On the Relationship Between Stability of Extreme Order Statistics and Convergence of the Maximum Likelihood Kernel Density Estimate
Michel Broniatowski, Paul Deheuvels, Luc Devroye
The Annals of Statistics · 1989 · Volume 17 · Issue 3
Abstract
Let $f$ be a density on the real line and let $f_n$ be the kernel estimate of $f$ in which the smoothing factor is obtained by maximizing the cross-validated likelihood product according to the method of Duin and Habbema, Hermans and Vandenbroek. Under mild regularity conditions on the kernel and $f$, we show, among other things that $\int|f_n - f| \rightarrow 0$ almost surely if and only if the sample extremes of $f$ are strongly stable.
Keywords
62G05, consistency, density function, kernel estimate, nonparametric estimation, order statistics, strong stability
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