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Limiting distributions of Kolmogorov‐Lévy‐type statistics under the alternative
Author(s) -
Kozek Andrzej
Publication year - 1987
Publication title -
canadian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.804
H-Index - 51
eISSN - 1708-945X
pISSN - 0319-5724
DOI - 10.2307/3314864
Subject(s) - mathematics , independent and identically distributed random variables , kolmogorov–smirnov test , distribution (mathematics) , empirical distribution function , random variable , type (biology) , combinatorics , limiting , limit (mathematics) , distribution function , metric (unit) , function (biology) , statistics , mathematical analysis , physics , statistical hypothesis testing , quantum mechanics , mechanical engineering , ecology , engineering , biology , operations management , evolutionary biology , economics
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}.