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

Let f be a density on the real line and let fₙ 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 ∫|fₙ - f| → 0 almost surely if and only if the sample extremes of f are strongly stable.