In this spaper, our aim is to revisit the nonparametric estimation of a square integrable density f on R, by using projection estimators on a Hermite basis. These estimators are studied from the point of view of their mean integrated squared error on R. A model selection method is described and proved to perform an automatic bias variance compromise. Then, we present another collection of estimators, of deconvolution type, for which we define another model selection strategy. Although the minimax asymptotic rates of these two types of estimators are mainly equivalent, the complexity of the Hermite estimators is usually much lower than the complexity of their deconvolution (or kernel) counterparts. These results are illustrated through a small simulation study.
Various members of the class of weighted insurance premiums and risk capital allocation rules have been researched from a number of perspectives. Corresponding formulas in the case of parametric families of distributions have been derived, and they have played a pivotal role when establishing parametric statistical inference in the area. Nonparametric inference results have also been derived in special cases such as the tail conditional expectation, distortion risk measure, and several members of the class of weighted premiums. For weighted allocation rules, however, nonparametric inference results have not yet been adequately developed. In the present paper, therefore, we put forward empirical estimators for the weighted allocation rules and establish their consistency and asymptotic normality under practically sound conditions. Intricate statistical considerations rely on the theory of induced order statistics, known as concomitants.