Abstract
In estimating quantiles with a sample of sizeN obtained from a distributionF, the perturbed sample quantiles based on a kernel functionk have been investigated by many authors. It is well known that their behaviour depends on the choices of “window-width”, saywN. Under suitable and reasonably mild assumptions onF andk, Ralescu and Sun (1993) have recently proven that limN→∞N1/4wN=0 is the necessary and sufficient condition for the asymptotic normality of the perturbed sample quantiles. In this paper, their rate of convergence is investigated. It turns out that the optimal Berry-Esseen rate ofO(N−1/2) can be achieved by choosing the window-width suitably, saywN=O(N−1/2). The obtained results, in addition to being explicit enough to verify the sufficient condition for the asymptotic normality, improve Ralescu's (1992) result of which the rate is of order (logN)N−1/2.
Published Version
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