Estimating a density, a hazard rate, and a transition intensity via the ρ-estimation method

We propose a unified study of three statistical settings by widening the ρ-estimation method developed in [BBS17]. More specifically, we aim at estimating a density, a hazard rate (from censored data), and a transition intensity of a time inhomogeneous Markov process. We relate the performance of ρ-estimators to deviations of a (possibly unbounded) empirical process. We deduce non-asymptotic risk bounds for an Hellinger-type loss on possibly random models. When the models are convex, maximum likelihood estimators coincide with ρ-estimators, and satisfy therefore our risk bounds. However, our results also apply to some models where the maximum likelihood method does not work. Besides, the robustness properties of ρ-estimators are not, in general, shared by maximum likelihood estimators. Subsequently, we present an alternative procedure to ρ-estimation, more numerically friendly, that yields a piecewise polynomial estimator. We prove theoretical results and carry out some numerical simulations that show the benefits of our approach compared with a more classical one based on maximum likelihood.