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

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 $\theta ⇝{\Psi }_n(\theta ,{\hat{h}}_n)=0$ 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.

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

References in corpus (1)

• Donsker theorems for diffusions: Necessary and sufficient conditions