Abstract: | Let Xi, 1 ≤ i ≤ n, 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 □{ρα(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 {? ≤ 0: G(x - α?) - ? F(x) ≤ G(x + α?) + ? for all x ?}. |