### Abstract

This paper generalizes a part of the theory of Z-estimation which has been developed mainly in the context of modern empirical processes to the case of stochastic processes, typically, semimartingales. We present a general theorem to derive the asymptotic behavior of the solution to an estimating equation θ ~+? ψ_{n}(θ, ĥ_{n}) = 0 with an abstract nuisance parameter h when the compensator of ψ_{n} is random. As its application, we consider the estimation problem in an ergodic diffusion process model where the drift coefficient contains an unknown, finite-dimensional parameter θ and the diffusion coefficient is indexed by a nuisance parameter h from an infinite-dimensional space. An example for the nuisance parameter space is a class of smooth functions. We establish the asymptotic normality and efficiency of a Z -estimator for the drift coefficient. As another application, we present a similar result also in an ergodic time series model.

Original language | English |
---|---|

Pages (from-to) | 3555-3579 |

Number of pages | 25 |

Journal | Annals of Statistics |

Volume | 37 |

Issue number | 6 A |

DOIs | |

Publication status | Published - 2009 Dec |

Externally published | Yes |

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### Keywords

- Asymptotic efficiency
- Discrete observation
- Ergodic diffusion
- Estimating function
- Metric entropy
- Nuisance parameter

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

**Asymptotic theory of semiparametric z-estimators for stochastic processes with applications to ergodic diffusions and time series.** / Nishiyama, Yoichi.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Asymptotic theory of semiparametric z-estimators for stochastic processes with applications to ergodic diffusions and time series

AU - Nishiyama, Yoichi

PY - 2009/12

Y1 - 2009/12

N2 - This paper generalizes a part of the theory of Z-estimation which has been developed mainly in the context of modern empirical processes to the case of stochastic processes, typically, semimartingales. We present a general theorem to derive the asymptotic behavior of the solution to an estimating equation θ ~+? ψn(θ, ĥn) = 0 with an abstract nuisance parameter h when the compensator of ψn is random. As its application, we consider the estimation problem in an ergodic diffusion process model where the drift coefficient contains an unknown, finite-dimensional parameter θ and the diffusion coefficient is indexed by a nuisance parameter h from an infinite-dimensional space. An example for the nuisance parameter space is a class of smooth functions. We establish the asymptotic normality and efficiency of a Z -estimator for the drift coefficient. As another application, we present a similar result also in an ergodic time series model.

AB - This paper generalizes a part of the theory of Z-estimation which has been developed mainly in the context of modern empirical processes to the case of stochastic processes, typically, semimartingales. We present a general theorem to derive the asymptotic behavior of the solution to an estimating equation θ ~+? ψn(θ, ĥn) = 0 with an abstract nuisance parameter h when the compensator of ψn is random. As its application, we consider the estimation problem in an ergodic diffusion process model where the drift coefficient contains an unknown, finite-dimensional parameter θ and the diffusion coefficient is indexed by a nuisance parameter h from an infinite-dimensional space. An example for the nuisance parameter space is a class of smooth functions. We establish the asymptotic normality and efficiency of a Z -estimator for the drift coefficient. As another application, we present a similar result also in an ergodic time series model.

KW - Asymptotic efficiency

KW - Discrete observation

KW - Ergodic diffusion

KW - Estimating function

KW - Metric entropy

KW - Nuisance parameter

UR - http://www.scopus.com/inward/record.url?scp=69949151781&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=69949151781&partnerID=8YFLogxK

U2 - 10.1214/09-AOS693

DO - 10.1214/09-AOS693

M3 - Article

AN - SCOPUS:69949151781

VL - 37

SP - 3555

EP - 3579

JO - Annals of Statistics

JF - Annals of Statistics

SN - 0090-5364

IS - 6 A

ER -