Curvature and inference for maximum likelihood estimates

Abstract

Maximum likelihood estimates are sufficient statistics in exponential families, but not in general. The theory of statistical curvature was introduced to measure the effects of MLE insufficiency in one-parameter families. Here, we analyze curvature in the more realistic venue of multiparameter families—more exactly, curved exponential families, a broad class of smoothly defined nonexponential family models. We show that within the set of observations giving the same value for the MLE, there is a “region of stability” outside of which the MLE is no longer even a local maximum. Accuracy of the MLE is affected by the location of the observation vector within the region of stability. Our motivating example involves “$g$-modeling,” an empirical Bayes estimation procedure.