No. 1074

Title:

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

Author(s):

Nishiyama, Yoichi (The Institute of Statistical Mathematics)

Key words:

Estimating function, nuisance parameter, metric entropy, asymptotic efficiency, ergodic diffusion, discrete observation.

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 function $ \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.