## The Annals of Statistics

### Simultaneous Estimation in Discrete Multivariate Exponential Families

Jine-Phone Chou

#### Abstract

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

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

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176347984

Digital Object Identifier
doi:10.1214/aos/1176347984

Mathematical Reviews number (MathSciNet)
MR1091853

Zentralblatt MATH identifier
0728.62033

JSTOR