On Symmetrized Pearson's Type Test in Autoregression with Outliers: Robust Testing of Normality
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We consider a stationary linear AR(p) model with observations subject to gross errors (outliers). The autoregression parameters are unknown as well as the distribution and moments of innoovations. The distribution of outliers Π is unknown and arbitrary, their intensity is γ n^-1/2 with an unknown γ, n is the sample size. The autoregression parameters are estimated by any estimator which is n^1/2-consistent uniformly in γ≤Γ<∞. 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 symmetrized test of Pearson's chi-square type for the normality of distribution of innovations. We establish qualitative robustness of these tests in terms of uniform equicontinuity of the limiting levels (as functions of γ and Π) with respect to γ in a neighborhood of γ=0.
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