The exact distribution of the maximizing point of the two-sample empirical process
Dietmar Ferger · Journal of nonparametric statistics · 2003
This paper deals with the smallest maximizing point \tau_{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 \tau_{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 \tau_{mn}^+ has distribution function F (x) .