The Effect of Dependence on Chi-Squared and Empiric Distribution Tests of Fit

Abstract

Suppose that a test of fit to a parametric family of distributions is employed, with critical points determined from the limiting null distribution of the test statistic for IID observations. It is shown that if the observations are in fact a stationary process satisfying a positive dependence condition, the test will reject a true null hypothesis too often. This result is established for a broad class of chi squared and empiric df tests, including the Pearson, Kolmogorov-Smirnov and Cramer-von Mises tests with general estimators of unknown parameters. Furthermore, the method of proof is sufficiently general to apply also to other classes of tests. Confounding of positive dependence with lack of fit is therefore a general phenomenon in the use of omnibus tests of fit.