# Equivalence of the Higher-order Asymptotic Efficiency of k-step and Extremum Statistics

Discussion Paper

7-1-2000

1269

42

## Abstract

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 > 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 provided k > 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.

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