Abstract
If X2 is the Pearson chi-squared statistic for testing fit, then X2n 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 X2 as a member. The first is a class of directed divergence statistics discussed by Cressie and Read, the second consists of nonnegative definite quadratic forms in the standardized cell frequencies. We investigate the large sample behavior of Tn, 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 Stigler.
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