Some applications of the Gnedenko–Korolyuk method to empirical distributions

E. O. Lutsenko, O. V. Marinich, I. K. Matsak
2009 Theory of Probability and Mathematical Statistics
A new proof of the Kolmogorov theorem on the asymptotic behavior of the deviation between a theoretical and an empirical distribution function is presented. We use the Gnedenko-Korolyuk approach based on some combinatorial properties of the merged sample constructed from two other independent samples. Some statistical applications of the Gnedenko-Korolyuk theorem are discussed. 2000 Mathematics Subject Classification. Primary 60B12. License or copyright restrictions may apply to redistribution;
more » ... to redistribution; see http://www.ams.org/journal-terms-of-use 134 E. O. LUTSENKO, O. V. MARINICH, AND I. K. MATSAK where D + m,n = sup −∞<t<∞ n = mn/(m + n), and F * n (t) and G * m (t) are the empirical distribution functions constructed from two independent identically distributed samples of sizes n and m, respectively, m ≤ n. Kolmogorov and Smirnov used lengthy calculations when proving relations (1) and (2) , respectively. Gnedenko and Korolyuk [3] found in 1951 an ingenious combinatorial proof of equality (2) for m = n. Moreover, the explicit distribution of the statistic D n,n is obtained in [3] for finite n. Below is their main result. Theorem 1.2 (Gnedenko-Korolyuk). If F (t) is a continuous distribution function of two independent samples and c = z √ 2n is the minimal integer that does not exceed z √ 2n, then, for (2n) −1/