Estimation of the average number of continuous crossings for non-stationary non-diffusion processes

Romain Azaïs (1, 2) , Alexandre Genadot (3, 4)

1 MOSAIC - Simulation et Analyse de la morphogenèse in siliCo
2 RDP - Reproduction et développement des plantes
3 CQFD - Quality control and dynamic reliability
4 IMB - Institut de Mathématiques de Bordeaux

Assume that you observe trajectories of a non-diffusive non-stationary process and that you are interested in the average number of times where the process crosses some threshold (in dimension d = 1) or hypersurface (in dimension d ≥ 2). Of course, you can actually estimate this quantity by its empirical version counting the number of observed crossings. But is there a better way? In this paper, for a wide class of piecewise smooth processes, we propose estimators of the average number of continuous crossings of an hypersurface based on Kac-Rice formulae. We revisit these formulae in the uni-and multivariate framework in order to be able to handle non-stationary processes. Our statistical method is tested on both simulated and real data.

Mots clés: Average number of crossings; Piecewise deterministic process; Plug-in estimators

Domaines: Méthodologie [stat.ME]; Probabilités [math.PR]

Romain Azaïs, Alexandre Genadot. Estimation of the average number of continuous crossings for non-stationary non-diffusion processes. Journal of Statistical Planning and Inference, 2019, 198, pp.119-138. ⟨10.1016/j.jspi.2018.04.002⟩. ⟨hal-01499914v3⟩