Abstract
A piecewise-deterministic Markov process is a stochastic process whose behavior is governed by an ordinary differential equation punctuated by random jumps occurring at random times. We focus on the nonparametric estimation problem of the jump rate for such a stochastic model observed within a long time interval under an ergodicity condition. We introduce an uncountable class (indexed by the deterministic flow) of recursive kernel estimates of the jump rate and we establish their strong pointwise consistency as well as their asymptotic normality. We propose to choose among this class the estimator with the minimal variance, which is unfortunately unknown and thus remains to be estimated. We also discuss the choice of the bandwidth parameters by cross-validation methods.
Highlights
Piecewise-deterministic Markov processes (PDMP’s in abbreviated form) have been introduced in the literature by Davis in [17] as a general class of continuous-time non-diffusion stochastic models, suitable for modeling deterministic phenomena in which the randomness appears as point events
We focus on the recursive nonparametric estimation of the jump rate of a PDMP from the observation of only one trajectory within a long time interval
We introduce a three-dimensional kernel estimator computed from the observation of the embedded Markov chain of a PDMP composed of the post-jump locations Zn and the travel times Sn+1 along the path Φ(Zn, t)
Summary
Piecewise-deterministic Markov processes (PDMP’s in abbreviated form) have been introduced in the literature by Davis in [17] as a general class of continuous-time non-diffusion stochastic models, suitable for modeling deterministic phenomena in which the randomness appears as point events. The authors rely on the specific form of the features of the process of interest in order to derive the asymptotic behavior of their estimation procedure These techniques have been generalized in [28] to introduce a nonparametric method for estimating the jump rate in a specific class of one-dimensional PDMP’s with monotonic motion and deterministic breaks, that is to say when the transition measure Q(x, dy) is a Dirac mass at some location depending on x. We state in (19) that this procedure is equivalent to maximize the criterion ν∞(ξ)G(ξ, τx(ξ)) along the curve Φ(x, −t), i.e. ξ = Φ(x, −τx(ξ)), where ν∞(ξ) denotes the invariant measure of the post-jump locations Zn and τx(ξ) is the only deterministic time to reach x following Φ(ξ, t) The choice of this criterion is far to be obvious without precisely computing the limit variance in the central limit theorem presented in Corollary 3.10. The proofs and the technicalities are postponed in Appendix A, B and C at the end of the paper
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