Second-order sufficiency and statistical invariants

Abstract

For samples from a regular distribution depending on one parameter a second-order sufficient statistic is introduced, and its characteristic function obtained correct to this order. With a canonical choice of parameter it is shown that the cumulants of this statistic are functions of basic statistical invariants including ‘curvature’ terms. The distribution of the maximum likelihood estimator given an approximate ancillary statistic is discussed for translation families and general distributions.