On the Power of Symmetrized Pearson's Type Test under Local Alternatives in Autoregression with Outliers
英文摘要：
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 function $G$ of innovations. The distribution of outliers $Π$ is unknown and arbitrary, their intensity is $γn^{-1/2}$ with an unknown $γ$, $n$ is the sample size. We test the hypothesis $H_0\colon G=G_0$ with simmetric $G_0$. We find the power of the test under local alternatives $H_{1n}(ρ)\colon G=(1-ρn^{-1/2})G_0+ρn^{-1/2}H$. Our test is the special symmetrized Pearson's type test. Namely, first of all we estimate the autoregression parameters and then 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. and its symmetrized variant under $H_{1n}(ρ)$ , which enables us to construct and investigate our symmetrized test of Pearson's type for $H_0$. We establish qualitative robustness of this test in terms of uniform equicontinuity of the limiting power (as functions of $γ,ρ$ and $Π$) with respect to $γ$ in a neighborhood of $γ=0$.
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