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

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 θ ~+? ψ_n(θ, ĥ_n) = 0 with an abstract nuisance parameter h when the compensator of ψ_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 θ 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.