Do Components of Smooth Tests of Fit Have Diagnostic Properties?

Smooth goodness of fit tests were introduced by Neyman (1937). They can be regarded as a compromise between globally consistency ("omnibus") tests of fit and procedures having high tests like, e.g., skewness and kurtosis measures in the context of testing for normality, have special diagnostic properties in case of rejection of a hypothesis in the sense that they constitute direct measures of the kind of departure from the hypothesis. Recent years, however, have witnessed a complete change of attitude towards the diagnostic capabilities of skewness and kurtosis measures in connection with normality testing. In this paper, we argue that any component of any smooth test of fit is strictly non-diagnostic when used conventionally. However, a proper rescaling of components does indeed achieve the desired "directed diagnosis."