Asymptotically Optimal Estimator of the Parameter of Semi-Linear Autoregression

TL;DR

This paper develops a family of estimators for semi-linear autoregression models and investigates their asymptotic optimality, applicable to both difference and differential equations driven by martingales.

Contribution

It introduces a new class of estimators depending on random Lipschitz functions and proves their asymptotic optimality for semi-linear autoregressive models.

Findings

Establishes asymptotic optimality of the proposed estimators.

Applies to models driven by square integrable martingales.

Provides a unified approach for difference and differential equations.

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 ${\rm d}\xi=-af(\xi){\rm d}t+{\rm d}\eta$, 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.