ABSTRACT

Our approach requires the model to be 'locally asymptotically normal'. In Sec. 2 we introduce this concept for general Markov chain models with possibly infinite-dimensional parameter and illustrate it with a few simple examples. In Sec. 3 we consider the problem of estimating a onedimensional function of the parameter and determine an optimal estimator within a simple class of estimators: the 'asymptotically linear' and 'regular' ones. Section 4 considers martingale estimating equations and indicates when they lead to asymptotically linear and regular estimators. Section 5 shows that the optimal asymptotically linear and regular estimator is already 'efficient', i.e., optimal among all regular estimators. The presentation is rigorous in Sections 2, 3 and 5, and heuristic in the others.