Optimal convergence rates of wavelet estimators for a hidden density in a mixture model

Abstract

This paper investigates nonparametric estimations of a density function in a mixture model. Firstly, a lower bound estimation under $L^{p}(1\leq p<+\infty)$ error of an arbitrary density estimator is discussed. Secondly, a linear estimator and an adaptive nonlinear estimator of the unknown density function are constructed by wavelet method. The rates of convergence of those two wavelet estimators are discussed with some mild conditions. Combining with the lower bound estimations, two wavelet estimator all can attain the optimal convergence rate. Finally, numerical examples are given to verify the performance of the two wavelet estimators.