The Asymptotic Efficiency of a Maximum Likelihood Estimator
The Asymptotic Efficiency of a Maximum Likelihood Estimator
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最大似然估计器的渐近效率
DOI：
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发表时间：
1961
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通讯作者：
Guy Lebanon
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作者：
Guy Lebanon
极大似然估计量的相合性是由Wald[6]和Wolfowitz[7]在非常一般的条件下建立的。它是渐近有效的需要更严格的条件，即一致和渐近正态，方差等于Cram6r-Rao下界。Cramer [2], Gurland [3]， Kulldorf[4]给出了典型条件，它们都限制了似然函数至少二阶导数的行为。例如，Le Cam[5]和Bahadur[1]等作者在更一般的背景下讨论了大样本估计，但仍然要求最大似然估计量的二阶导数的正则性条件是渐近有效的。然而，已知的情况是不包括在这些规则条件。密度函数f(x, 0) = (1/2) exp -lx -Al提供了一个例子。样本中位数是6的最大似然估计量。已知它是渐近正态，方差为n-1，即Cram6r-Rao下界。但对于几乎所有的x， logf /a9是不连续的，而a2logf /la2是零。本文给出了不涉及似然二阶导数的较弱的渐近效率条件。给出了两组充分条件。从第一种开始，可以直接证明渐近效率，而不需要诉诸Wald-Wolfowitz结果，但存在一个经常不满足的凸性要求。第二组条件免除了这一要求，代价是其他方面的一些专业化，但必须通过沃尔福威茨方法建立一致性。最后考虑了一种更一般的情况，在这种情况下，一个改进的极大似然过程仍然可以产生一个渐近有效的估计量。给出了这种改进估计量与一类光滑估计量的关系。
The consistency of a maximum likelihood estimator has been established under very general conditions by Wald [6] and Wolfowitz [7]. Much more stringent conditions are needed for it to be asymptotically efficient, that is, consistent and asymptotically normal with variance equal to the Cram6r-Rao lower bound. Typical conditions are given by Cramer [2], Gurland [3], Kulldorf [4], all of which restrict the behavior of at least the second derivative of the likelihood function. Authors such as, for example, Le Cam [5] and Bahadur [1] discuss large sample estimation in a more general context but still require regularity conditions on the second derivative of the likelihood for the maximum likelihood estimator to be asymptotically efficient. However, cases are known which are not covered by these regularity conditions. The density function f(x, 0) = (1/2) exp -lx -Al provides an example. The sample median is a maximum likelihood estimator of 6. It is known to be asymptotically normal with variance n-1, which is the Cram6r-Rao lower bound. But a log f/a9 is discontinuous and a2 log f/la2 is zero for almost all x. In the present paper weaker conditions for asymptotic efficiency are given which do not involve the second derivative of the likelihood. Two sets of sufficient conditions are stated. From the first, asymptotic efficiency can be proved directly without appeal to the Wald-Wolfowitz result but there is a convexity requirement which is frequently not satisfied. The second set of conditions dispenses with this requirement at the cost of some specialization elsewhere, but consistency has to be established by the Wald-Wolfowitz method. Finally a more general situation is considered where a modified maximum likelihood procedure is shown still to yield an asymptotically efficient estimator. The relation of this modified estimator to a class of smoothed estimators is indicated.