The asymptotic minimax risk for the estimation of constrained binomial and multinomial probabilities

In this note we present a direct and simple approach to obtain bounds on the asymptotic minimax risk for the estimation of restrained binominal and multinominal proportions. Quadratic, normalized quadratic and entropy loss are considered and it is demonstrated that in all cases linear estimators are asymptotically minimax optimal. For the quadratic loss function the asymptotic minimax rsik does not change unless a neighborhood of the point 1/2 is excluded by the restrictions on the parameter space. For the two other loss functions the asymptotic minimax risks remain unchanged if additional knowledge about the location of the unknown probability of success is imposed. The results are also extended to the problem of minimax estimation of a vector of contrained multinominal propabilities.