On the estimation of the quantile density function by orthogonal series

Abstract

The classical estimator of a quantile density function by orthogonal series depends on the empirical distribution function estimator Hn. The fact that Hn is a step function even when the underlying cumulative distribution function is continuous, has called for the need (in certain areas of application like estimating the quantile density function ) for smooth estimators of Hn. The present work has two goals. The first one is to introduce a new technique for estimating by orthogonal series for any orthonormal system in , a smooth nonparametric estimators of and are proposed. Asymptotic properties of the proposed estimators are studied. The second is to introduce a new method for selection of a smoothing parameter. A simulation study is done to compare the performances of the new approach with the (Chesneau et al 2016) one, when comparing mean integrated square error of the two estimators.