  A general theory of efficient estimation for ergodic diffusion processes
sampled at high frequency with an infinite time horizon is presented. High
frequency sampling is common in many applications, with finance as a prominent
example. The theory is formulated in term of approximate martingale estimating
functions and covers a large class of estimators including most of the
previously proposed estimators for diffusion processes. Easily checked
conditions ensuring that an estimating function is an approximate martingale
are derived, and general conditions ensuring consistency and asymptotic
normality of estimators are given. Most importantly, simple conditions are
given that ensure rate optimality and efficiency. Rate optimal estimators of
parameters in the diffusion coefficient converge faster than estimators of
drift coefficient parameters because they take advantage of the information in
the quadratic variation. The conditions facilitate the choice among the
multitude of estimators that have been proposed for diffusion models. Optimal
martingale estimating functions in the sense of Godambe and Heyde and their
high frequency approximations are, under weak conditions, shown to satisfy the
conditions for rate optimality and efficiency. This provides a natural feasible
method of constructing explicit rate optimal and efficient estimating functions
by solving a linear equation.
