On the estimation of the quantile density function by orthogonal series

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 H(.) is continuous, has called for the need (in certain areas of application like estimating the quantile density function ψ(.)) for smooth estimators of Hn. The present work has two goals. The first one is to introduce a new technique for estimating ψ(.) by orthogonal series for any orthonormal system in L2[0,1], a smooth nonparametric estimators of ψ(.) and H(.) 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.