Distributions of Random Partitions and Their Applications

Assume that a random sample of size m is selected from a population containing a countable number of classes (subpopulations) of elements (individuals). A partition of the set of sample elements into (unordered) subsets, with each subset containing the elements that belong to same class, induces a random partition of the sample size m, with part sizes {Z 1,Z 2,...,Z N } being positive integer-valued random variables. Alternatively, if N j is the number of different classes that are represented in the sample by j elements, for j=1,2,...,m, then (N 1,N 2,...,N m ) represents the same random partition. The joint and the marginal distributions of (N 1,N 2,...,N m ), as well as the distribution of $N=\sum^m_{j=1}N_{\!j}$ are of particular interest in statistical inference. From the inference point of view, it is desirable that all the information about the population is contained in (N 1,N 2,...,N m ). This requires that no physical, genetical or other kind of significance is attached to the actual labels of the population classes. In the present paper, combinatorial, probabilistic and compound sampling models are reviewed. Also, sampling models with population classes of random weights (proportions), and in particular the Ewens and Pitman sampling models, on which many publications are devoted, are extensively presented.