Adaptive and non-adaptive estimation for degenerate diffusion processes

TL;DR

This paper develops and analyzes adaptive and joint estimation methods for parameters in degenerate diffusion systems from discrete data, demonstrating improved efficiency and faster convergence of estimators.

Contribution

It introduces new estimation procedures for degenerate diffusion models, proving their asymptotic normality and showing they outperform existing methods in variance reduction.

Findings

Estimators for  have asymptotic normality under various conditions.

Incorporating both components' increments reduces estimator variance.

Convergence for  is significantly faster than for other parameters.

Abstract

We discuss parametric estimation of a degenerate diffusion system from time-discrete observations. The first component of the degenerate diffusion system has a parameter $\theta_1$ in a non-degenerate diffusion coefficient and a parameter $\theta_2$ in the drift term. The second component has a drift term parameterized by $\theta_3$ and no diffusion term. Asymptotic normality is proved in three different situations for an adaptive estimator for $\theta_3$ with some initial estimators for ($\theta_1$ , $\theta_2$), an adaptive one-step estimator for ($\theta_1$ , $\theta_2$ , $\theta_3$) with some initial estimators for them, and a joint quasi-maximum likelihood estimator for ($\theta_1$ , $\theta_2$ , $\theta_3$) without any initial estimator. Our estimators incorporate information of the increments of both components. Thanks to this construction, the asymptotic variance of the estimators for $\theta_1$ is smaller than the standard one based only on the first component. The convergence of the estimators for $\theta_3$ is much faster than the other parameters. The resulting asymptotic variance is smaller than that of an estimator only using the increments of the second component.