Measures of lack of fit from tests of chi-squared type
If \(X^ 2\) is the Pearson chi-squared statistic for testing fit, then \(X^ 2/n\) has long been considered an associated measure of the degree of lack of fit. Here we consider two classes of statistics of chi-squared type, each having \(X^ 2\) as a member. The first is a class of directed divergence statistics discussed by \textit{N. Cressie} and \textit{T. R. C. Read} [J. R. Stat. Soc., Ser. B 46, 440-464 (1984; Zbl 0571.62017)], the second consists of nonnegative definite quadratic forms in the standardized cell frequencies. We investigate the large sample behavior of T/n, where T is any of these statistics. A number of auxiliary results on the Cressie-Read statistics are also obtained. The measures are illustrated by application to data from classical physics compiled by \textit{S. M. Stigler} [Ann. Stat. 5, 1055-1098 (1977; Zbl 0374.62050)].
scientific article; zbMATH DE number 3965198
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