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 show non-asymptotic risk bounds for an Hellinger-type loss when the models consist, for instance, of piecewise polynomial functions, multimodal functions, or functions whose square root is piecewise convex-concave. Under convex-type assumptions on the models, maximum likelihood estimators coincide with $\rho$-estimators, and satisfy therefore our risk bounds. However, our results also apply to some models where the maximum likelihood method does not work. Subsequently, we present an alternative way, based on estimator selection, to define a piecewise polynomial estimator. We control the risk of the estimator and carry out some numerical simulations to compare our approach with a more classical one based on maximum likelihood only