论文标题
在对称的皮尔逊（Pearson）的类型测试中，对自动估计的正态性：本地替代方案下的功率
On Symmetrized Pearson's Type Test for Normality of Autoregression: Power under Local Alternatives
论文作者
论文摘要
我们考虑使用具有总数错误（离群值）的观测值的固定线性AR（$ p $）型号。自动进展参数以及分布功能（D.F.）$ g $的创新尚不清楚。异常值的分布$π$是未知的，任意的，它们的强度为$γn^{ - 1/2} $，未知$γ$，$ n $是样本量。我们测试了创新正常的假设$ \ mathbf {h}_φ\ colon g \ in \ {φ（x/θ），\，θ> 0 \}，$ $ $ $（x）$是d.f. $ \ mathbf {n}（0,1）$。我们的测试是特殊对称的皮尔逊类型测试。我们在本地替代方案下找到该测试的功能$ \ mathbf {h} _ {1n}（ρ）（ρ）\ colon g（x）= a_n（x）：=（1-ρn^{ - 1/2}）φ（x/θ_0） （在$ \ mathbf {h}_φ$下）创新的差异。首先，我们估计自动估计参数，然后使用估计自动化的残差，我们构建了一种经验分布函数（R.E.D.F.），这是（不可访问）E.D.F.的对应物。自动估计创新。之后，我们构建了对称的变体R.E.D.F.我们的测试统计量是对称R.E.D.F.的功能。我们获得了这种对称的R.E.D.F.的随机扩展。在$ \ mathbf {h} _ {1n}（ρ）$下，这使我们能够研究测试。我们在$γ= 0 $的社区中，就限制能力的均匀等准（作为$γ，ρ$和$π$的功能）建立了该测试的定性鲁棒性。
We consider a stationary linear AR($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 $γ=0$.
