On the Relationship Between Stability of Extreme Order Statistics and Convergence of the Maximum Likelihood Kernel Density Estimate

BY MICHEL BRONIATOWSKI, PAUL DEHEUVELS AND LUC DEVROYEIUniversite Pierre et Marie Curie, Universite Pierre et Marie Curie andMcGill UniversityLet f be a density on the real line and let f,~ be the kernel estimate of fin which the smoothing factor is obtained by maximizing the cross-validatedlikelihood product according to the method of Duin and Habbema, Hermansand Vandenbroek. Under mild regularity conditions on the kernel and f, weshow, among other things that f Jf,~ - f ( --~ 0 almost surely if and only if thesample extremes of f are strongly stable.

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