Piecewise linear continuous estimators of the quantile function

Delphine Blanke (1) , Denis Bosq (2)

1 LMA - EA2151 Laboratoire de Mathématiques d'Avignon
2 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation

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 $F_n^{-1}$.

Mots clés: Quantile function estimation; Piecewise linear estimator; Sample quantile estimator; Empirical quantile estimator; Order statistics; Mean integrated squared error

Domaines: Statistiques [math.ST]

Delphine Blanke, Denis Bosq. Piecewise linear continuous estimators of the quantile function. Advances in Contemporary Statistics and Econometrics, 2021, Festschrift in Honor of Christine Thomas-Agnan. ⟨hal-02917464⟩