Ratio F test for testing simultaneous hypotheses in models with blocked compound symmetric covariance structure

This article deals with testing simultaneous hypotheses about the mean structure and the covariance structure in models with blocked compound symmetric (BCS) covariance structure. Considered models are used for double multivariate data, which means that m -variate vector of observation is measured r...

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Published in:	Statistical papers (Berlin, Germany) Vol. 62; no. 5; pp. 2109 - 2118
Main Authors:	Zmyślony, Roman, Kozioł, Arkadiusz
Format:	Journal Article
Language:	English
Published:	Berlin/Heidelberg Springer Berlin Heidelberg 01.10.2021 Springer Nature B.V
Subjects:	Covariance matrix Economic Theory/Quantitative Economics/Mathematical Methods Economics Finance Hypotheses Insurance Management Mathematical analysis Mathematics and Statistics Mean Multivariate analysis Normality Operations Research/Decision Theory Probability Theory and Stochastic Processes Research Article Statistical tests Statistics Statistics for Business Structure of mean vector Structure of covariance matrices Positive and negative part of estimator Best unbiased estimator Block compound symmetric covariance structure Double multivariate data Testing hypotheses
ISSN:	0932-5026, 1613-9798
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Abstract	This article deals with testing simultaneous hypotheses about the mean structure and the covariance structure in models with blocked compound symmetric (BCS) covariance structure. Considered models are used for double multivariate data, which means that m -variate vector of observation is measured repeatedly over u levels of some factor on each of n individual. Additionally, the assumption of multivariate normality for this type of data is made. We use framework of ratio of positive and negative parts of best unbiased estimators to obtain simultaneous F test. The test statistic is constructed as a ratio of test statistics for testing single hypotheses about the mean vector and the covariance matrix. In simulation study power of obtained test is compared with powers of three other F tests—two for testing single hypotheses and one for testing simultaneous hypotheses, whose test statistic is convex combination of test statistics of these two single F tests. The problem of simultaneous testing of the mean vector and covariance matrix was also consider in paper (Hyodo and Nishiyama, Commun Stat Theory Methods, https://doi.org/10.1080/03610926.2019.1639751 , 2019).
AbstractList	This article deals with testing simultaneous hypotheses about the mean structure and the covariance structure in models with blocked compound symmetric (BCS) covariance structure. Considered models are used for double multivariate data, which means that m -variate vector of observation is measured repeatedly over u levels of some factor on each of n individual. Additionally, the assumption of multivariate normality for this type of data is made. We use framework of ratio of positive and negative parts of best unbiased estimators to obtain simultaneous F test. The test statistic is constructed as a ratio of test statistics for testing single hypotheses about the mean vector and the covariance matrix. In simulation study power of obtained test is compared with powers of three other F tests—two for testing single hypotheses and one for testing simultaneous hypotheses, whose test statistic is convex combination of test statistics of these two single F tests. The problem of simultaneous testing of the mean vector and covariance matrix was also consider in paper (Hyodo and Nishiyama, Commun Stat Theory Methods, 10.1080/03610926.2019.1639751 , 2019). This article deals with testing simultaneous hypotheses about the mean structure and the covariance structure in models with blocked compound symmetric (BCS) covariance structure. Considered models are used for double multivariate data, which means that m -variate vector of observation is measured repeatedly over u levels of some factor on each of n individual. Additionally, the assumption of multivariate normality for this type of data is made. We use framework of ratio of positive and negative parts of best unbiased estimators to obtain simultaneous F test. The test statistic is constructed as a ratio of test statistics for testing single hypotheses about the mean vector and the covariance matrix. In simulation study power of obtained test is compared with powers of three other F tests—two for testing single hypotheses and one for testing simultaneous hypotheses, whose test statistic is convex combination of test statistics of these two single F tests. The problem of simultaneous testing of the mean vector and covariance matrix was also consider in paper (Hyodo and Nishiyama, Commun Stat Theory Methods, https://doi.org/10.1080/03610926.2019.1639751 , 2019). This article deals with testing simultaneous hypotheses about the mean structure and the covariance structure in models with blocked compound symmetric (BCS) covariance structure. Considered models are used for double multivariate data, which means that m-variate vector of observation is measured repeatedly over u levels of some factor on each of n individual. Additionally, the assumption of multivariate normality for this type of data is made. We use framework of ratio of positive and negative parts of best unbiased estimators to obtain simultaneous F test. The test statistic is constructed as a ratio of test statistics for testing single hypotheses about the mean vector and the covariance matrix. In simulation study power of obtained test is compared with powers of three other F tests—two for testing single hypotheses and one for testing simultaneous hypotheses, whose test statistic is convex combination of test statistics of these two single F tests. The problem of simultaneous testing of the mean vector and covariance matrix was also consider in paper (Hyodo and Nishiyama, Commun Stat Theory Methods, https://doi.org/10.1080/03610926.2019.1639751, 2019).
Author	Kozioł, Arkadiusz Zmyślony, Roman
Author_xml	– sequence: 1 givenname: Roman orcidid: 0000-0001-8029-0578 surname: Zmyślony fullname: Zmyślony, Roman email: rzmyslony@op.pl, r.zmyslony@wmie.uz.zgora.pl organization: Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra – sequence: 2 givenname: Arkadiusz orcidid: 0000-0002-5347-4109 surname: Kozioł fullname: Kozioł, Arkadiusz organization: Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra
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Cites_doi	10.13001/1081-3810.3743 10.13001/1081-3810.3748 10.1080/03610926.2019.1639751 10.1080/02331889708802533 10.1016/j.jmva.2015.11.001 10.1080/03610918.2019.1634205 10.1080/03610921003672212 10.1016/j.laa.2018.02.019
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References	Kozioł, Roy, Zmyślony, Leiva, Fonseca (CR4) 2018; 547C Roy, Leiva (CR7) 2011; 40 Roy, Zmyślony, Fonseca, Leiva (CR8) 2016; 144 Hyodo, Nishiyama (CR2) 2019 Johnson, Wichern (CR3) 2007 Michalski, Zmyślony (CR5) 1996; 27 Seely (CR9) 1977; 39 Zmyślony (CR10) 1980; 6 Zmyślony, Kozioł (CR11) 2019 Michalski, Zmyślony (CR6) 1999; 17 Zmyślony, Žežula, Kozioł (CR12) 2018; 33 Fonseca, Zmyślony, Kozioł (CR1) 2018; 33 JF Seely (1182_CR9) 1977; 39 A Roy (1182_CR8) 2016; 144 R Zmyślony (1182_CR11) 2019 A Michalski (1182_CR5) 1996; 27 R Zmyślony (1182_CR12) 2018; 33 M Hyodo (1182_CR2) 2019 A Kozioł (1182_CR4) 2018; 547C A Michalski (1182_CR6) 1999; 17 R Zmyślony (1182_CR10) 1980; 6 RA Johnson (1182_CR3) 2007 M Fonseca (1182_CR1) 2018; 33 A Roy (1182_CR7) 2011; 40
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SubjectTerms	Covariance matrix Economic Theory/Quantitative Economics/Mathematical Methods Economics Finance Hypotheses Insurance Management Mathematical analysis Mathematics and Statistics Mean Multivariate analysis Normality Operations Research/Decision Theory Probability Theory and Stochastic Processes Research Article Statistical tests Statistics Statistics for Business
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Title	Ratio F test for testing simultaneous hypotheses in models with blocked compound symmetric covariance structure
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