In this article, we study the statistical properties of the goodness‐of‐fit measure m_pp proposed by (Eshima & Tabata 2007, Statistics & Probability Letters 77, 583–593) for generalised linear models. Focusing on the special case of Poisson regression using the canonical log link function, and assuming a random vector X of covariates, we obtain an explicit form for m_pp that enables us to study its properties and construct a new estimator for the measure by utilising information about the shape of the covariate distribution. Simulations show that the newly proposed estimator for m_pp exhibits better performance in terms of mean squared error than the simple unbiased covariance estimator, especially for larger absolute values of the slope coefficients. In contrast, it may be more unstable when the value of the slope coefficient is close to boundary of the domain of the moment generating function for the corresponding covariate. We illustrate the application of m_pp on a data set of counts of complaints against doctors working in an emergency unit in hospital, in particular, showing how our proposed estimator can be efficiently computed across a series of candidate models.

中文翻译：

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关于泊松回归模型的拟合优度度量

在这篇文章中，我们研究的拟合优度衡量的统计特性米_PP由（埃希马和2007畑，统计与概率快报77，583-593）广义线性模型提出。着眼于使用典范对数链接函数的泊松回归的特殊情况，并假设协变量的随机向量X，我们获得m _pp的显式形式，这使我们能够研究其性质并通过利用信息构造该度量的新估计量关于协变量分布的形状。仿真表明，新提出的m _pp估计量与简单的无偏协方差估计器相比，在均方误差方面具有更好的性能，尤其是对于较大的斜率系数绝对值而言。相反，当斜率系数的值接近相应协变量的矩生成函数的域的边界时，它可能会更加不稳定。我们举例说明了m _pp在针对医院急诊部门工作的医生的投诉计数数据集上的应用，特别是说明了如何在一系列候选模型中有效地计算出我们提出的估计量。