关于自回归正态性的对称化Pearson类型检验：局部选择下的功效
On Symmetrized Pearson's Type Test for Normality of Autoregression: Power under Local Alternatives
摘要：
本文考虑了观测值具有粗差（离群值）的平稳线性AR（$p$）模型.自回归参数以及分布函数（d.f.）$G$的创新是未知的。离群值$Π$的分布未知且任意，其强度为$γn^{-1/2}$，$γ$未知，$n$为样本量。我们检验了创新$$\mathbf{H}_Φ\colon G \in \{Φ(x/θ),\,θ>0\},$$ $Φ(x)$的正态性假设。$\mathbf{N}(0,1)$。我们的检验是特殊的对称化皮尔逊类型检验。我们发现，在本地替代方案下，该测试的威力$$\mathbf{H}_{1n}(ρ)\colon G(x)=A_n(x):=(1-ρn^{-1/2})Φ(x/θ_0)+ρn^{-1/2}H(x), $$ $ρ\geq 0,\,θ_0$是未知的（在$\mathbf{H}_Φ$下）创新方差。首先我们估计自回归参数，然后利用估计的自回归的残差构造一种经验分布函数（r.e.d.f.），其是（不可接近的）E.D.F.的对应物。自我回归的创新。在此之后，我们构造对称化变体r.e.d.f.。我们的检验统计量是来自对称r.e.d.f.的泛函。我们得到一个随机扩展的对称化r.e.d. f。在$\mathbf{H}_{1n}(ρ)$下，这使我们能够研究我们的测试。我们建立了定性的鲁棒性，这个测试的限制功率（作为$γ,ρ$和$Π$的函数）的均匀等度连续性方面的$γ$在附近的$γ=0$。
英文摘要：
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 $Π$ is unknown and arbitrary, their intensity is $γn^{-1/2}$ with an unknown $γ$, $n$ is the sample size. We test the hypothesis for normality of innovations $$\mathbf{H}_Φ\colon G \in \{Φ(x/θ),\,θ>0\},$$ $Φ(x)$ 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 $$\mathbf{H}_{1n}(ρ)\colon G(x)=A_n(x):=(1-ρn^{-1/2})Φ(x/θ_0)+ρn^{-1/2}H(x), $$ $ρ\geq 0,\,θ_0$ is the unknown (under $\mathbf{H}_Φ$) 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 $\mathbf{H}_{1n}(ρ)$ , 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 $γ,ρ$ and $Π$) with respect to $γ$ in a neighborhood of $γ=0$.
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