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

DOI10.1051/PS/2016013arXiv1406.2845WikidataQ105584646 ScholiaQ105584646MaRDI QIDQ6252255

Nathalie Krell

Publication date: 11 June 2014

Abstract: 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) window size 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 ${mathcalH}^s(mathcalD)$, the squared-loss error of the estimator is asymptotically close to the rate of $n^{-s/(2s+1)}$. Simulations illustrate the behavior of our estimator.

Mathematics Subject Classification ID

Asymptotic properties of nonparametric inference (62G20) Nonparametric estimation (62G05) Markov processes: estimation; hidden Markov models (62M05) Continuous-time Markov processes on general state spaces (60J25)

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