A Graphical Procedure for Comparing Goodness-Of-Fit Tests

SUMMARY A method for comparing tests of normality using their ‘isotones‘, i.e. contours on the surfaces representing p-values, is proposed. Select a family of distributions labelled by two parameters λ1 and λ2 to represent the null and alternative hypotheses. For each (λ1, λ2) take an ‘ideal sample’ from that distribution, calculate the value of the goodness-of-fit statistic and determine the corresponding P-value under the normality assumption. This generates a surface of the p-values of the ‘values of the test statistic' in the (λ1, λ2) plane. In this paper the construction and interpretation of the isotones, the contours on the above surfaces, for competing tests, is illustrated using the Shapiro–Wilk W, Vasicek's entropy test and Lin and Mudholkar's Zp-test of the composite hypothesis of normality.