In this paper, we consider the problem of estimating the density function of a Chi-squared variable on the basis of observations of another Chi-squared variable and a normal variable under the Kullback–Leibler divergence. We assume that these variables have a common unknown scale parameter and that the mean of the normal variable is also unknown. We compare the risk functions of two Bayesian predictive densities: one with respect to a hierarchical shrinkage prior and the other based on a noninformative prior. The hierarchical Bayesian predictive density depends on the normal variable while the Bayesian predictive density based on the noninformative prior does not. Sufficient conditions for the former to dominate the latter are obtained. These predictive densities are compared by simulation.

在本文中，我们考虑在 Kullback-Leibler 散度下根据另一个卡方变量和一个正态变量的观察来估计卡方变量的密度函数的问题。我们假设这些变量有一个共同的未知尺度参数，并且正常变量的均值也是未知的。我们比较了两种贝叶斯预测密度的风险函数：一种是关于分层收缩先验，另一种是基于非信息先验。分层贝叶斯预测密度取决于正态变量，而基于非信息先验的贝叶斯预测密度则不。获得了前者支配后者的充分条件。这些预测密度通过模拟进行比较。