The effect of dependence on chi-squared and empiric distribution tests of fit (Q800065)

Let \(X_ 1,...,X_ n\) be a sequence of observations with an unknown distribution function (d.f.) G and \(\{\) \(F(\cdot,\theta)\), \(\theta\in\Omega \}\) be a family of d.f.'s depending on a parameter \(\theta\) taking values in an open subset \(\Omega\) of an m-dimensional Euclidean space. Suppose that a test of fit is employed with null hypothesis about \(X_ 1,...,X_ n\) being independent and identically distributed (iid) r.v.'s and \(G=F(\cdot,\theta_ 0)\) for some \(\theta_ 0\in\Omega.\) As for the estimates \(\theta_ n\) of the unknown parameter \(\theta_ 0\) the following assumption is made: under \(F(\cdot,\theta_ 0) \theta_ n\) have an asymptotic representation \[ n^{1/2}(\theta_ n-\theta_ 0)=n^{-1/2}\sum^{n}_{i=1}g(X_ i,\theta_ 0)+o_ p(1), \] with \(\{X_ i\}\) being iid r.v.'s and being an observation on a stationary stochastic process (SSP). It is proved that if the observations \(\{X_ i\}\) are in fact a SSP satisfying a positive dependence condition, then the limiting level of the test is at least as large in the SSP case as in the iid case. The class of tests for which this result holds includes chi-square, Pearson, empirical d.f., Kolmogorov-Smirnov and Cramér-von Mises tests.