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# ASYMPTOTIC THEORY OF SEMIPARAMETRIC Z -ESTIMATORS FOR STOCHASTIC PROCESSES WITH APPLICATIONS TO ERGODIC DIFFUSIONS AND TIME SERIES

Yoichi Nishiyama
The Annals of Statistics
Vol. 37, No. 6A (December 2009), pp. 3555-3579
Stable URL: http://www.jstor.org/stable/25662203
Page Count: 25
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## 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 $\theta \rightsquigarrow \Psi _{n}(\theta,\hat{h}_{n})=0$ 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.

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