Dirichlet analysis for inverse multinomial with both quota and quota-free cells

Abstract

For a broad variety of quota-sampling problems, the moment structure for the total frequency in various cells at stopping time is studied, as well as the cross moments, covariances, and correlations. Previously, we considered two different types of quotas: frequency vs. run quotas. In this paper, the effect of having quota-free cells present is considered. Our main strategy is to relate the cell frequencies at stopping time to the total number of observations needed to reach the stopping time. For the case of α frequency quota and γ quota-free cells, we consider the problem of stopping as soon as any J of the α frequency quotas are satisfied. The main tool that we use is the Dirichlet integrals which are defined here, but the reader should consult [3] and [4] for further background. Another method, called the generalized probability generating function (gpgf), is also used for confirmation of the various results. The paper terminates with an example of how to use the results of this paper in a two-stage sampling (with a fixed sample size n in the first stage) to make the covariances between certain total cell frequencies equal to zero (i.e., uncorrelated) at stopping time.