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

2012 ◽  
Vol 122 (6) ◽  
pp. 2480-2485
Author(s):  
Claire Lacour
Keyword(s):  
Markov Chain ◽  
Nonparametric Estimation ◽  
Transition Density ◽  
Stationary Density

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 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.

Tail behaviour of the stationary density of general non-linear autoregressive processes of order 1

1993 ◽  
Vol 30 (2) ◽  
pp. 315-329 ◽  
Author(s):  
Jean Diebolt ◽  
Dominique Guégan
Keyword(s):  
Markov Chain ◽  
Autoregressive Processes ◽  
Stationary Density ◽  
Tail Behaviour ◽  
Non Linear ◽  
The Common

We examine the main properties of the Markov chain Xt = T(Xt– 1) + σ(Xt– 1)ε t. Under general and tractable assumptions, we derive bounds for the tails of the stationary density of the process {Xt} in terms of the common density of the ε t's.

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