On the Power of Symmetrized Pearson's Type Test under Local Alternatives in Autoregression with Outliers

Abstract

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 $\Pi$ is unknown and arbitrary, their intensity is $\gamma n^{-1/2}$ with an unknown $\gamma$, $n$ 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 $H_{1n}(\rho)\colon G=(1-\rho n^{-1/2})G_0+\rho n^{-1/2}H$. 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 $H_{1n}(\rho)$ , 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 $\gamma,\rho$ and $\Pi$) with respect to $\gamma$ in a neighborhood of $\gamma=0$.