A test that the population being sampled has a specified distribution. It was introduced by Shapiro and Wilk in 1965. The test compares the ordered sample values with the corresponding order statistics from the specified distribution. The test is most commonly used to test for a normal distribution, in which case the test statistic, W, is given by where x(j) is the jth largest of n observations, s2 is the unbiased estimate of the population variance, and wj is a function of the means and variances and covariances of the order statistics. The Shapiro–Francia test uses a simpler substitute for wj.
In the case where the hypothesized distribution is exponential the statistic takes the simple form where x̄ is the sample mean. In this case, W is closely related to the inverse of the Darling test statistic, K, given by which, for n > 500, has an approximate normal distribution with mean n(n-1)n+1 and variance
Subjects: Probability and Statistics.