Optimality and small ͉-optimality of martingale estimating functions

Abstract

Martingale estimating functions determined from a given collection (the base) of conditional expectations are considered for estimating the parameters of a discretely observed diffusion. Small ͉-optimality of these functions (i.e. near-efficiency when the observations are close together) is discussed, and in particular it is shown that this can be achieved provided the base is large enough. It is also shown that the optimal martingale estimating function with a given base is automatically small ͉-optimal, provided only that the base is sufficiently large. In both cases the critical dimension of the base is the same and determined by the dimension of the diffusion, and by whether the squared diffusion matrix is parameter-dependent or not; the critical number does not depend, however, on the dimension of the parameter.