Likelihood and Observed Geometries

In the differential geometric approach to parametric statistics, developed by Chentsov, Efron, Amari, and others, the parameter space is set up as a differentiable manifold with expected information as metric tensor and with a family of affine connections, the a-connections, determined from the expected information and the skewness tensor of the score vector. The usefulness of this approach is particularly notable in connection with Edgeworth expansions of estimators. Motivated by the conditionality viewpoint, an "observed" parallel to that theory is established in the present paper using obseved information and an "observed skewness" tensor instead of the above expected quantities. The formula c\|hatj\|^1/2-L for the conditional distribution of the maximum likelihood estimator is expanded (to third order) asymptotically and the "observed geometries" are shown to have a role in this type of expansion similar to that of the "expected geometries" in the Edgeworth expansions mentioned above. In these new developments "mixed derivatives of the log model function, " defined by means of an auxiliary statistic complementing the maximum likelihood estimator, take the place of moments of derivatives of the log likelihood function.