Simultaneous Estimation in Discrete Multivariate Exponential Families

Chou, Jine-Phone

Ann. Statist., Tome 19 (1991) no. 1, p. 314-328 / Harvested from Project Euclid

Résumé

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 $(m = 0)$.

Détail

Publié le : 1991-03-14
Classification:  Discrete multivariate exponential families,  admissibility,  uniformly minimum variance unbiased estimators,  maximum likelihood estimators,  negative multinomial distribution,  quota fulfillment,  62C15,  62F10,  62H99