Canadian Journal of Statistics

Limiting distributions of Kolmogorov‐Lévy‐type statistics under the alternative

Journal Article

Abstract

Let Xi, 1 ≤ in, be independent identically distributed random variables with a common distribution function F, and let G be a smooth distribution function. We derive the limit distribution of α(Fn, G) ‐ α(F, G)}, where Fn is the empirical distribution function based on X1,…,Xn and α is a Kolmogorov‐Lévy‐type metric between distribution functions. For α ≤ 0 and two distribution functions F and G the metric pα is given by pα(F, G) = inf {ϵ ≤ 0: G(x ‐ αϵ) ‐ ϵ F(x)G(x + αϵ) + ϵ for all x ℝ}.

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