Abstract
SUMMARY Differential geometry is applied to the problems of defining higher-order asymptotic ancillarity and of obtaining the asymptotic conditional distribution of an efficient estimator in multiparameter curved exponential families. It is shown that a fundamental role is played in the asymptotic theory of estimation by a one-parameter family of affine connexions and curvatures of subspaces. Asymptotic ancillary statistics of higher order are explicitly constructed with the help of the geometry. The conditional distribution of an estimator is given in terms of the exponential curvature of the model and the mixture curvature of the ancillary subspaces associated with the estimator.
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