# Asymptotic theory of semiparametric $Z$-estimators for stochastic processes with applications to ergodic diffusions and time series - Mathematics > Statistics Theory

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Abstract: This paper generalizes a part of the theory of $Z$-estimation which has beendeveloped mainly in the context of modern empirical processes to the case ofstochastic processes, typically, semimartingales. We present a general theoremto derive the asymptotic behavior of the solution to an estimating equation$\theta\leadsto \Psi n\theta,\widehat{h} n=0$ with an abstract nuisanceparameter $h$ when the compensator of $\Psi n$ is random. As its application,we consider the estimation problem in an ergodic diffusion process model wherethe 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 spaceis a class of smooth functions. We establish the asymptotic normality andefficiency of a $Z$-estimator for the drift coefficient. As anotherapplication, we present a similar result also in an ergodic time series model.

Autor: Yoichi Nishiyama

Fuente: https://arxiv.org/