Goodness-of-Fit Tests (Tests for Normality)
Many statistical procedures are, strictly speaking, only appropriate when the corresponding parametric assumptions that are made regarding data distributions are sufficiently accurate. Among these tests, the Shapiro-Wilk test is highly efficient. For more details regarding goodness-of-fit tests, the authors refer the reader to Huber-Carol et al., Vexler and Gurevich, Claeskens and Hjort and Vexler. In order to test a complete sample for normality, Shapiro and Wilk introduced a new statistical procedure in which the test statistic was obtained by dividing the square of an appropriate linear combination of the sample order statistics by the usual symmetric estimate of variance. Stephens provided a practical guide to goodness-of-fit tests using statistics based on the empirical distribution function. The authors investigated powers of the tests under certain alternatives and revealed that some of them are some of the most powerful goodness-of-fit tests available for the related hypotheses.