Limiting Distributions of Kolmogorov-Smirnov Type Statistics Under the Alternative

Abstract

Let $X_1, X_2, \cdots$ be a sequence of independent and identically distributed random variables with the common distribution being uniform on [0, 1]. Let $Y_1, Y_2, \cdots$ be a sequence of i.i.d. variables with continuous $\operatorname{cdf}F(t)$ and with [0, 1] support. Let $F_n(t, \omega)$ denote the empirical distribution function based on $Y_1(\omega), \cdots, Y_n(\omega)$ and let $G_m(t, \omega)$ the empirical $\operatorname{cdf}$ pertaining to $X_1(\omega), \cdots, X_m(\omega)$. Let $\sup_{0\leqq t \leqq 1}|F(t) - t| = \lambda$ and $D_n = \sup_{0 \leqq t \leqq 1}|F_n(t, \omega) - t|$. The limiting distribution of $n^{\frac{1}{2}}(D_n - \lambda)$ 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.