Electronic Journal of Statistics

Optimal choice among a class of nonparametric estimators of the jump rate for piecewise-deterministic Markov processes

Romain Azaïs and Aurélie Muller-Gueudin

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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.

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Electron. J. Statist. Volume 10, Number 2 (2016), 3648-3692.

Received: October 2015
First available in Project Euclid: 3 December 2016

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Zentralblatt MATH identifier

Primary: 62M05: Markov processes: estimation 62G20: Asymptotic properties 60J25: Continuous-time Markov processes on general state spaces

Cross-validation jump rate kernel method nonparametric estimation piecewise-deterministic Markov process


Azaïs, Romain; Muller-Gueudin, Aurélie. Optimal choice among a class of nonparametric estimators of the jump rate for piecewise-deterministic Markov processes. Electron. J. Statist. 10 (2016), no. 2, 3648--3692. doi:10.1214/16-EJS1207. https://projecteuclid.org/euclid.ejs/1480734074

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  • [1] Odd Olai, Aalen.Statistical inference for a family of counting processes. ProQuest LLC, Ann Arbor, MI, 1975. Thesis (Ph.D.)–University of California, Berkeley.
  • [2] Terje Aven and Uwe, Jensen.Stochastic models in reliability, volume 41 ofApplications of Mathematics (New York). Springer-Verlag, New York, 1999.
  • [3] Romain Azaïs. A recursive nonparametric estimator for the transition kernel of a piecewise-deterministic Markov, process.ESAIM: Probability and Statistics, 18:726–749, January 2014.
  • [4] Romain Azaïs, François Dufour, and Anne Gégout-Petit. Nonparametric estimation of the jump rate for non-homogeneous marked renewal, processes.Ann. Inst. H. Poincaré Probab. Statist., 49(4) :1204–1231, 11 2013.
  • [5] Romain Azaïs, François Dufour, and Anne Gégout-Petit. Non-parametric estimation of the conditional distribution of the interjumping times for piecewise-deterministic markov, processes.Scandinavian Journal of Statistics, 41(4):950–969, 2014.
  • [6] Romain Azaïs and Alexandre Genadot. Semi-parametric inference for the absorption features of a growth-fragmentation, model.TEST, 24(2):341–360, 2015.
  • [7] Anis Ben Abdessalem, Romain Azaïs, Marie Touzet-Cortina, Anne Gégout-Petit, and Monique Puiggali. Stochastic modelling and prediction of fatigue crack propagation using piecewise-deterministic markov, processes.Accepted for publication in Journal of Risk and Reliability, 2016.
  • [8] Patrice Bertail, Stephan Clemençon, and Jessica Tressou. A storage model with random release rate for modelling exposure to food, contaminants.Mathematical Bioscience and Engineering, 5(1):35–60, 2008.
  • [9] Patrice Bertail, Stephan Clemençon, and Jessica Tressou. Statistical analysis of a dynamic model for dietary contaminant, exposure.Journal of Biological Dynamics, 4(2):212–234, 2010.
  • [10] Adrien Brandejsky, Benoîte de Saporta, and François Dufour. Numerical method for expectations of piecewise-determistic markov, processes.CAMCoS, 7(1):63–104, 2012.
  • [11] Adrien Brandejsky, Benoîte De Saporta, and François Dufour. Numerical methods for the exit time of a piecewise-deterministic markov, process.Adv. in Appl. Probab., 44(1):196–225, 03 2012.
  • [12] Djalil Chafai, Florent Malrieu, and Katy Paroux. On the long time behavior of the TCP window size, process.Stochastic Processes and their Applications, 120(8) :1518–1534, April 2010.
  • [13] Julien Chiquet and Nikolaos Limnios. A method to compute the transition function of a piecewise deterministic Markov process with application to, reliability.Statist. Probab. Lett., 78(12) :1397–1403, 2008.
  • [14] Julien Chiquet, Nikolaos Limnios, and Mohamed Eid. Piecewise deterministic markov processes applied to fatigue crack growth, modelling.Journal of Statistical Planning and Inference, 139(5) :1657–1667, 2009.
  • [15] Gerda Claeskens and Peter Hall. Effect of dependence on stochastic measures of accuracy of density, estimations.The Annals of Statistics, 30(2):431–454, 2002.
  • [16] Alina Crudu, Arnaud Debussche, Aurélie Muller, Ovidiu Radulescu, et al. Convergence of stochastic gene networks to hybrid piecewise deterministic, processes.The Annals of Applied Probability, 22(5) :1822–1859, 2012.
  • [17] Mark H. A., Davis.Markov models and optimization, volume 49 ofMonographs on Statistics and Applied Probability. Chapman & Hall, London, 1993.
  • [18] Benoîte de Saporta, François Dufour, Huilong Zhang, and Charles Elegbede. Optimal stopping for the predictive maintenance of a structure subject to, corrosion.Journal of Risk and Reliability, 226 (2):169–181, 2012.
  • [19] Marie Doumic, Marc Hoffmann, Nathalie Krell, and Lydia Robert. Statistical estimation of a growth-fragmentation model observed on a genealogical, tree.Bernoulli (to appear), 2015.
  • [20] Marie Doumic, Marc Hoffmann, Patricia Reynaud-Bouret, and Vincent Rivoirard. Nonparametric estimation of the division rate of a size-structured, population.SIAM Journal on Numerical Analysis, 50(2):925–950, 2012.
  • [21] Marie, Duflo.Random iterative models. Applications of Mathematics. Springer-Verlag, Berlin, 1997.
  • [22] Alexandre Genadot and Michèle Thieullen. Averaging for a fully coupled piecewise-deterministic markov process in infinite, dimensions.Adv. in Appl. Probab., 44(3):749–773, 09 2012.
  • [23] Jeffrey D. Hart and Philippe Vieu. Data-driven bandwidth choice for density estimation based on dependent, data.The Annals of Statistics, pages 873–890, 1990.
  • [24] Jianghai Hu, Wei-Chung Wu, and Shankar, Sastry.Modeling subtilin production in bacillus subtilis using stochastic hybrid systems. In R. Alur and G.J. Pappas, editors,Hybrid Systems: Computation and Control, number 2993 in LNCS, Springer-Verlag, Berlin, 2004.
  • [25] Martin, Jacobsen.Point Process Theory and Applications: Marked Point and Piecewise Deterministic Processes. Probability and its Applications. Birkhäuser, Boston-Basel-Berlin, 2006.
  • [26] Tae Yoon Kim. Asymptotically optimal bandwidth selection rules for the kernel density estimator with dependent, observations.Journal of Statistical Planning and Inference, 59(2):321–336, 1997.
  • [27] Tae Yoon Kim and Dennis D. Cox. Bandwidth selection in kernel smoothing of time, series.Journal of Time Series Analysis, 17(1):49–63, 1996.
  • [28] Nathalie Krell. Statistical estimation of jump rates for a specific class of piecewise deterministic markov, processes.arXiv :1406.2845, March 2015.
  • [29] Sean Meyn and Richard L., Tweedie.Markov chains and stochastic stability. Cambridge University Press, Cambridge, second edition, 2009.
  • [30] N.R., Norris.Exploring the optimality of various bacterial motility strategies: a stochastic hybrid systems approach. PhD thesis, Massachusetts Institute of Technology, 2013.
  • [31] H. G. Othmer, X. Xin, and C. Xue. Excitation and adaptation in bacteria. a model signal transduction system that controls taxis and spatial pattern, formation.International Journal of Molecular Sciences, 14(5) :9205–9248, 2013.
  • [32] Ovidiu Radulescu, Aurélie Muller, and Alina Crudu. Théorèmes limites pour les processus de markov à, sauts.Technique et Science Informatiques, 26(3–4):443–469, 2007.
  • [33] Lydia Robert, Marc Hoffmann, Nathalie Krell, Stephane Aymerich, Jerome Robert, and Marie Doumic. Division in escherichia coli is triggered by a size-sensing rather than a timing, mechanism.BMC Biology, 12(1):17, 2014.
  • [34] M. J. Tindall, P. K. Maini, S. L. Porter, and J. P. Armitage. Overview of mathematical approaches used to model bacterial chemotaxis ii: bacterial, populations.Bulletin of Mathematical Biology, 70(6) :1570–1607, 2008.
  • [35] D. Virkler, B. Hillberry, and P. Goel. The statistical nature of fatigue crack, propagation.J Engng Mater Technol, 101(2):148–153, 1979.