Nonparametric estimation of jump rates for a specific class of Piecewise Deterministic Markov Processes

Nathalie Krell (1) , Emeline Schmisser (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 LPP - Laboratoire Paul Painlevé - UMR 8524

In this paper, we consider a piecewise deterministic Markov process (PDMP), with known flow and deterministic transition measure, and unknown jump rate $\lambda$. To estimate nonparametrically the jump rate, we first construct an adaptive estimator of the stationary density, then we derive a quotient estimator $\hat{\lambda}_n$ of $\lambda$. We provide uniform bounds for the risk of these estimators, and prove that the estimator of the jump rate is nearly minimax (up to a $\ln^2(n)$ factor). Simulations illustrate the behavior of our estimator.

Mots clés: piecewise deterministic Markov processes; model selection; 62G05; Nonparametric estimation; 62G07; 62G20; 60J75; 60J25; 62M05

Domaines: Statistiques [math.ST]; Probabilités [math.PR]

Nathalie Krell, Emeline Schmisser. Nonparametric estimation of jump rates for a specific class of Piecewise Deterministic Markov Processes. 2019. ⟨hal-01996064v1⟩