A Nonparametric Two-Sample Test for Scale Using Linear Combination of Quadratic and Linear Function of Placements

Abstract

We propose a distribution-free test for the nonparametric two-sample scale problem. We discuss the problem of testing the homogeneity of two populations against scale alternatives. Contrary to the usual assumption that both populations have a common median, we assume both populations to have a common quantile of order α; not necessarily half. Without loss of generality, the common quantile is assumed to be equal to zero. This statistics uses the concept of placement of observations from a population having more spread around common quantile as compared to the observations from the other population. The construction of the statistics is similar to the construction of statistics proposed by Sukhatme (Ann Math Stat 28:188–194, 1957) and Deshpande and Kusum (Austrl J Stat 26:16–24, 1984). It shows that the new test is more efficient than the Sukhatme, Deshpande-Kusum, and Mood Test (Mood in Ann Math Stat 34:973–983, 1954) in Pitmen ARE. We further show through Monte Carlo simulation that the Test has good power when the common quantile is in the tails of the distribution than Sukhatme Test and Deshpande-Kusum test.