Abstract
Let F be a distribution function with a density f. For non-negative integers r, statistics Fr based on a random sample of size n from F are exhibited. Under no assumption whatsoever on f, it is shown that sup with probability one as n If r > 1 and is integrable, then sup F(x) is at least of the order O(n-1 loglog n)1/2 W.p.1. The same conclusion holds with regard to is bounded. Uniform mean square consistencies of Fo and Fr are also proved and rates of convergence are investigated. Exact asymptotic values for the bias and variance of the estimators Fr are obtained. The central limit theorem with the rate of convergence for our estimators is established.
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