An Algorithm for Optimal Bandwidth Selection for Smooth Nonparametric Quantile Estimation

Abstract

We address the problem of bandwidth selection for kernel estimates of the marginal probability distribution of a time series Y _i , i = 1,2, …, n that is a transformation G (Z _i ⁿ _i of a stationary zero mean Gaussian process Z _i . Here G is an unknown function. Within this setup, it is known that if the underlying Gaussian process Z _i has long-memory correlations, then the optimal bandwith depends on the long-memory parameter δ (0 ≤; δ < 1/2), as well as several unknown parameters related to the function G and the underlying Gaussian process Z _i . Clearly, even with short-memory correlations, the optimal bandwidth will continue to depend on various unknown quantities. The aim of this paper is to propose a bandwidth selection procedure that estimates various relevant functions directly from the data to be used in an iterative procedure.