Rank-based optimal tests of the adequacy of an elliptic VARMA model
椭圆分布 VARMA 模型适合性的秩最优检验AI 解读 ↓
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椭圆分布 VARMA 模型适合性的秩最优检验
- 研究问题
- 对创新服从椭圆对称分布的向量自回归移动平均模型，如何检验模型适合性，并在重尾甚至无限方差情形下保持检验有效？
- 方法与思路
- 作者利用伪马氏距离的秩和基于 Tyler 散布矩阵的标准化残差构造检验，分别研究固定得分与估计得分程序的局部渐近性质。
- 材料陈述的贡献
- 摘要称，检验在椭圆对称创新下有效，并建立局部渐近极小极大性质与相对效率结果；估计得分程序在满足技术条件的密度类中自适应。
- 阅读关注点
- 阅读全文时应核对残差估计、散布矩阵及得分构造的条件，区分局部渐近最优性、相对效率比较和有限样本功效结论。
- 阅读边界
- 仅据版本未注明的 OpenAlex 索引摘要；技术条件未列全，效率与功效陈述应结合局部渐近语境，不能推出所有样本及非椭圆分布下均占优。
原文摘要
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.