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 s tatistic 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 consid ered, the obtained accurate distribution of the Kolmogorov, statistic has the same form as that of Zhang's. By this result, the bootstrap ap proximation of the distribution of the Kolmogorov statistic is investi gated, and the square-root n -asymptotic rate is obtained. (C) 1994 Ac ademic Press, Inc.