Multivariate wavelet density estimation for strong mixing stratified size-biased sample

This paper considers wavelet estimations of a multivariate density function based on stratified size-biased and strong mixing data. We provide upper bounds of the mean integrated squared error for linear and nonlinear wavelet estimators in Besov space B-p,q(s)(Rd). It is shown that the linear estimator achieves the optimal convergence rate in the case of p >= 2. Moreover, the convergence rate of nonlinear estimator coincides with the optimal convergence rate up to a ln N factor for p is an element of[1,+infinity). In addition, the nonlinear wavelet estimator is adaptive.