Wavelet optimal estimations for a two-dimensional continuous-discrete density function over L p risk

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 $L^2$ risk for a two-dimensional continuous-discrete density function over Besov spaces $B_{r,q}^s$ . This paper deals with $L^p$ ( $1\le p<\infty$ ) risk estimations over Besov space, which generalizes Chesneau-Dewan-Doosti's theorems. In addition, we firstly provide a lower bound of $L^p$ risk. It turns out that the linear wavelet estimator attains the optimal convergence rate for $r\ge p$ , and the nonlinear one offers optimal estimation up to a logarithmic factor.

Keywords: Continuous-discrete density; Density estimation; Optimality; Wavelets.