  The transition density of a diffusion process does not admit an explicit
expression in general, which prevents the full maximum likelihood estimation
(MLE) based on discretely observed sample paths. A\"{\i}t-Sahalia [J. Finance
54 (1999) 1361--1395; Econometrica 70 (2002) 223--262] proposed asymptotic
expansions to the transition densities of diffusion processes, which lead to an
approximate maximum likelihood estimation (AMLE) for parameters. Built on
A\"{\i}t-Sahalia's [Econometrica 70 (2002) 223--262; Ann. Statist. 36 (2008)
906--937] proposal and analysis on the AMLE, we establish the consistency and
convergence rate of the AMLE, which reveal the roles played by the number of
terms used in the asymptotic density expansions and the sampling interval
between successive observations. We find conditions under which the AMLE has
the same asymptotic distribution as that of the full MLE. A first order
approximation to the Fisher information matrix is proposed.
