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

Details Asymptotic theory of semiparametric $Z$-estimators for stochastic processes with applications to ergodic diffusions and time series Asymptotic theory of semiparametric $Z$-estimators for stochastic processes with applications to ergodic diffusions and time series

2009-09-02

arxiv.org/abs/0909.0439v1

Annals of Statistics 2009, Vol. 37, No. 6A, 3555-3579

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.1214/09-AOS693 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of

Scientific paper

10.1214/09-AOS693

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 $\theta\leadsto \Psi_n(\theta,\widehat{h}_n)=0$ with an abstract nuisance parameter $h$ when the compensator of $\Psi_n$ 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 $\theta$ 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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