Abstract:
Let $\widehat F_n$ be the smooth empirical estimator obtained by integrating
a kernel type density estimator based on a random sample of size $n$ from
continuous distribution function $F$. The almost sure deviation between
smooth empirical and smooth quantile processes is investigated under
$\phi$-mixing and strong mixing conditions. We derive a pointwise as well as a
uniform Bahadur–Kieffer type representation for smooth quantiles under cases of
$\phi$-mixing and strong mixing. These results extend those of Babu and Singh
[J. Multivariate Anal., 8 (1978), pp. 532–549] and Ralescu
[J. Statist. Plann. Inference, 32 (1992), pp. 243–258].
Keywords:kernel density estimator, almost sure deviation, smooth empirical process, smooth quantile process.