Note on the tail behavior of general weighted empirical processes.(English)Zbl 0725.62047

Summary: Under minimal conditions precise bounds are obtained for the expectation of the supremum of the weighted empirical process over the interval (0,1/(n(log n)$${}^{d-1}))$$, where d is the dimension of the underlying random vectors. The allowed growth of the weight function is optimal in the iid case.
The results will have broad applications in the theory of all kinds of nonstandard weighted empirical processes, such as empirical processes based on uniform spacings or U-statistics, where it is often not so easy to show directly (as in the iid case) that the considered suprema converge to 0 in probability.

MSC:

 62G20 Asymptotic properties of nonparametric inference 62E20 Asymptotic distribution theory in statistics 62G30 Order statistics; empirical distribution functions 62G99 Nonparametric inference
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