EQUIVALENCE OF THE HIGHER ORDER ASYMPTOTIC EFFICIENCY OF k-STEP AND EXTREMUM STATISTICS
证明，在光滑性和矩条件满足时，k步估计与极值估计的高阶渐近效率等价，例如牛顿-拉夫森k步估计在k=1、2、3时分别达到一阶、三阶和七阶渐近等价。
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 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 ≥ 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.