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 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.
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Length: 42 pages Date of creation: Jul 2000 Date of revision: Publication status: Published in Econometric Theory (2002), 18: 1040-1085 Handle: RePEc:cwl:cwldpp:1269
Find related papers by JEL classification: C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: General - - - Hypothesis Testing C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: General - - - Estimation
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