Abstract:
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 $\gamma=0$.
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