7. Probability and Statistics Soviet Essays - Sheynin, Oscar

determined the precise asymptotic characteristic of the maximal deviation of F N (x) from F(x)(for a continuous F(x)). DenoteD N = sup |F N (x) – F(x)|, |x| < + .Then Kolmogorov’s finding means that, as N ,lim P[D N < (/N)] = 1 – 2 ∞k =1(– 1) k–1 exp(–2k 2 2 ), > 0. (2)This fact is being used as a test of goodness-of-fit for checking whether, given a large N, F(x)is the true distribution.The problem solved by Kolmogorov served Smirnov as a point of departure for a numberof wide and deep studies. In short, his results are as follows. He [3; 7] determined thelimiting distribution of the 2 goodness-of-fit (Cramér – Mises – Smirnov) test 2 = N ∞ −∞[F N (x) – F(x)] 2 g[F(x)]dF(x).Later he [17] essentially simplified his initial proof, also see Anderson & Darling (1952).Smirnov [8] examined the limiting distribution of the variableN1N2N N1N 2+1 ,N2D =N1N2N + N12sup | FN x)− FN( ) | (3)( x1 2as N 1 , N 2 . Here, the empirical distribution functions corresponded to two independentsequences of observations of a random variable with a continuous distribution. The limitingdistribution coincided with the Kolmogorov law (2). Smirnov’s finding is widely used forchecking the homogeneity of two samples of large sizes.Smirnov [9] also extended the just described Kolmogorov theorem. His main result is this.Let us consider the curvesy 1 (x) = F(x) + /N, y 2 (x) = F(x) – /N, |x| < + ,and denote the number of times that the empirical distribution function passes beyond thezone situated between them by N (). Suppose also that N (t; ) = P{[ N ()/N] < t}, t > 0, > 0.Then, as N , N (t; ) (t; ) = 1 – 2 ∞k =1[(– 1) k k2d k − [ t + 2( k + 1) λ]/k!] { t exp} . (4)kdt2In his later investigations Smirnov [18; 19] considered the convergence of the histograms (ofempirical densities) to the density of the random variable. One of his relevant theorems