Ghosh Malay, Ghosh Tamal, Hirose Masayo Y
Department of Statistics, University of Florida, 223 Griffin-Floyd Hall, Gainesville, FL USA.
Institute of Mathematics for Industry, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka Japan.
Sankhya B (2008). 2022;84(2):449-471. doi: 10.1007/s13571-021-00269-8. Epub 2021 Oct 11.
The paper intends to serve two objectives. First, it revisits the celebrated Fay-Herriot model, but with homoscedastic known error variance. The motivation comes from an analysis of count data, in the present case, COVID-19 fatality for all counties in Florida. The Poisson model seems appropriate here, as is typical for rare events. An empirical Bayes (EB) approach is taken for estimation. However, unlike the conventional conjugate gamma or the log-normal prior for the Poisson mean, here we make a square root transformation of the original Poisson data, along with square root transformation of the corresponding mean. Proper back transformation is used to infer about the original Poisson means. The square root transformation makes the normal approximation of the transformed data more justifiable with added homoscedasticity. We obtain exact analytical formulas for the bias and mean squared error of the proposed EB estimators. In addition to illustrating our method with the COVID-19 example, we also evaluate performance of our procedure with simulated data as well.
本文旨在实现两个目标。首先，它重新审视了著名的费伊 - 赫里奥特模型，但误差方差已知且为同方差。其动机源于对计数数据的分析，在当前情况下，是对佛罗里达州所有县的新冠死亡人数的分析。泊松模型在此似乎适用，这对于罕见事件来说很典型。估计采用经验贝叶斯（EB）方法。然而，与传统的共轭伽马分布或泊松均值的对数正态先验不同，这里我们对原始泊松数据进行平方根变换，同时对相应均值也进行平方根变换。使用适当的逆变换来推断原始泊松均值。平方根变换使得变换后数据的正态近似在增加同方差的情况下更合理。我们得到了所提出的EB估计量的偏差和均方误差的精确解析公式。除了用新冠疫情的例子说明我们的方法外，我们还使用模拟数据评估了我们方法的性能。
