Abstract The negative multinomial distribution is a multivariate generalization of the negative binomial distribution. In this paper, we consider the problem of estimating an unknown matrix of probabilities on the basis of observations of negative multinomial variables under the standardized squared error loss. First, a general sufficient condition for a shrinkage estimator to dominate the UMVU estimator is derived and an empirical Bayes estimator satisfying the condition is constructed. Next, a hierarchical shrinkage prior is introduced, an associated Bayes estimator is shown to dominate the UMVU estimator under some conditions, and some remarks about posterior computation are presented. Finally, shrinkage estimators and the UMVU estimator are compared by simulation.

中文翻译：

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负多项式参数向量的贝叶斯收缩估计

摘要 负多项式分布是负二项式分布的多元推广。在本文中，我们考虑在标准化平方误差损失下基于负多项式变量的观察来估计未知概率矩阵的问题。首先，推导出收缩估计量支配 UMVU 估计量的一般充分条件，并构造满足该条件的经验贝叶斯估计量。接下来，引入了分层收缩先验，相关联的贝叶斯估计器在某些条件下会主导 UMVU 估计器，并提出一些关于后验计算的评论。最后，通过模拟比较收缩估计量和 UMVU 估计量。