Abstract
A higher-order asymptotic theory of estimation is presented in this Chapter in the framework of the geometry of the model M and the ancillary family A associated with the estimator. Conditions for the consistency and efficiency of an estimator are given in geometrical terms of A. The higher-order terms of the covariance of an efficient estimators are decomposed into the sum of three non-negative geometrical terms. This proves that the bias corrected maximum likelihood estimator is the best estimator from the point of view of the third order asymptotic evaluation. The effect of parametrization is elucidated from the geometrical viewpoint.
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