We consider a system of dependent Poisson variables, where each variable is the sum of an independent variate and a common variate. It is the common variate that creates the dependence. Within this system, a test of independence may be constructed where the null hypothesis is that the common variate is identically zero. In the present paper, we consider the maximum log likelihood ratio test. For this test, it is well-known that the asymptotic distribution of the test statistic is an equal mixture of zero and a chi square distribution with one degree of freedom. We examine a Bartlett correction of this test, in the hope that we will get better approximation of the nominal size for moderately large sample sizes. This correction is explicitly derived, and its usefulness is explored in a simulation study. For practical purposes, the correction is found to be useful in dimension two, but not in higher dimensions.

翻译：我们考虑一个依附的 Poisson 变量系统, 其中每个变量是独立变数和共同变数的和。 这是创造依赖性的常见变量。 在这个系统中, 独立测试可以构建, 无效的假设是通用变数为零。 在本文中, 我们考虑最大日志概率比测试。 对于这个测试, 众所周知, 测试统计的无症状分布是零的等量混合, 和具有某种自由度的香味平方分布。 我们检查了这个测试的巴特利特修正, 希望我们能更好地接近中等大样本大小的名义大小。 这一修正是明确的推算, 并在模拟研究中探索其有用性。 为了实际目的, 测试统计的无症状分布在二维中是有用的, 但不在更高的维度上。