Minimax estimation of a bounded parameter of a discrete distribution

For a vast class of discrete model families with cdf's FθFθ, and for estimating θθ under squared error loss under a constraint of the type θ∈[0,m]θ∈[0,m], we present a general and unified development concerning the minimaxity of a boundary supported prior Bayes estimator. While the sufficient conditions obtained are of the expected form m⩽m(F)m⩽m(F), the approach presented leads, in many instances, to both necessary and sufficient conditions, and/or explicit values for m(F)m(F). Finally, the scope of the results is illustrated with various examples that, not only include several common distributions (e.g., Poisson, Binomial, Negative Binomial), but many others as well.