The law of the iterated logarithm for the integrated squared deviation of a kernel density estimator

Abstract

Let $f_{n,K}$ denote a kernel estimator of a density f in $R$ such that ${\int }_R{f}^p\left(x\right)dx<\mathrm{\infty }$ for some p>2. It is shown, under quite general conditions on the kernel K and on the window sizes, that the centred integrated squared deviation of $f_{n,K}$ from its mean, $\parallel f_{n,K}-E{f}_{n,K}{\parallel }_2^2-E\parallel f_{n,K}-E{f}_{n,K}{\parallel }_2^2$ satisfies a law of the iterated logarithm (LIL). This is then used to obtain an LIL for the deviation from the true density, $\parallel f_{n,K}-f{\parallel }_2^2-E\parallel f_{n,K}-f{\parallel }_2^2$. The main tools are the Komlós-Major-Tusnády approximation, a moderate-deviation result for triangular arrays of weighted chi-square variables adapted from work by Pinsky, and an exponential inequality due to Giné, Latała and Zinn for degenerate U-statistics applied in combination with decoupling and maximal inequalities.