Asymptotically Optimal Estimator of the Parameter of Semi-Linear Autoregression

Abstract

The difference equations ${\xi }_k=af({\xi }_{k-1})+{\epsilon }_k$, where $({\epsilon }_k)$ is a square integrable difference martingale, and the differential equation $\mathrm{d}\xi =-af(\xi )\mathrm{d}t+\mathrm{d}\eta$, where $\eta$ is a square integrable martingale, are considered. A family of estimators depending, besides the sample size $n$ (or the observation period, if time is continuous) on some random Lipschitz functions is constructed. Asymptotic optimality of this estimators is investigated.

10 pages