Semi-parametric inference for the absorption features of a growth-fragmentation model
In the present paper, we focus on semi-parametric methods for estimating the absorption probability and the distribution of the absorbing time of a growth-fragmentation model observed within a long time interval. We establish that the absorption probability is the unique solution in an appropriate space of a Fredholm equation of the second kind whose parameters are unknown. We estimate this important characteristic of the underlying process by solving numerically the estimated Fredholm equation. Even if the study has been conducted for a particular model, our method is quite general.
KeywordsNon-ergodic piecewise-deterministic Markov process Growth-fragmentation process Semi-parametric estimation Absorption probability Fredholm integral equation
Mathematics Subject Classification62M05 93C30 62G05
The referees deserve thanks for careful reading of the original version of the manuscript and many helpful suggestions for improvement in the article. The authors also acknowledge Alexandre Boumezoued for fruitful discussions about hybrid processes and Poisson random measures.
- Azaïs R (2014) A recursive nonparametric estimator for the transition kernel of a piecewise-deterministic Markov process. ESAIM Probab Stat. doi: 10.1051/ps/2013054 (in press)
- Azaïs R, Dufour F, Gégout-Petit A (2014) Nonparametric estimation of the conditional distribution of the inter-jumping times for piecewise-deterministic Markov processes. Scand J Stat. doi: 10.1111/sjos.12076 (in press)
- Cloez B, Hairer M (2014) Exponential ergodicity for markov processes with random switching. Bernoulli (in press)Google Scholar
- Costa O, Dufour F (2013) Continuous average control of piecewise deterministic Markov processes., Springer briefs in mathematics. SpringerGoogle Scholar
- De Saporta B, Dufour F, Zhang H, Elegbede C (2012) Optimal stopping for the predictive maintenance of a structure subject to corrosion. J Risk Reliab 226(2):169–181Google Scholar
- Doumic M, Hoffmann M, Krell N, Robert L (2014) Statistical estimation of a growth-fragmentation model observed on a genealogical tree. Bernoulli (in press)Google Scholar
- Jacobsen M (2006) Point process theory and applications : marked point and piecewise deterministic processes., Probability and its applicationsBirkhäuser, BostonGoogle Scholar
- Murray JD (2002) Mathematical biology I: an introduction, interdisciplinary applied mathematics, vol 17. Springer, New YorkGoogle Scholar