We consider a stationary linear AR($p$) model with unknown mean. The autoregression parameters as well as the distribution function (d.f.) $G$ of innovations are unknown. The observations contain gross errors (outliers). The distribution of outliers is unknown and arbitrary, their intensity is $\gamma n^{-1/2}$ with an unknown $\gamma$, $n$ is the sample size. The assential problem in such situation is to test the normality of innovations. Normality, as is known, ensures the optimality properties of widely used least squares procedures. To construct and study a Pearson chi-square type test for normality we estimate the unknown mean and the autoregression parameters. Then, using the estimates, we find the residuals in the autoregression. Based on them, 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. Our Pearson's satatistic is the functional from r.e.d.f. Its asymptotic distributions under the hypothesis and the local alternatives are determined by the asymptotic behavior of r.e.d.f. %Therefore, the study of the asymptotic properties of r.e.d.f. is a natural and meaningful task. In the present work, we find and substantiate in details the stochastic expansions of the r.e.d.f. in two situations. In the first one d.f. $ G (x) $ of innovations does not depend on $ n $. We need this result to investigate test statistic under the hypothesis. In the second situation $ G (x) $ depends on $ n $ and has the form of a mixture $ G (x) = A_n (x) = (1-n ^ {- 1/2}) G_0 (x) + n ^ { -1/2} H (x). $ We need this result to study the power of test under the local alternatives.
翻译:我们考虑的是固定线性AR($ p$) 模式,其平均值未知。 常态性确保广泛使用的最小方程式的最佳性能( d.f.) 。 用于构建和研究用于正常性的Pearson chi- quare 测试,我们估算出未知的平均值和回溯参数。 外部值分布为未知的任意性, 其强度为$\gamma n ⁇ 1/ /2}$, 样本大小为$gamma, 样本大小为$. 。 此情况下存在的问题在于测试创新的正常性能。 正常性能确保广泛使用的最小方程式的优化性能( d. f.) 。 我们的Searson CH/2 类型测试中,我们估算出未知的平均值和回溯值参数值参数值参数值值值值值参数。 然后,我们根据这些估算,我们构建一种经验性分布功能(r. de. d. f.) (we. de. f.),这是(in. f.) 对(不易获取的) 自动变现变数的变变数的变数研究结果。