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.
@article{1176344613,
author = {Olkin, Ingram and Sobel, Milton},
title = {Admissible and Minimax Estimation for the Multinomial Distribution and for K Independent Binomial Distributions},
journal = {Ann. Statist.},
volume = {7},
number = {1},
year = {1979},
pages = { 284-290},
language = {en},
url = {http://dml.mathdoc.fr/item/1176344613}
}
Olkin, Ingram; Sobel, Milton. Admissible and Minimax Estimation for the Multinomial Distribution and for K Independent Binomial Distributions. Ann. Statist., Tome 7 (1979) no. 1, pp. 284-290. http://gdmltest.u-ga.fr/item/1176344613/