Functional limit laws for the increments of the quantile process; with applications

Abstract

We establish a functional limit law of the logarithm for the increments of the normed quantile process based upon a random sample of size n→∞. We extend a limit law obtained by Deheuvels and Mason [12], showing that their results hold uniformly over the bandwidth h, restricted to vary in [h'_n,h''_n], where {h'_n}_n≥1 and {h''_n}_n≥1 are appropriate non-random sequences. We treat the case where the sample observations follow possibly non-uniform distributions. As a consequence of our theorems, we provide uniform limit laws for nearest-neighbor density estimators, in the spirit of those given by Deheuvels and Mason [13] for kernel-type estimators.