An exponential inequality for the distribution function of the kernel density estimator, with applications to adaptive estimation

It is shown that the uniform distance between the distribution function $F_n^K(h)$ of the usual kernel density estimator (based on an i.i.d. sample from an absolutely continuous law on ${\mathbb{R}}$ ) with bandwidth h and the empirical distribution function F _n satisfies an exponential inequality. This inequality is used to obtain sharp almost sure rates of convergence of $\|F_n^K(h_n)-F_n\|_\infty$ under mild conditions on the range of bandwidths h _n , including the usual MISE-optimal choices. Another application is a Dvoretzky–Kiefer–Wolfowitz-type inequality for $\|F_n^{K}(h)-F\|_\infty$ , where F is the true distribution function. The exponential bound is also applied to show that an adaptive estimator can be constructed that efficiently estimates the true distribution function F in sup-norm loss, and, at the same time, estimates the density of F—if it exists (but without assuming it does)—at the best possible rate of convergence over Hölder-balls, again in sup-norm loss.