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Almost sure convergence of smoothed empirical distribution functions. (English) Zbl 0798.62063

Summary: A smoothed version \(\widetilde F_ n\) of the empirical distribution function \(F_ n\) is generated using a sequence of kernels \(\{k_ n\}\). Under certain hypotheses it is shown that the supremum norm \(\| \widetilde F_ n-F_ n \| = o(n^{-1/2})\) with probability one. A rate of convergence for the bias term \(\| E (\widetilde F_ n)-F \|\) is also obtained. These results are applied to derive the asymptotic distributions of \(\sqrt n (\widetilde F_ n(x) - F(x))\) and \(\sqrt n \| \widetilde F_ n - F \|\).

MSC:

62G20 Asymptotic properties of nonparametric inference
62G07 Density estimation
62G30 Order statistics; empirical distribution functions
62E20 Asymptotic distribution theory in statistics
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