The exact and near-exact distributions of the main likelihood ratio test statistics used in the complex multivariate normal setting

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• 作者：Carlos A. Coelho ; Barry C. Arnold ; Filipe J. Marques
• 关键词：Covariance matrix ; Equality of covariance matrices ; Equality of mean vectors ; Fourier transforms ; Generalized integer gamma (GIG) distribution ; Generalized near ; integer gamma (GNIG) distribution ; Independence ; Mixtures ; Expected value matrix ; Sphericity ; Statistical distributions (distribution functions) ; 62H05 ; 62H10 ; 62E15 ; 62H15 ; 62E20
• 刊名：TEST
• 出版年：2015
• 出版时间：June 2015
• 年：2015
• 卷：24
• 期：2
• 页码：386-416
• 全文大小：679 KB
• 参考文献：Anderson TW (2003) An introduction to multivariate statistical analysis, 3rd edn. Wiley, New YorkMATH
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• 作者单位：Carlos A. Coelho (1)
  Barry C. Arnold (2)
  Filipe J. Marques (1)
  
  1. Departamento de Matem谩tica, Faculdade de Ci锚ncias e Tecnologia, Centro de Matem谩tica e Aplica莽玫es (CMA-FCT/UNL), Universidade Nova de Lisboa, Caparica, Portugal
  2. Statistics Department, University of California, Riverside, USA
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Statistics
  Statistics
  Statistical Theory and Methods
  Statistics for Business, Economics, Mathematical Finance and Insurance
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1863-8260

文摘

In this paper the authors show how it is possible to establish a common structure for the exact distribution of the main likelihood ratio test (LRT) statistics used in the complex multivariate normal setting. In contrast to what happens when dealing with real random variables, for complex random variables it is shown that it is possible to obtain closed-form expressions for the exact distributions of the LRT statistics to test independence, equality of mean vectors and the equality of an expected value matrix to a given matrix. For the LRT statistics to test sphericity and the equality of covariance matrices, cases where the exact distribution has a non-manageable expression, easy to implement and very accurate near-exact distributions are developed. Numerical studies show how these near-exact distributions outperform by far any other available approximations. As an example of application of the results obtained, the authors develop a near-exact approximation for the distribution of the LRT statistic to test the equality of several complex normal distributions.