Limiting distributions of Kolmogorov-Lévy-type statistics under the alternative. (English) Zbl 0617.62014
Let \(X_ i\), \(1\leq i\leq n\), be independent random variables with the common distribution function F(x). Let \(F_ n(x)\) be the empirical distribution function of the \(X_ i\). Define the metric
\[
\rho_{\alpha}(F,G)=\inf \{\epsilon:\epsilon >0,\quad G(x-\alpha \epsilon)-\epsilon \leq F(x)\leq G(x+\alpha \epsilon)+\epsilon,\quad all\quad x\}.
\]
The main result of the paper provides the limiting distribution of
\[
\sqrt{n}[\rho_{\alpha}(F_ n,G)- \rho_{\alpha}(F,G)],
\]
where G is a smooth distribution function.
Reviewer: J.Galambos
MSC:
| 62E20 | Asymptotic distribution theory in statistics |
| 60F05 | Central limit and other weak theorems |
| 62G30 | Order statistics; empirical distribution functions |
Keywords:
Kolmogorov-Smirnov metric; Prohorov-Lévy metric; Kolmogorov-Lévy-type metric; empirical distribution functionReferences:
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