Fernholz, Luisa Turrin Almost sure convergence of smoothed empirical distribution functions. (English) Zbl 0798.62063 Scand. J. Stat. 18, No. 3, 255-262 (1991). 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 \(\sqrt n \| \widetilde F_ n - F \|\). Cited in 7 Documents MSC: 62G20 Asymptotic properties of nonparametric inference 62G07 Density estimation 62G30 Order statistics; empirical distribution functions 62E20 Asymptotic distribution theory in statistics Keywords:kernel estimator; density estimation; smoothing; almost sure convergence; empirical distribution function; sequence of kernels; supremum norm; rate of convergence; bias PDFBibTeX XMLCite \textit{L. T. Fernholz}, Scand. J. Stat. 18, No. 3, 255--262 (1991; Zbl 0798.62063)