Nonparametric estimation of the derivatives of the stationary density for stationary processes

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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Erratum to “Nonparametric estimation of the stationary density and the transition density of a Markov chain” [Stoch. Process. Appl. 118 (2008) 232–260]

Stochastic Processes and their Applications ◽

10.1016/j.spa.2012.03.001 ◽

2012 ◽

Vol 122 (6) ◽

pp. 2480-2485

Author(s):

Claire Lacour

Keyword(s):

Markov Chain ◽

Nonparametric Estimation ◽

Transition Density ◽

Stationary Density

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Nonparametric Estimation of Partial Derivatives of a Multivariate Probability Density by the Method of Wavelets

Asymptotics in Statistics and Probability ◽

10.1515/9783110942002-022 ◽

2000 ◽

pp. 321-330 ◽

Cited By ~ 1

Author(s):

B. L. S. Prakasa Rao

Keyword(s):

Probability Density ◽

Nonparametric Estimation ◽

Partial Derivatives ◽

Multivariate Probability ◽

Derivatives Of

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Nonparametric Estimation for a Class of Piecewise-Deterministic Markov Processes

Journal of Applied Probability ◽

10.1239/jap/1389370091 ◽

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.

Download Full-text