椭圆型VARMA模型充分性的秩最优检验
Rank-based optimal tests of the adequacy of an elliptic VARMA model
摘要：
我们推导出最佳的基于秩的测试的充分性的向量自回归移动平均（VARMA）模型与椭圆形轮廓的创新密度。这些测试是基于伪马哈拉诺比斯距离的秩和从Tyler的[Ann. Statist. 15（1987）234-251]散布矩阵;它们概括了由Hallin和Puri [J. Multivariate Anal. 39（1991）1-29]。两种类型的最优性属性被认为是，无论是在本地和渐近意义上，一拉乐凸轮：（a）（固定分数程序）局部渐近minimaxity在选定的径向密度，和（b）（估计分数程序）局部渐近minimaxity均匀超过一类F的径向密度。与经典的检验方法不同，基于互协方差阵，这些检验方法在任意椭圆对称新息密度下都是有效的，包括具有无穷大方差和重尾的新息密度。我们表明，相对于传统（高斯）方法，我们的固定分数程序的ARE与Hallin和Paindaveine [Bernoulli 8（2002 b）787-815]中提出的随机性检验相同。因此，在这里得到的经典的Dauff-Savage和Hodges-Lehmann结果的多变量序列扩展也成立;特别是，我们检验的van der Waerden版本一致地比基于互协方差的检验更强大。至于我们的估计分数的程序，他们是完全自适应的，因此，一致最优的创新密度满足所需的技术假设。
英文摘要：
We are deriving optimal rank-based tests for the adequacy of a vector autoregressive-moving average (VARMA) model with elliptically contoured innovation density. These tests are based on the ranks of pseudo-Mahalanobis distances and on normed residuals computed from Tyler's [Ann. Statist. 15 (1987) 234-251] scatter matrix; they generalize the univariate signed rank procedures proposed by Hallin and Puri [J. Multivariate Anal. 39 (1991) 1-29]. Two types of optimality properties are considered, both in the local and asymptotic sense, a la Le Cam: (a) (fixed-score procedures) local asymptotic minimaxity at selected radial densities, and (b) (estimated-score procedures) local asymptotic minimaxity uniform over a class F of radial densities. Contrary to their classical counterparts, based on cross-covariance matrices, these tests remain valid under arbitrary elliptically symmetric innovation densities, including those with infinite variance and heavy-tails. We show that the AREs of our fixed-score procedures, with respect to traditional (Gaussian) methods, are the same as for the tests of randomness proposed in Hallin and Paindaveine [Bernoulli 8 (2002b) 787-815]. The multivariate serial extensions of the classical Chernoff-Savage and Hodges-Lehmann results obtained there thus also hold here; in particular, the van der Waerden versions of our tests are uniformly more powerful than those based on cross-covariances. As for our estimated-score procedures, they are fully adaptive, hence, uniformly optimal over the class of innovation densities satisfying the required technical assumptions.
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