Unified Large-Sample Theory of General Chi-Squared Statistics for Tests of Fit

Abstract

We present a unified large-sample theory of general chi-squared tests of fit under composite null hypotheses and Pitman alternatives. The sta- tistics are quadratic forms in the standardized cell frequencies, and we allow random cells, k-variate observations from not necessarily continuous distributions, and quite general estimates of unknown parameters. Gener- alizations of published results on a number of specific chi-squared tests follow.