Cross-validation method for bivariate measure with certain mixture

We consider a pair of random variables (X, Y) whose probability measure is the sum of an absolutely continuous measure, a discrete measure and a finite number of absolutely continuous measures on several lines. An asymptotically unbiased and consistent estimate of the density of the continuous part is given in [13]. In this work, we focus on the choice of these parameters so that this estimate will be optimal and the rate of convergence will be better. we as well as its rate of convergence. To achieve this we use the cross-validation technics.