Complete asymptotic expansions for the maximum deviation of the empirical density function

Complete asymptotic expansion (19) is obtained for the properly normalized maximum deviation of the empirical density function. As a consequence, the exact rate of convergence to the limiting distribution is found. Some approximations, more useful in practice, and their rates of convergence are also given. In the special histogram case the results obtained are refinements and generalizations of some results of N. V. Smirnov ([5], [7]).