Wavelet optimal estimations for a two-dimensional continuous-discrete density function over Lp $L^{p}$ risk

• Main Authors: Lin Hu, Xiaochen Zeng, Jinru Wang
• Format: Article
• Language: English
• Published: SpringerOpen 2018-10-01
• Series: Journal of Inequalities and Applications
• Subjects: Wavelets; Density estimation; Continuous-discrete density; Optimality
• Online Access: http://link.springer.com/article/10.1186/s13660-018-1868-7

Abstract The mixed continuous-discrete density model plays an important role in reliability, finance, biostatistics, and economics. Using wavelets methods, Chesneau, Dewan, and Doosti provide upper bounds of wavelet estimations on L2 $L^{2}$ risk for a two-dimensional continuous-discrete density function over Besov spaces Br,qs $B^{s}_{r,q}$. This paper deals with Lp $L^{p}$ ( 1≤p<∞ $1\leq p < \infty$) risk estimations over Besov space, which generalizes Chesneau–Dewan–Doosti’s theorems. In addition, we firstly provide a lower bound of Lp $L^{p}$ risk. It turns out that the linear wavelet estimator attains the optimal convergence rate for r≥p $r \geq p$, and the nonlinear one offers optimal estimation up to a logarithmic factor.