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

Andrzej Kozek*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)

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 {ϵ ≤ 0: G(x ‐ αϵ) ‐ ϵ F(x) ≤ G(x + αϵ) + ϵ for all x ℝ}.

Original languageEnglish
Pages (from-to)77-85
Number of pages9
JournalCanadian Journal of Statistics
Volume15
Issue number1
DOIs
Publication statusPublished - 1987
Externally publishedYes

Keywords

  • Asymptotic distribution
  • empirical distribution function
  • Kolmogorov‐Smirnov metric
  • Prohorov‐Lévy metric

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