Statistical estimation of jump rates for a piecewise deterministic Markov processes with deterministic increasing motion and jump mechanism

Krell, Nathalie

doi:10.1051/ps/2016013

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Statistical estimation of jump rates for a piecewise deterministic Markov processes with deterministic increasing motion and jump mechanism

Krell, Nathalie ¹

¹ Université de Rennes 1, Institut de Recherche mathématique de Rennes, CNRS-UMR 6625, Campus de Beaulieu, Bâtiment 22, 35042 Rennes cedex, France.

ESAIM: Probability and Statistics, Tome 20 (2016), pp. 196-216

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Résumé

We consider the class of Piecewise deterministic Markov processes (PDMP), whose state space is $(0, \infty)$ , that possess an increasing deterministic motion and with a deterministic jump mechanism. 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” Escherichia coli cell. Having observed the PDMP until its $n$ th jump, we construct a nonparametric estimator of the jump rate $λ$ . Our main result is that for $𝒟$ a compact subset of $(0, \infty)$ , if $λ$ is in the Hölder space $ℋ^{8} (𝒟)$ , the squared-loss error of the estimator is asymptotically close to the speed of $n^{- s / (2 s + 1)}$ . Simulations illustrate the behavior of our estimator.

MR Zbl

DOI : 10.1051/ps/2016013

Classification : 62M05, 62G05, 62G20, 60J25
Keywords: Piecewise deterministic markov processes, nonparametric estimation, jump rate estimation, ergodicity of Markov chains

Affiliations des auteurs :

Krell, Nathalie ¹

¹ Université de Rennes 1, Institut de Recherche mathématique de Rennes, CNRS-UMR 6625, Campus de Beaulieu, Bâtiment 22, 35042 Rennes cedex, France.

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     author = {Krell, Nathalie},
     title = {Statistical estimation of jump rates for a piecewise deterministic {Markov} processes with deterministic increasing motion and jump mechanism},
     journal = {ESAIM: Probability and Statistics},
     pages = {196--216},
     publisher = {EDP-Sciences},
     volume = {20},
     year = {2016},
     doi = {10.1051/ps/2016013},
     mrnumber = {3528624},
     zbl = {06674053},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/ps/2016013/}
}

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Krell, Nathalie. Statistical estimation of jump rates for a piecewise deterministic Markov processes with deterministic increasing motion and jump mechanism. ESAIM: Probability and Statistics, Tome 20 (2016), pp. 196-216. doi: 10.1051/ps/2016013

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