Nonparametric wavelet Quantile density estimations based on biased data

Estimation of a quantile density function from biased data is a frequent problem in industrial life testingexperiments 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, anadaptive hard thresholding wavelet estimator is constructed. Under mild assumptions on the model, weprove that it enjoys powerful mean integrated squared error properties over Besov balls. The performanceof proposed estimator is investigated by a numerical study.In this study, we develop two types of wavelet estimators for the quantile density function when datacomes from a biased distribution function. Our wavelet hard thresholding estimator which is introducedas a nonlinear estimator, has the feature to be adaptive according to q(x). We show that these estimatorsattain optimal and nearly optimal rates of convergence over a wide range of Besov function classes.