A combined strategy for multivariate density estimation

Non-linear aggregation strategies have recently been proposed in response to the problem of how to combine, in a non-linear way, estimators of the regression function (see for instance [Biau, G., Fischer, A., Guedj, B., and Malley, J. (2016), ‘COBRA: A Combined Regression Strategy’, Journal of Multivariate Analysis, 146, 18–28.]), classification rules (see [Cholaquidis, A., Fraiman, R., Kalemkerian, J., and Llop, P. (2016), ‘A Nonlinear Aggregation Type Classifier’, Journal of Multivariate Analysis, 146, 269–281.]), among others. Although there are several linear strategies to aggregate density estimators, most of them are hard to compute (even in moderate dimensions). Our approach aims to overcome this problem by estimating the density at a point x using not just sample points close to x but in a neighbourhood of the (estimated) level set $ f(x) $ f(x). We show that the mean squared error of our proposal is at most equal to the mean squared error of the best density estimator used for the aggregation plus a second term that tends to zero. This fact is illustrated through a simulation study. A Central Limit Theorem is also proven.