Is the Ewens distribution the only one-parameter family of partition structures where the total number of types sampled is a sufficient statistic? In general, the answer is no. It is shown that all counterexamples can be generated via an urn scheme. The urn scheme need only satisfy two general conditions. In fact, the conditions are both necessary and sufficient. However, in particular, for a large class of partition structures that naturally arise in the infinite alleles theory of population genetics, the Ewens distribution is the only one in this class where the total number of types is sufficient for estimating the mutation rate. Finally, asymptotic sufficiency for parametric families of partition structures is discussed.

This paper is concerned with models for sampling from populations in which there exists a total order on the collection of types, but only the relative ordering of types which actually appear in the sample is known. The need for consistency between different sample sizes limits the possible models to what are here called ‘consistent ordered sampling distributions'. We give conditions under which weak convergence of population distributions implies convergence of sampling distributions and conversely those under which population convergence may be inferred from convergence of sampling distributions. A central result exhibits a collection of ‘ordered sampling functions', none of which is continuous, which separates measures in a certain class. More generally, we characterize all consistent ordered sampling distributions, proving an analogue of de Finetti's theorem in this context. These results are applied to an unsolved problem in genetics where it is shown that equilibrium age-ordered population allele frequencies for a wide class of exchangeable reproductive models converge weakly, as the population size becomes large, to the so-called GEM distribution. This provides an alternative characterization which is more informative and often more convenient than Kingman's (1977) characterization in terms of the Poisson–Dirichlet distribution.

For each n > 0, the Ewens sampling formula from population genetics is a measure on the set of all partitions of the integer n. To determine the limiting distributions for the part sizes of a partition with respect to the measures given by this formula, we associate to each partition a step function on [0, 1]. Each jump in the function equals the number of parts in the partition of a certain size. We normalize these functions and show that the induced measures on D[0, 1] converge to Wiener measure. This result complements Kingman's frequency limit theorem [10] for the Ewens partition structure.