Assessing the accuracy of normal approximations

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.