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

TL;DR

This paper develops a general asymptotic theory for semiparametric Z-estimators in stochastic processes, with applications to ergodic diffusions and time series, establishing their normality and efficiency.

Contribution

It introduces a new theorem for the asymptotic behavior of Z-estimators with random compensators in stochastic processes, extending existing empirical process theory.

Findings

Proves asymptotic normality of Z-estimators in ergodic diffusions.

Establishes efficiency of the proposed estimators.

Extends theory to infinite-dimensional nuisance parameters.

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 $\theta\leadsto \Psi_n(\theta,\widehat{h}_n)=0$ 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.