Convergence Rates of Wavelet Density Estimators for Strongly Mixing Samples

Abstract

This paper considers wavelet estimations for a multivariate density function based on strongly mixing data. We first construct a linear wavelet estimator and provide a convergence rate over $$L^{p} (1\le p<\infty )$$ risk in Besov space $$B^{s}_{r,q}(\mathbb {R}^{d})$$ . However, this estimator depends on the smoothness of density function, which means that the estimator is not adaptive. A nonlinear adaptive wavelet estimator is proposed by thresholding method. Moreover, the convergence rate of nonlinear estimator is better than the linear one in the case of $$r\le p$$ .