# Piecewise linear continuous estimators of the quantile function

Abstract : In 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 $(\frac{X_k^{\ast}+ X^{\ast}_{k+1}}{2}, \frac{k}{n})$, with $X_1^{\ast},\dotsc,X_n^{\ast}$ 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 $F_n^{-1}$.
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https://hal-univ-avignon.archives-ouvertes.fr/hal-02917464
Contributor : Delphine Blanke <>
Submitted on : Wednesday, August 19, 2020 - 12:12:46 PM
Last modification on : Thursday, December 10, 2020 - 12:32:17 PM
Long-term archiving on: : Monday, November 30, 2020 - 9:39:41 PM

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• HAL Id : hal-02917464, version 1

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Delphine Blanke, Denis Bosq. Piecewise linear continuous estimators of the quantile function. 2020. ⟨hal-02917464⟩

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