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Krell, Nathalie; Schmisser, Émeline

Nonparametric estimation of jump rates for a specific class of piecewise deterministic Markov processes. (English) Zbl 1472.60115

Bernoulli 27, No. 4, 2362-2388 (2021).
Summary: In this paper, we consider a unidimensional piecewise deterministic Markov process (PDMP), with homogeneous jump rate \(\lambda (x)\). This process is observed continuously, so the flow \(\phi\) is known. 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 \(\lambda \). Under some ergodicity conditions, we bound the risk of these estimators (and give a uniform bound on a small class of functions), and prove that the estimator of the jump rate is nearly minimax (up to a \(\ln^2(n)\) factor). The simulations illustrate our theoretical results.

Cited in 2 Documents

MSC:

60J05 Discrete-time Markov processes on general state spaces
60J25 Continuous-time Markov processes on general state spaces
62G05 Nonparametric estimation
60J76 Jump processes on general state spaces

Keywords:

piecewise deterministic Markov processes; model selection; nonparametric estimation; jump rate
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Full Text: DOI arXiv

References:

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