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

Abstract

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 nonasymptotic 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 convexconcave. Under convex-type assumptions on the models, 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. 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.

Density estimation under local differential privacy and Hellinger loss

“…We then simplify the right-hand side of this inequality. The proof of (20) comes from elementary computations, see Claim 8 of [Sar21a] for instance.…”

“…It has also been used in other frameworks such as the estimation of the intensity of a Poisson process, the conditional density, the hazard rate, the transition intensity of a Markov chain or a Markovian process, the means of non-negative data, see [BB09,Bar11,Sar21a] for some references. One of the goals of this manuscript is to find out the minimax rates for h 2 under smoothness constraints in an inverse problem context.…”