Rank order probabilities for the dispersion problem

Subhash C. Kochar and George Woodworth

Statistics & Probability Letters, 1992, vol. 13, issue 3, 203-208

Abstract: First we consider the two-sample dispersion problem. Let X and Y be independent random variables of continuous type with distribution functions F and G, respectively, satisfying G(x) = F([sigma]x) for all x where [sigma] > 0 and F(x) = 1 - F(- x), for all x. Let X1, X2 (Y1, Y2) be two independent copies of X (Y) and define p([sigma]) = P[X1 [less-than-or-equals, slant] Y1 [less-than-or-equals, slant] Y2 [less-than-or-equals, slant] X2]. It is shown that p([sigma]) is nondecreasing in [sigma] implying P[X1 [less-than-or-equals, slant] Y1 [less-than-or-equals, slant] Y2 [less-than-or-equals, slant] X2][greater-or-equal, slanted] P[Y1 [less-than-or-equals, slant] X1 [less-than-or-equals, slant] X2 [less-than-or-equals, slant] Y2] for [sigma] [greater-or-equal, slanted] 1. Next, we consider the bivariate scale problem where (X, Y) has joint c.d.f. F(x, y) satisfying F(x, y) = H(x, [sigma]y), where H(x, y) is bivariate symmetric and also H(x, y) = (- x, - y) for every (x, y). Sufficient conditions are given under which p([sigma]) is nondecreasing in [sigma]. These results have been applied to study the properties of some nonparametric tests for the two-sample dispersion and paired-sample problems.

Keywords: Mood's; test; for; dispersion; U-statistic; bivariate; symmetry (search for similar items in EconPapers)
Date: 1992
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