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The purpose of this paper is to pursue our study of rho-estimators built fromi.i.d. observations that we defined in Baraud et al. (2014). For a\rho-estimator based on some model S (which means that the estimator belongs toS) 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 adimension function of the model and is often related to the "metric dimension"of this model, as defined in Birg\'e (2006). This is a minimax point of viewand it is well-known that it is pessimistic. Typically, the bound is accuratefor most points in the model but may be very pessimistic when the truedistribution belongs to some specific part of it. This is the situation that wewant to investigate here. For some models, like the set of decreasing densitieson [0,1], there exist specific points in the model that we shall call"extremal" and for which the risk is substantially smaller than the typicalrisk. Moreover, the risk at a non-extremal point of the model can be bounded bythe sum of the risk bound at a well-chosen extremal point plus the square ofits distance to this point. This implies that if the true density is closeenough to an extremal point, the risk at this point may be smaller than theminimax risk on the model and this actually remains true even if the truedensity does not belong to the model. The result is based on some refinedbounds on the suprema of empirical processes that are established in Baraud(2016).

Rates of convergence of rho-estimators for sets of densities satisfying shape constraints