Comprehensive study of tests for normality and symmetry: extending the Spiegelhalter test

Statistical inference in the form of hypothesis tests and confidence intervals often assumes that the distribution(s) being sampled are normal or symmetric. As a result, numerous tests have been proposed in the literature for detecting departures from normality and symmetry. This article initially summarizes the research that has been conducted for developing such tests. The results of an extensive simulation study to compare the power of existing tests for normality is then presented. The effects on power of sample size, significance level, and in particular, alternative distribution shape are investigated. In addition, the power of three modifications to the tests for normality proposed by Spiegelhalter [ Spiegelhalter, D. J., 1977, A test for normality against symmetric alternatives. Biometrika, 64, 415 418; Spiegelhalter, D. J., 1980, An omnibus test for normality for small samples. Biometrika, 67, 493 - 496.], which are tailored to particular shape departures from the normal distribution is evaluated. The test for normality suggested by Spiegelhalter [ Spiegelhalter, D. J., 1980, An omnibus test for normality for small samples. Biometrika, 67, 493 - 496.] is also extended here to serve as a test for symmetry. The results of a simulation study performed to assess the power of this proposed test for symmetry and its comparison with existing tests are summarized and discussed. A key consideration in the assessment of the power of these various tests for symmetry is the ability of the test to maintain the nominal significance level.