title: 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.
authors.affiliation: Department of Psychology, The Ohio State University, 1835 Neil Avenue, Columbus, OH, 43210-1222, USA, browne.4@osu.edu.
authors.forename: Michael W
authors.initials: MW
authors.lastname: Browne
authors.forename: Alexander
authors.initials: A
authors.lastname: Shapiro
mesh_terms: D000465:Algorithms; D015233:Models, Statistical