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”, sayw N. Under suitable and reasonably mild assumptions onF andk, Ralescu and Sun (1993) have recently proven that lim N→∞ N 1/4wN=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-Esséen rate ofO(N−1/2) can be achieved by choosing the window-width suitably, sayw N=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.
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Research supported by the Office of Naval Research Contract N00014-91-J-1020.
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Seoh, M., Puri, M.L. Berry-Esséen rate in asymptotic normality for perturbed sample quantiles. Metrika 41, 83–98 (1994). https://doi.org/10.1007/BF01895308
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DOI: https://doi.org/10.1007/BF01895308