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Nonparametric Estimation for a Class of Piecewise-Deterministic Markov Processes

Part of: Nonparametric inference Markov processes

Published online by Cambridge University Press:  30 January 2018

Takayuki Fujii
Takayuki Fujii*
Affiliation:
Osaka University
*
∗ Current address: Faculty of Economics, Shiga University, 1-1-1 Banba, Hikone, Shiga 522-8522, Japan. Email address: takayuki-fujii@biwako.shiga-u.ac.jp
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Abstract

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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.

Keywords

Piecewise-deterministic Markov processlocal timestationary densityuniform consistencynonparametric estimation

MSC classification

Primary: 62G20: Asymptotic properties
Secondary: 60J55: Local time and additive functionals
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 4 , December 2013 , pp. 931 - 942
Copyright
© Applied Probability Trust 

References

Borovkov, K. and Last, G. (2008). On level crossings for a general class of piecewise-deterministic Markov processes. Adv. Appl. Prob. 40, 815–834.CrossRefGoogle Scholar
Bosq, D. (1998). Nonparametric Statistics for Stochastic Processes: Estimation and Prediction (Lecture Notes Statist. 110), 2nd edn. Springer, New York.CrossRefGoogle Scholar
Bosq, D. and Davydov, Y. (1999). Local time and density estimation in continuous time. Math. Meth. Statist. 8, 22–45.Google Scholar
Brémaud, P. (1981). Point Processes and Queues: Martingale Dynamics. Springer, New York.CrossRefGoogle Scholar
Brill, P. H. and Posner, M. J. M. (1977). Level crossings in point processes applied to queues: single-server case. Operat. Res. 25, 662–674.CrossRefGoogle Scholar
Davis, M. H. A. (1993). Markov Models and Optimization. Chapman and Hall, London.CrossRefGoogle Scholar
Fujii, T. and Nishiyama, Y. (2012). Some problems in nonparametric inference for the stress release process related to the local time. Ann. Inst. Statist. Math. 64, 991–1007.CrossRefGoogle Scholar
Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd edn. Springer, New York.Google Scholar
Kutoyants, Y. A. (1998). Efficient density estimation for ergodic diffusion processes. Statist. Infer. Stoch. Process. 1, 131–155.CrossRefGoogle Scholar
Kutoyants, Y. A. (2004). Statistical Inference for Ergodic Diffusion Processes. Springer, London.CrossRefGoogle Scholar
Last, G. (2004). Ergodicity properties of stress release, repairable system and workload models. Adv. Appl. Prob. 36, 471–498.CrossRefGoogle Scholar
Last, G. and Brandt, A. (1995). Marked Point Processes on the Real Line: The Dynamic Approach. Springer, New York.Google Scholar
Leadbetter, M. R. (1966). On crossings of levels and curves by a wide class of stochastic processes. Ann. Math. Statist. 37, 260–267.CrossRefGoogle Scholar
Liptser, R. S. and Shiryaev, A. N. (1989). Theory of Martingales. Kluwer, Dordrecht.CrossRefGoogle Scholar
Nishiyama, Y. (2000). Weak convergence of some classes of martingales with Jumps. Ann. Prob. 28, 685–712.CrossRefGoogle Scholar
Rice, S. O. (1944). Mathematical analysis of random noise. Bell System Tech. J. 23, 282–332.CrossRefGoogle Scholar
Van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes: With Applications to Statistics. Springer, New York.CrossRefGoogle Scholar
Van Zanten, J. H. (2001). On the uniform convergence of the empirical density of an ergodic diffusion. Statist. Infer. Stoch. Process. 3, 251–262.CrossRefGoogle Scholar