A New Approach to Sampling from Finite Populations. II Distribution-Free Sufficiency

Summary The idea that in some situations the prior knowledge about the unknown parameters could be formulated as a class of prior distributions, which could be used, not necessarily through Bayes posterior probability, for subsequent inference, is already present in Godambe (1955). In the present paper, the concept of distribution-free linear sufficiency or in short linear sufficiency, originally due to Barnard (1963b) but redefined by the present author (Part I), is extended by defining distribution-free sufficiency, removing the restriction of linearity. This extension again is based on the assumption that in some situations prior knowledge could be formulated as a class of prior distributions. A certain linear estimator of the population total, which in Part I was shown to satisfy the redefined criteria of linear sufficiency uniquely in the class of all linear estimators, is now shown to satisfy this extended criteria of distribution-free sufficiency in the entire class of estimators. Further the general relationship between the linear sufficiency of Part I and the distribution-free sufficiency introduced here is investigated. Broadly the result is that, if we restrict to linear estimators only, distribution-free sufficiency is identical with linear sufficiency. Finally, some remarks are offered by way of comparison between the result obtained by the author previously (Godambe, 1955) and the result here, about the utilization of the prior information. The approach of this paper clearly implies a generalization of Fisherian sufficiency suitable for the situations when prior knowledge consists of a class of prior distributions. An alternative generalization when the prior knowledge consists of a group structure is due to Barnard (1963a).