Admissible and Minimax Estimation for the Multinomial Distribution and for K Independent Binomial Distributions

Admissible and minimax estimation is discussed for estimating the parameters in the (a) multinomial distribution and in (b) k independent binomial distributions. In (a) the loss function is ∑ⁿ₀δᵢ(x) - θᵢ²/θᵢ, where θ₀, ⋯, θₖ(∑θᵢ = 1) are the parameters in the multinomial distribution, and the estimators are restricted to ∑ᵏ₀δᵢ(x) = 1. In (b) the loss functions considered are the weighted sum of quadratic losses. The method of proof is based on a multivariate analog of the Cramer-Rao inequality, and uses the divergence theorem in a novel way.