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:
[1] | Billingsley, Convergence of Probability Measures. (1968) · Zbl 0172.21201 |
[2] | Ibragimov, Statistical Estimation. (1981) · doi:10.1007/978-1-4899-0027-2 |
[3] | Kolmogorov, Sulla determinazione empirica di una legge di distribuzione, Giorn. Ist. Ital. Att. 4 pp 83– (1933) |
[4] | Kozek, Minimum Lévy distance estimation of a translation parameter (1982) |
[5] | Lévy, Calcul des probabilités. (1925) |
[6] | Prohorov, Probability distributions in functional spaces (in Russian), Uspekhi. Mat. Nauk 8 (3) pp 165– (1953) |
[7] | Raghavachari, Limiting distributions of Kolmogorov-Smirnov type statistics under the alternative, Ann. Statist. 1 pp 67– (1973) · Zbl 0276.62028 |
[8] | Smirnov, Approximate laws of distribution of random variables from empirical data (in Russian), Uspekhi Mat. Nauk 10 pp 179– (1944) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.