THE ACCURATE DISTRIBUTION OF THE KOLMOGOROV STATISTIC WITH APPLICATIONS TO BOOTSTRAP APPROXIMATION

Suppose that x1,...,x(m) are i.i.d. variables with distribution F(.). When F(.) is continuous, the accurate distribution of the Kolmogorov statistic was obtained by Zhang. In this paper, a discrete distribution F(.) which masses value 1/n at each point y(i), i = 1,...,n is considered, the obtained accurate distribution of the Kolmogorov, statistic has the same form as that of Zhang's. By this result, the bootstrap approximation of the distribution of the Kolmogorov statistic is investigated, and the square-root n -asymptotic rate is obtained. (C) 1994 Academic Press, Inc.