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 N1/2-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 > 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 towards 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, we obtain asymptotic equivalence to integer order s providedk > 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 (N1/2-normalized) k-step and extremum estimators lie in any convex set are o(1), o(N-3/2), and o(N-3) respectively.