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

摘要

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 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 parallel to F-n(K) (h(n)) - F-n parallel to(infinity) under mild conditions on the range of bandwidths hn, including the usual MISE-optimal choices. Another application is a Dvoretzky-Kiefer-Wolfowitz-type inequality for parallel to F-n(K) (h) - F parallel to infinity, 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 Holder-balls, again in sup-norm loss.