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.
EQUIVALENCE OF THE HIGHER ORDER ASYMPTOTIC EFFICIENCY OF
k-STEP AND EXTREMUM STATISTICS