Piecewise Linear Continuous Estimators of the Quantile Function

Abstract

AbstractIn Blanke and Bosq (2018), families of piecewise linear estimators of the distribution function F were introduced. It was shown that they reduce the mean integrated squared error (MISE) of the empirical distribution function \(F_n\) and that the minimal MISE was reached by connecting the midpoints \((\frac{X_k^{*}+ X^{*}_{k+1}}{2}, \frac{k}{n})\), with \(X_1^{*},\dotsc ,X_n^{*}\) the order statistics. In this contribution, we consider the reciprocal estimators, built respectively for known and unknown support of distribution, for estimating the quantile function \(F^{-1}\). We prove that these piecewise linear continuous estimators again strictly improve the MISE of the classical sample quantile function \(F_n^{-1}\).