  We study maximum-likelihood-type estimation for diffusion processes when the
coefficients are nonrandom and observation occurs in nonsynchronous manner. The
problem of nonsynchronous observations is important when we consider the
analysis of high-frequency data in a financial market. Constructing a
quasi-likelihood function to define the estimator, we adaptively estimate the
parameter for the diffusion part and the drift part. We consider the asymptotic
theory when the terminal time point $T_n$ and the observation frequency goes to
infinity, and show the consistency and the asymptotic normality of the
estimator. Moreover, we show local asymptotic normality for the statistical
model, and asymptotic efficiency of the estimator as a consequence. To show the
asymptotic properties of the maximum-likelihood-type estimator, we need to
control the asymptotic behaviors of some functionals of the sampling scheme.
Though it is difficult to directly control those in general, we study tractable
sufficient conditions when the sampling scheme is generated by mixing
processes.
