### 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 |
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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 1 |

Externally published | Yes |

### Keywords

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

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty