Wavelet optimal estimations for a two-dimensional continuous-discrete density function over [Formula: see text] 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 [Formula: see text] risk for a two-dimensional continuous-discrete density function over Besov spaces [Formula: see text] . This paper deals with [Formula: see text] ([Formula: see text] ) risk estimations over Besov space, which generalizes Chesneau–Dewan–Doosti’s theorems. In addition, we firstly provide a lower bound of [Formula: see text] risk. It turns out that the linear wavelet estimator attains the optimal convergence rate for [Formula: see text] , and the nonlinear one offers optimal estimation up to a logarithmic factor.