Adaptive and non-adaptive estimation for degenerate diffusion processes.

Summary We discuss parametric estimation of a degenerate diffusion system from time-discrete observations. The first component of the degenerate diffusion system has a parameter θ_1 in a non-degenerate diffusion coefficient and a parameter θ_2 in the drift term. The second component has a drift term parameterized by θ_3 and no diffusion term. Asymptotic normality is proved in three different situations for an adaptive estimator for θ_3 with some initial estimators for (θ_1 , θ_2), an adaptive one-step estimator for (θ_1 , θ_2 , θ_3) with some initial estimators for them, and a joint quasi-maximum likelihood estimator for (θ_1 , θ_2 , θ_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 θ_1 is smaller than the standard one based only on the first component. The convergence of the estimators for θ_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.