Characterizations of linear sufficient statistics

A necessary and sufficient condition is developed such that there exists a continous linear sufficient statistic T for a dominated collection of totally finite measures defined on the Borel field generated by the open sets of a Banach space X. In particular, corollary necessary and sufficient conditions are given so that there exists a rank K linear sufficient statistic T for any finite collection of probability measures having n-variate normal densities. In this case a simple calculation, involving only the population means and covariances, determines the smallest integer K for which there exists a rank K linear sufficient statistic T (as well as an associated statistic T itself).