A note on the asymptotically optimal bandwidth for Nadaraya's quantile estimator

Let F(x) be a distribution function and let [xi]p denote the pth quantile for 0 < p < 1. In this note Nadaraya's estimator of [xi]p is considered. It is shown that the bandwidth h0 = C(k, f, K, p)n-1/(2k - 1) for k [greater-or-equal, slanted] 2 is asymptotically optimal in the sense that [p(n, h0) - [xi]p]2/infh[p(n, h) - [xi]p]2 --> 1 in probability as n --> [infinity], where p(n, h) denotes the estimator depending on the sample size n and the bandwidth h and C(k, f, K, p) is a constant depending on k, p, the density function f of F, and the kernel function K. When k = 2, this optimal bandwidth is identical to that obtained by Azzalini (1981) for the mean squared error criterion and is of the order n, which is different from the optimal bandwidth (in the mean squared error sense) for the kernel estimator of the density function.