Comment on the Asymptotics of a Distribution-Free Goodness of Fit Test Statistic

Abstract

In a recent article Jennrich and Satorra (Psychometrika 78: 545–552, 2013) showed that a proof by Browne (British Journal of Mathematical and Statistical Psychology 37: 62–83, 1984) of the asymptotic distribution of a goodness of fit test statistic is incomplete because it fails to prove that the orthogonal component function employed is continuous. Jennrich and Satorra (Psychometrika 78: 545–552, 2013) showed how Browne’s proof can be completed satisfactorily but this required the development of an extensive and mathematically sophisticated framework for continuous orthogonal component functions. This short note provides a simple proof of the asymptotic distribution of Browne’s (British Journal of Mathematical and Statistical Psychology 37: 62–83, 1984) test statistic by using an equivalent form of the statistic that does not involve orthogonal component functions and consequently avoids all complicating issues associated with them.