On the symmetrized chi-square tests in autoregression with outliers in data

A linear stationary model $\mathrm{AR}(p)$ with unknown expectation, coefficients, and the distribution function of innovations $G(x)$ is considered. Autoregression observations contain gross errors (outliers, contaminations). The distribution of contaminations $\Pi$ is unknown, their intensity is $\gamma n^{-1/2}$ with unknown $\gamma$, and $n$ is the number of observations. The main problem here (among others) is to test the hypothesis on the normality of innovations $\boldsymbol H_{\Phi}\colon G (x)\in \{\Phi(x/\theta),\,\theta>0\}$, where $\Phi(x)$ is the distribution function of the normal law $\boldsymbol N(0,1)$. In this setting, the previously constructed tests for autoregression with zero expectation do not apply. As an alternative, we propose special symmetrized chi-square type tests. Under the hypothesis and $\gamma=0$, their asymptotic distribution is free. We study the asymptotic power under local alternatives in the form of the mixture $G(x)=A_{n,\Phi}(x):=(1-n^{-1/2})\Phi(x/\theta_0)+n^{-1/2}H(x)$, where $H(x)$ is a distribution function, and $\theta_0^2$ is the unknown variance of the innovations under $\boldsymbol H_{\Phi}$. The asymptotic qualitative robustness of the tests is established in terms of equicontinuity of the family of limit powers (as functions of $\gamma$, $\Pi,$ and $H(x)$) relative to $\gamma$ at the point $\gamma=0$.