{"title":"On testing the equality of latent roots of scatter matrices under ellipticity","authors":"Gaspard Bernard , Thomas Verdebout","doi":"10.1016/j.jmva.2023.105232","DOIUrl":null,"url":null,"abstract":"<div><p>In the present paper, we tackle the problem of testing <span><math><mrow><msub><mrow><mi>H</mi></mrow><mrow><mn>0</mn><mi>q</mi></mrow></msub><mo>:</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>></mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>q</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mo>⋯</mo><mo>=</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>p</mi></mrow></msub></mrow></math></span>, where <span><math><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>p</mi></mrow></msub></mrow></math></span> are the scatter matrix eigenvalues of an elliptical distribution on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>. This is a classical problem in multivariate analysis which is very useful in dimension reduction. We analyse the problem using the Le Cam asymptotic theory of experiments and show that contrary to the testing problems on eigenvalues and eigenvectors of a scatter matrix tackled in Hallin et al. (2010), the non-specification of nuisance parameters has an asymptotic cost for testing <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>0</mn><mi>q</mi></mrow></msub></math></span>. We moreover derive signed-rank tests for the problem that enjoy the property of being asymptotically distribution-free under ellipticity. The van der Waerden rank test uniformly dominates the classical pseudo-Gaussian procedure for the problem. Numerical illustrations show the nice finite-sample properties of our tests.</p></div>","PeriodicalId":16431,"journal":{"name":"Journal of Multivariate Analysis","volume":null,"pages":null},"PeriodicalIF":1.4000,"publicationDate":"2023-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Multivariate Analysis","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0047259X23000787","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 0
Abstract
In the present paper, we tackle the problem of testing , where are the scatter matrix eigenvalues of an elliptical distribution on . This is a classical problem in multivariate analysis which is very useful in dimension reduction. We analyse the problem using the Le Cam asymptotic theory of experiments and show that contrary to the testing problems on eigenvalues and eigenvectors of a scatter matrix tackled in Hallin et al. (2010), the non-specification of nuisance parameters has an asymptotic cost for testing . We moreover derive signed-rank tests for the problem that enjoy the property of being asymptotically distribution-free under ellipticity. The van der Waerden rank test uniformly dominates the classical pseudo-Gaussian procedure for the problem. Numerical illustrations show the nice finite-sample properties of our tests.
在本文中,我们解决了H0q:λq>;λq+1=…=λp,其中λ1,…,λp是Rp上椭圆分布的散射矩阵特征值。这是多元分析中的一个经典问题,在降维中非常有用。我们使用Le Cam渐近实验理论分析了这个问题,并表明与Hallin等人(2010)中解决的关于散射矩阵的特征值和特征向量的测试问题相反,扰动参数的非规范性对于测试H0q具有渐近代价。此外,我们还导出了椭圆度下具有渐近分布自由性质的问题的有符号秩检验。van der Waerden秩检验一致地支配了该问题的经典伪高斯过程。数值示例显示了我们测试的良好有限样本特性。
期刊介绍:
Founded in 1971, the Journal of Multivariate Analysis (JMVA) is the central venue for the publication of new, relevant methodology and particularly innovative applications pertaining to the analysis and interpretation of multidimensional data.
The journal welcomes contributions to all aspects of multivariate data analysis and modeling, including cluster analysis, discriminant analysis, factor analysis, and multidimensional continuous or discrete distribution theory. Topics of current interest include, but are not limited to, inferential aspects of
Copula modeling
Functional data analysis
Graphical modeling
High-dimensional data analysis
Image analysis
Multivariate extreme-value theory
Sparse modeling
Spatial statistics.