The Annals of Statistics

Simultaneous Estimation in Discrete Multivariate Exponential Families

Jine-Phone Chou

Full-text: Open access


Let $X$ have a discrete density of the form $f(x) = t(x)\xi(\theta)\theta^{x_1}_1 \cdots \theta^{x_p}_p$, where $t(x)$ is nonzero on some infinite subset of $Z^p$. Consider simultaneous estimation of the $\theta_i$ under the loss $\mathscr{L}_m(\theta, \delta) = \sum^p_{i = 1} \theta^{-m}_i(\theta_i - \delta_i)^2, m \geq 0$. For integers $m \geq 1$, estimators are found which improve on the maximum likelihood estimator or uniformly minimum variance unbiased estimator. The improved estimators are distinguished by the property that they do not depend on $m$ for "large values" of the observed vector. On the other hand, we prove admissibility of a class of estimators, including the MLE and UMVUE, for some discrete densities of the indicated form under squared error loss $(m = 0)$.

Article information

Ann. Statist., Volume 19, Number 1 (1991), 314-328.

First available in Project Euclid: 12 April 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62C15: Admissibility
Secondary: 62F10: Point estimation 62H99: None of the above, but in this section

Discrete multivariate exponential families admissibility uniformly minimum variance unbiased estimators maximum likelihood estimators negative multinomial distribution quota fulfillment


Chou, Jine-Phone. Simultaneous Estimation in Discrete Multivariate Exponential Families. Ann. Statist. 19 (1991), no. 1, 314--328. doi:10.1214/aos/1176347984.

Export citation