On Symmetrized Pearson's Type Test for Normality of Autoregression: Power under Local Alternatives

We consider a stationary linear AR($p$) model with observations subject to gross errors (outliers). The autoregression parameters as well as the distribution function (d.f.) $G$ of innovations are unknown. The distribution of outliers $\mathrm{\Pi }$ is unknown and arbitrary, their intensity is $\gamma n^{-1/2}$withanunknown$\gamma$,$n$ is the sample size. We test the hypothesis for normality of innovations $${\mathbf{H}}_{\mathrm{\Phi }}\text{\hspace{0.17em}⁣}:G\in \{\mathrm{\Phi }(x\mathrm{/}\theta ),\theta >0\},$$ $\mathrm{\Phi }(x)$ is the d.f. $\mathbf{N}(0,1)$. Our test is the special symmetrized Pearson’s type test. We find the power of this test under local alternatives $${\mathbf{H}}_{1n}(\rho )\text{\hspace{0.17em}⁣}:G(x)=A_n(x):=(1-\rho n^{-1\mathrm{/}2})\mathrm{\Phi }(x\mathrm{/}{\theta }_0)+\rho n^{-1\mathrm{/}2}H(x),$$ $\rho\geq 0,,\theta_0$istheunknown(under$$\mathbf{H}$_\Phi$) variance of innovations. 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 (r.e.d.f.), which is a counterpart of the (inaccessible) e.d.f. of the autoregression innovations. After this we construct the symmetrized variant r.e.d.f. Our test statistic is the functional from symmetrized r.e.d.f. We obtain a stochastic expansion of this symmetrized r.e.d.f. under ${\mathbf{H}}_{1n}(\rho )$ , which enables us to investigate our test. We establish qualitative robustness of this test in terms of uniform equicontinuity of the limiting power (as functions of $\gamma ,\rho$ and $\mathrm{\Pi }$) with respect to $\gamma$ in a neighborhood of $\gamma =0$.