Finite de Finetti-Type Results as Approximation Results by the Expectation of Sufficient Statistics

We show that finite de Finetti-type results may be viewed as results on the approximation of certain continuous functions of a parameter by a sequence of positive operators (Ln) . For distribtions that depend on a finite-dimensional statistic (Tn) only, Ln is the expectation operator of (Tn) under the extremal infinite exchangeable distributions. The rate of approximation of finite exchangeable distributions by mixtures of marginals of infinite exchangeable distributions is the rate of approximation of a single function of the parameter, namely the second indefinite integral of the Fisher information. Our results include a major part of what is known about finite de Finetti theorems. The theory is, however, not only valid for the case when the extremal infinite exchangeable distributions are products of identical distributions. It applies as well to Markov-exchangeable distributions or the linear model. Moreover, the metric is not restricted to the supremum norm. (author's abstract)