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 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 2k 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.
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Article provided by Cambridge University Press in its journal Econometric Theory.
Volume (Year): 18 (2002) Issue (Month): 05 (October) Pages: 1040-1085 Download reference. The following formats are available: HTML
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