Limiting distributions of kolmogorov-smirnov type statistics under the alternative

Abstract

Let XI, X2, - - - be a sequence of independent and identically distributed random variables with the common distribution being uniform on [0, 1]. Let Y1, Y2, * * - be a sequence of i.i.d. variables with continuous cdf F(t) and with [0, 1] support. Let Fn(t, wo) denote the empirical distribution function based on Yi(w), * * *, Yn(w) and let Gm(t, w) the empirical cdf pertaining to X1(@), * * *, Xm(a(). Let supogt?j JF(t) - tj = i and Dn = supostsl i Fn(t, o1) - tl. The limiting distribution of ni(Dn - i) is obtained in this paper. The limiting distributions under the alternative of the corresponding one-sided statistic in the one-sample case and the corresponding Smirnov statistics in the two-sample case are also derived. The asymptotic distributions under the alternative of Kuiper's statistic are also obtained.