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

Krell, Nathalie

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

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 $\lambda$. Our main result is that for $𝒟$ a compact subset of $(0,\infty )$, if $\lambda$ is in the Hölder space ${\mathscr{H}}^8\left(𝒟\right)$, the squared-loss error of the estimator is asymptotically close to the speed of $n^{-s/(2s+1)}$. Simulations illustrate the behavior of our estimator.

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