An alternative formulation of Neyman’s smooth goodness of fit tests under composite alternatives

Abstract

We propose a natural extension of Neyman’s smooth goodness of fit tests under composite hypotheses. The components of our smooth tests are immediate analogues of the corresponding components in the completely specified null case. We show that, when testing for univariate normality, one of our smooth tests is similar to D’Agostino and Pearson’s (1973) statistic; when testing for multivariate normality, our smooth test for skewness is identical to Mardia’s (1970) measure of multivariate skewness; and, when testing for exponentiality, our simplest smooth test is equivalent to Greenwood’s (1946) statistic.