Approximations for the Null Distribution of the W Statistic

The present note deals with the development of an approximation to the null distribution of W, a statistic suggested for testing normality by Shapiro and Wilk (1965). This approximation is based on fitting and smoothing empirical sampling results. The KT statistic is defined as the ratio of the square of a linear combination of the ordered sample to the usual sum of squares of deviations about the mean. For a sample from a normal distribution, the ratio is statistically independent of its denominator and so the moments of the ratio are equal to the ratio of the moments. This enables the simple computation of the 3 and 1 moments of W. Higher moments of W are not available and hence the Cornish-Fisher expansion could not be used as an approximation method. Good approximation was attained, after preliminary investigations, using Johnson’s (1949) S, distribution, which is defined as that of the random variable u, where 2 = y + 6 In u - e X+e-u is distributed as standard normal, and where E and X + e are the minimum and maximum attainable values of U, respectively. For IV, X + E = 1 for all n, while e is a known function of sample size (see Shapiro and Wilk (1965)). Values of e are given in Table 1 of the present note for n = 3(1)50. To obtain suitable values of y and 6, in the case when the bounds are known, Johnson (1949) recommends matching chosen percentage points. An alternative method might be to match two moments, but would require heavy computation. Also, while the matching of two moments could be done solely using theoretical values, in principle, it would not necessarily provide for weighting the fit so as to be good in the tails of the distribution-which is what is wanted. The procedure actually used here was to do, for each 72, the simple least squares regression of the empirical sampling value of