The exact distribution of the maximizing point of the two-sample empirical process

Abstract

This paper deals with the smallest maximizing point \ au_{mn}^+ of the two-sample empirical process \\{F_m (x) - G_n (x)\\colon x \\in {\\open R} \\} where F_m (x) and G_n (x) are the empirical distribution functions of X_1,\\ldots,X_m and Y_1,\\ldots,Y_n , respectively, which are two independent samples of i.i.d. random variables with common distribution function F (x) . We determine the distribution function of \ au_{mn}^+ for finite subsample sizes m and n . It turns out to be a polynomial of degree m + n in the variable F (x) . If m and n are relatively prime then \ au_{mn}^+ has distribution function F (x) .