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Non-Parametric Estimation of the Conditional Distribution of the Interjumping Times for Piecewise-Deterministic Markov Processes

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  • Romain Azaïs
  • François Dufour
  • Anne Gégout-Petit

Abstract

type="main" xml:id="sjos12076-abs-0001"> This paper presents a non-parametric method for estimating the conditional density associated to the jump rate of a piecewise-deterministic Markov process. In our framework, the estimation needs only one observation of the process within a long time interval. Our method relies on a generalization of Aalen's multiplicative intensity model. We prove the uniform consistency of our estimator, under some reasonable assumptions related to the primitive characteristics of the process. A simulation study illustrates the behaviour of our estimator.

Suggested Citation

  • Romain Azaïs & François Dufour & Anne Gégout-Petit, 2014. "Non-Parametric Estimation of the Conditional Distribution of the Interjumping Times for Piecewise-Deterministic Markov Processes," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 41(4), pages 950-969, December.
    • Handle: RePEc:bla:scjsta:v:41:y:2014:i:4:p:950-969
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    File URL: http://hdl.handle.net/10.1111/sjos.12076
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    References listed on IDEAS

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    1. Chiquet, Julien & Limnios, Nikolaos, 2008. "A method to compute the transition function of a piecewise deterministic Markov process with application to reliability," Statistics & Probability Letters, Elsevier, vol. 78(12), pages 1397-1403, September.
    2. Utikal, Klaus J., 1993. "Nonparametric inference for a doubly stochastic Poisson process," Stochastic Processes and their Applications, Elsevier, vol. 45(2), pages 331-349, April.
    3. Utikal, Klaus J., 1997. "Nonparametric inference for Markovian interval processes," Stochastic Processes and their Applications, Elsevier, vol. 67(1), pages 1-23, April.
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    Cited by:

    1. Frédéric Lavancier & Ronan Le Guével, 2021. "Spatial birth–death–move processes: Basic properties and estimation of their intensity functions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 83(4), pages 798-825, September.
    2. Hoffmann, Marc & Marguet, Aline, 2019. "Statistical estimation in a randomly structured branching population," Stochastic Processes and their Applications, Elsevier, vol. 129(12), pages 5236-5277.
    3. Josef Anton Strini & Stefan Thonhauser, 2020. "On Computations in Renewal Risk Models—Analytical and Statistical Aspects," Risks, MDPI, vol. 8(1), pages 1-20, March.

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