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

Nathalie Krell (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

We consider the class of Piecewise Deterministic Markov Processes (PDMP), whose state space is $\R_{+}^{*}$, that possess an increasing deterministic motion and that shrink deterministically when they jump. Well known examples for this class of processes are Transmission Control Protocol (TCP) windowsize process and the processes modeling the size of a "marked" {\it Escherichia coli} cell. Having observed the PDMP until its $n$th jump, we construct a nonparametric estimator of the jump rate $\lambda$. Our main result is that for $D$ a compact subset of $\R_{+}^{*}$, if $\lambda$ is in the H{\"{o}}lder space ${\mathcal H}^s({\mathcal D})$, the squared-loss error of the estimator is asymptotically close to the rate of $n^{-s/(2s+1)}$. Simulations illustratethe behavior of our estimator.

Mots clés: Piecewise Deterministic Markov processes; ergodicity of Markov chains; Nonparametric estimation; jump rate estimation

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Nathalie Krell. Statistical estimation of jump rates for a specific class of Piecewise Deterministic Markov Processes.. ESAIM: Probability and Statistics, 2016, 20, pp.196-216. ⟨10.1051/ps/2016013⟩. ⟨hal-01004230v3⟩