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