Wavelet Density Estimation For Strong Mixing Stratified Size-biased Sample

Density estimation for biased data is an important research direction in nonparametric statistics,which plays an important role in statistics,economics and big data processing.Wavelet is widely used in density estimation because of its unique time-frequency local analysis characteristics.Inspired by the work of Donoho,Doosti,Ramirez,Chesneau and Chaubey,this paper constructs wavelet density estimators by using wavelet method for the stratified size-biased samples,and discusses the convergence order of L^p?1?p??? risk of wavelet estimators in Besov space B_r,q^s?H?.Firstly,we define a linear wavelet estimator for density functions based on the strongly mixing stratified size-biased samples,and investigate its convergence order on L^p?1?p??? risk in Besov space.Especially,when the stratified size M=1 and the strongly mixing samples degenerate into independent cases,our results reduce to the theorem in Shirazi and Doosti.Secondly,in order to get the adaptive estimator,the stratified nonlinear wavelet estimator is constructed by using the threshold method,and the convergence order of the estimator on L^p?1?p??? risk is discussed.Finally,the convergence order of the two wavelet estimators is analyzed and compared.The results show that the convergence order of the nonlinear wavelet estimator is better than the linear ones in the sense of one ln N factor difference for r?p.