scholarly journals Nonparametric estimation of jump rates for a specific class of piecewise deterministic Markov processes

Bernoulli ◽  
2021 ◽  
Vol 27 (4) ◽  
Author(s):  
Nathalie Krell ◽  
Émeline Schmisser
Keyword(s):  
Markov Processes ◽  
Nonparametric Estimation ◽  
Specific Class ◽  
Piecewise Deterministic Markov Processes

scholarly journals A New Proof of the Wiener-Hopf Factorization via Basu's Theorem

2012 ◽  
Vol 49 (03) ◽  
pp. 876-882
Author(s):  
Brian Fralix ◽  
Colin Gallagher
Keyword(s):  
Random Walks ◽  
Markov Processes ◽  
Lévy Processes ◽  
Levy Processes ◽  
Specific Class ◽  
Piecewise Deterministic Markov Processes

We illustrate how Basu's theorem can be used to derive the spatial version of the Wiener-Hopf factorization for a specific class of piecewise-deterministic Markov processes. The classical factorization results for both random walks and Lévy processes follow immediately from our result. The approach is particularly elegant when used to establish the factorization for spectrally one-sided Lévy processes.

Nonparametric Estimation for a Class of Piecewise-Deterministic Markov Processes

2013 ◽  
Vol 50 (04) ◽  
pp. 931-942
Author(s):  
Takayuki Fujii
Keyword(s):  
Local Time ◽  
Markov Processes ◽  
Nonparametric Estimation ◽  
Sample Path ◽  
Point Of View ◽  
Statistical Point ◽  
Jump Size ◽  
Stationary Density ◽  
Piecewise Deterministic Markov Processes ◽  
Estimation Problems

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.

scholarly journals A New Proof of the Wiener-Hopf Factorization via Basu's Theorem

2012 ◽  
Vol 49 (3) ◽  
pp. 876-882
Author(s):  
Brian Fralix ◽  
Colin Gallagher
Keyword(s):  
Random Walks ◽  
Markov Processes ◽  
Lévy Processes ◽  
Levy Processes ◽  
Specific Class ◽  
Piecewise Deterministic Markov Processes

We illustrate how Basu's theorem can be used to derive the spatial version of the Wiener-Hopf factorization for a specific class of piecewise-deterministic Markov processes. The classical factorization results for both random walks and Lévy processes follow immediately from our result. The approach is particularly elegant when used to establish the factorization for spectrally one-sided Lévy processes.

scholarly journals Nonparametric Estimation for a Class of Piecewise-Deterministic Markov Processes

2013 ◽  
Vol 50 (4) ◽  
pp. 931-942
Author(s):  
Takayuki Fujii
Keyword(s):  
Local Time ◽  
Markov Processes ◽  
Nonparametric Estimation ◽  
Sample Path ◽  
Point Of View ◽  
Statistical Point ◽  
Jump Size ◽  
Stationary Density ◽  
Piecewise Deterministic Markov Processes ◽  
Estimation Problems

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.

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