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

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 ≤ p < ∞ ) 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 ≥ p , and the nonlinear one offers optimal estimation up to a logarithmic factor.