L2 consistency of the kernel quantile estimator

Let F be a continuous distribution function and let Q be its associated quantile function. Let F-h be the kernel estimator of F and Q(h) that of Q. In this article the L-2 right inversion distance between Q(h) and Q is introduced. It is shown that this distance can be represented in terms of F-h and F, more precisely it is established that the right inversion distance is equal to the conventional integrated squared error between F-h and F. This representation shows that any good bandwidth for F is a reasonable bandwidth for Q(h) and, this fact enables us to suggest methods to choose the smoothing parameter of Q(h). Let Q((h) over cap cv) be the kernel, estimator of Q equipped with the global cross-validation bandwidth (h) over cap (c)(v), designed for F-h. Let Q((h) over cap rho i) be the linear kernel estimator of Q, (h) over cap (rho i), being the plug-in bandwidth function. A small scale simulation study presented in this paper contains some examples of distributions for which Q((h) over cap cv) appears to be superior to Q((h) over cap rho i) This paper also contains some properties of the classical L-2 distance between Q(h) and Q.