Note: this paper is an older technical report, and remains here for archival purposes. There are two updated manuscripts that build on this work. (1) On the half-Cauchy prior for a global scale parameter. (2) Large-scale simultaneous testing with hypergeometric inverted-beta priors.

This paper introduces an approach to estimation in possibly sparse data sets using shrinkage priors based upon the class of hypergeometric-beta distributions. These widely applicable priors turn out to be a four-parameter generalization of the beta family, and are pseudo-conjugate: they cannot themselves be expressed in closed form, but they do yield tractable moments and marginal likelihoods when used as priors for the mean of a normal distribution. These priors are useful in situations where standard priors are inappropriate or ill-behaved. Non-Bayesians will find these priors useful for generating easily computable shrinkage estimators that have excellent risk properties. Bayesians will find them useful for generating computationally tractable priors for a variance parameter. We illustrate the use of these priors on a variety of global and local shrinkage problems, and we prove a theorem that characterizes their risk proprieties when used for estimation of a normal mean under a quadratic loss function.