In this talk, we consider the inverse problem of estimating the product of two  densities, given a d-dimensional n-sample of i.i.d. observations drawn from each distribution.

We propose a general method of estimation encompassing both projection estimators with model selection device and kernel estimators with bandwidth selection strategies. The procedures do not consist in making the product of two density estimators, but in plugging an overfitted estimator of one of the two densities, in an estimator based on the second sample. Our findings are a first step toward a better understanding of the good performances of overfitting in regression Nadaraya-Watson estimator.