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A piecewise-deterministic Markov process is a stochastic process whosebehavior is governed by an ordinary differential equation punctuated by randomjumps occurring at random times. We focus on the nonparametric estimationproblem of the jump rate for such a stochastic model observed within a longtime interval under an ergodicity condition. We introduce an uncountable class(indexed by the deterministic flow) of recursive kernel estimates of the jumprate and we establish their strong pointwise consistency as well as theirasymptotic normality. We propose to choose among this class the estimator withthe minimal variance, which is unfortunately unknown and thus remains to beestimated. We also discuss the choice of the bandwidth parameters bycross-validation methods.

Optimal choice among a class of nonparametric estimators of the jump rate for piecewise-deterministic Markov processes