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

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

2006-12-10

arxiv.org/abs/math/0612260v3

Electronic Journal of Statistics 2007, Vol. 1, 496-518

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.1214/07-EJS099 the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by t

Scientific paper

10.1214/07-EJS099

We establish a functional limit law of the logarithm for the increments of the normed quantile process based upon a random sample of size $n\to\infty$. 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 $\{h'_n\}_{n\geq1}$ and $\{h''_n\}_{n\geq 1}$ 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.

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