  Motivated by the fundamental problem of modeling the frequency of frequencies
(FoF) distribution, this paper introduces the concept of a cluster structure to
define a probability function that governs the joint distribution of a random
count and its exchangeable random partitions. A cluster structure, naturally
arising from a completely random measure mixed Poisson process, allows the
probability distribution of the random partitions of a subset of a population
to be dependent on the population size, a distinct and motivated feature that
makes it more flexible than a partition structure. This allows it to model an
entire FoF distribution whose structural properties change as the population
size varies. A FoF vector can be simulated by drawing an infinite number of
Poisson random variables, or by a stick-breaking construction with a finite
random number of steps. A generalized negative binomial process model is
proposed to generate a cluster structure, where in the prior the number of
clusters is finite and Poisson distributed, and the cluster sizes follow a
truncated negative binomial distribution. We propose a simple Gibbs sampling
algorithm to extrapolate the FoF vector of a population given the FoF vector of
a sample taken without replacement from the population. We illustrate our
results and demonstrate the advantages of the proposed models through the
analysis of real text, genomic, and survey data.
