it was developed by Wilcoxon and extended by Mann and Whitney. But further, it turns out that there exist simple adaptive rank tests that can discover differences between distributions more easily than WMW tests. These adaptive non parametric procedures display significant improvements in power over the parametric t- test in samples of large and moderate sizes. The purpose of this chapter is two folds , first to introduce the selector statistics , secondly compare the t-test with adaptive distribution-free test like Wilcoxon test, test based on scores under normality and under different models of nonnormality, like heavy tailed or asymmetric distributions Adaptive tests are important in applications because the practicing statistician usually has no information about the underlying distribution . The adaptive testing procedures that are truly nonparametric distribution-free. That is, the two stages of the inference process are constructed in such a way that it control the overall  -level . Monte-Carlo simulations are used for comparison of the tests with respect to level  and power . 2. Selector statistics for selection of test We apply the concept of Hogg(1974) that is based on following lemma: (i) Let F denote the class of distributions under consideration. Suppose that each of k tests T 1 , T 2 , …, T k is distribution –free over F , that is     ∈   =  for each F∈F , h = 1, . . ., k. (ii) Let S be some statistic (called a selector statistic) that is, under Ho, independent of T 1 , . . . , T k for each F∈F . Suppose we use S to decide which test T h to conduct. Specially, let M s denote the set of all values of S with the following decomposition: M s =  1 ∪ 2 ∪… ∪  ,   ∩   = ∅  ≠. So that ∈  corresponds to the decision to use test   . The overall testing procedure is then defined by: If ∈  then reject Ho if   ∈  . This two-staged adaptive test is distribution-free under Ho over the class F ,i.e. it maintains the level  for each F∈F . The proof of this lemma is given by Randle and Wolfe(1979).Using the lemma, as a selector statistic, we use a function of order statistics of combined sample. We choose the selector statistic as S = (  1  ,  2  ) Table 1.1: Theoretical values of Q 1 and Q 2 for selected distributions Distribution Q 1 Q 2 Uniform(0,1) Normal Exponential(with λ=1) 1 1 4.569 1.9 2.585 2.864 Where  1  =   0.05 −  0.5   0.5 −  0.05 and   2 =    0.05 −  0.05   0.5 −  0.5 And Hogg’s(1974) measures for skewness and tailweight, and    ,    and    denote the average of the smallest , middle and largest  order statistics, respectively, in the combined sample; fractional items are used when  is not an integer. Obviously   1 = 1 if the data are symmetric and   1 <1 (>1) if the data are skewed to the left(right) . The Paper ID: ART20162024 104