Non-adaptive estimation for degenerate diffusion processes

Abstract

We consider a degenerate system of stochastic differential equations. The first component of the system has a parameter θ 1 \theta _1 in a non-degenerate diffusion coefficient and a parameter θ 2 \theta _2 in the drift term. The second component has a drift term with a parameter θ 3 \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 ( θ 1 , θ 2 , θ 3 ) (\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 θ 1 \theta _1 is smaller than the one for standard estimator based on the first component only, and that the convergence of the estimator for θ 3 \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].