In this paper we study nonparametric estimation problems for a class of piecewise-deterministic Markov processes (PDMPs). Borovkov and Last (2008) proved a version of Rice's formula for PDMPs, which explains the relation between the stationary density and the level crossing intensity. From a statistical point of view, their result suggests a methodology for estimating the stationary density from observations of a sample path of PDMPs. First, we introduce the local time related to the level crossings and construct the local-time estimator for the stationary density, which is unbiased and uniformly consistent. Secondly, we investigate other estimation problems for the jump intensity and the conditional jump size distribution.
"Nonparametric estimation for a class of piecewise-deterministic Markov processes." J. Appl. Probab. 50 (4) 931 - 942, December 2013. https://doi.org/10.1239/jap/1389370091