Equivalence of the higher order asymptotic efficiency of k-step and extremum statistics

It is well known that a one-step scoring estimator that starts from any N-1/2- consistent estimator has the same first-order asymptotic efficiency as the maximum likelihood estimator. This paper extends this result to k-step estimators and test statistics for k greater than or equal to 1, higher order asymptotic efficiency, and general extremum estimators and test statistics. The paper shows that a k-step estimator has the same higher order asymptotic efficiency, to any given order, as the extremum estimator toward which it is stepping, provided (I) k is sufficiently large, (ii) some smoothness and moment conditions hold, and (iii) a condition on the initial estimator holds. For example, for the Newton-Raphson k-step estimator based on an initial estimator in a wide class, we obtain asymptotic equivalence to integer order s provided 2(k) greater than or equal to s + 1. Thus, for k = 1, 2, and 3, one obtains asymptotic equivalence to first, third, and seventh orders, respectively. This means that the maximum differences between the probabilities that the (N-1/2 -normalized) k-step and extremum estimators lie in any convex set are o(1), o(N-3/2), and o(N-3), respectively.