Assessing the Accuracy of Normal Approximations
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正态近似精度的评估
- 研究问题
- 如何以可操作的方式评估正态近似的准确性，尤其是近似似然区域的几何偏差与预测区域的实际概率含量？
- 方法与思路
- 作者分别比较精确似然与正态近似的等高线，以及精确密度与近似密度的选定线积分，并与既有评估方法对照。
- 材料陈述的贡献
- 摘要提出两种诊断方法，一种给出似然区域的内外包含关系，另一种在牙齿测量示例中量化预测区域概率偏差。
- 阅读关注点
- 应核对等高线和积分路径的选择、精确密度的可获得性，以及示例中概率含量偏差区间的计算方式与解释。
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原文摘要
The widespread applicability and use of normal approximations creates a need for methods for assessing their accuracy in an operational fashion. In this article, two new methods are proposed and exemplified. These new methods are based on a comparison of the level curves of an exact likelihood with the level curves of the usual normal approximation to it and on a comparison of selected line integrals of an exact density and a normal approximation to it. The operational usefulness of these methods is compared with the operational usefulness of two well-known existing approaches to assessing the accuracy of approximations, namely convergence rates and the curvature methods of Bates and Watts (1980). As an example, the second method is applied to the Lee—Geisser (1972, 1975) approximation to the predictive distribution of dental measurements on girls at the ages of 8, 10, 12, and 14 years. In this application, the method shows that the 50% central (predictive) probability region has actual (predictive) probability content between 3% and 6% higher, that the approximate 80% central region has actual probability content between 1.7% lower and 4.4% higher, and that the approximate 95% central region has actual probability content between 3.7% lower and 5.6% higher. The first method gives the smallest approximate likelihood region that contains the exact likelihood region and the largest approximate likelihood region contained in the exact region.