Nonparametric wavelet Quantile density estimations based on biased data

Abstract

Estimation of a quantile density function from biased data is a frequent problem in industrial life testing experiments and medical studies. The estimation of a quantile density function in the biased nonparametric regression model is inves- tigated. We propose and develop a new wavelet-based methodology for this problem. In particular, an adaptive hard thresholding wavelet estimator is constructed. Under mild assumptions on the model, we prove that it enjoys powerful mean integrated squared error properties over Besov balls. The performance of proposed estimator is investigated by a numerical study. In this study, we develop two types of wavelet estimators for the quantile density function when data comes from a biased distribution function. Our wavelet hard thresholding estimator which is introduced as a nonlinear estimator, has the feature to be adaptive according to q(x). We show that these estimators attain optimal and nearly optimal rates of convergence over a wide range of Besov function classes.