On the empirical distribution function of residuals in autoregression with outliers and Pearson's chi-square type tests

We consider a stationary linear AR($p$) model with observations subject to gross errors (outliers). The distribution of outliers $\Pi$ is unknown and arbitrary, their intensity is $\gamma n^{-1/2}$ with an unknown $\gamma$, $n$ is the sample size. The autoregression parameters are unknown, they are estimated by any estimator which is $n^{1/2}$-consistent uniformly in $\gamma\leq \Gamma<\infty$. Using the residuals from the estimated autoregression, we construct a kind of empirical distribution function (e.d.f.), which is a counterpart of the (inaccessible) e.d.f. of the autoregression innovations. We obtain a stochastic expansion of this e.d.f., which enables us to construct the tests of Pearson's chi-square type for testing hypotheses about the distribution of innovations. We establish qualitative robustness of these tests in terms of uniform equicontinuity of the limiting levels (as functions of $\gamma$ and $\Pi$) with respect to $\gamma$ in a neighborhood of $\gamma=0$.