  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.
