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

Let X i , 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 □ {ρ α ( F n , G ) ‐ α ( F, G )}, where F n is the empirical distribution function based on X 1 ,…, X n and α is a Kolmogorov‐Lévy‐type metric between distribution functions. For α ≤ 0 and two distribution functions F and G the metric p α is given by p α ( F, G ) = inf {ϵ ≤ 0: G ( x ‐ αϵ) ‐ ϵ F(x) ≤ G ( x + αϵ) + ϵ for all x ℝ}.