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

Abstract

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 $\sum^n_0\lbrack\delta_i(x) - \theta_i\rbrack^2/\theta_i$, where $\theta_0, \cdots, \theta_k(\sum\theta_i = 1)$ are the parameters in the multinomial distribution, and the estimators are restricted to $\sum^k_0\delta_i(x) = 1$. 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.