Rho-estimators for shape restricted density estimation

The purpose of this paper is to pursue our study of ρ-estimators built from i.i.d. observations that we defined in Baraud et al. (2014). For a ρ-estimator based on some model S¯ (which means that the estimator belongs to S¯) and a true distribution of the observations that also belongs to S¯, the risk (with squared Hellinger loss) is bounded by a quantity which can be viewed as a dimension function of the model and is often related to the “metric dimension” of this model, as defined in Birgé (2006). This is a minimax point of view and it is well-known that it is pessimistic. Typically, the bound is accurate for most points in the model but may be very pessimistic when the true distribution belongs to some specific part of it. This is the situation that we want to investigate here. For some models, like the set of decreasing densities on [0,1], there exist specific points in the model that we shall call extremal and for which the risk is substantially smaller than the typical risk. Moreover, the risk at a non-extremal point of the model can be bounded by the sum of the risk bound at a well-chosen extremal point plus the square of its distance to this point. This implies that if the true density is close enough to an extremal point, the risk at this point may be smaller than the minimax risk on the model and this actually remains true even if the true density does not belong to the model. The result is based on some refined bounds on the suprema of empirical processes that are established in Baraud (2016).