Non-adaptive estimation for degenerate diffusion processes

We consider a degenerate system of stochastic differential equations. The first component of the 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 with a parameter $\theta _3$ and no diffusion term. Parametric estimation of the degenerate diffusion system is discussed under a sampling scheme. We investigate the asymptotic behavior of the joint quasi-maximum likelihood estimator for $(\theta _1,\theta _2,\theta _3)$. The estimation scheme is non-adaptive. The estimator incorporates information of the increments of both components, and under this construction, we show that the asymptotic variance of the estimator for $\theta _1$ is smaller than the one for standard estimator based on the first component only, and that the convergence of the estimator for $\theta _3$ is much faster than for the other parameters. By simulation studies, we compare the performance of the joint quasi-maximum likelihood estimator with the adaptive and one-step estimators investigated in Gloter and Yoshida [Electron. J. Statist 15 (2021), no. 1, 1424–1472].