High-Dimensional Consistencies of KOO Methods for the Selection of Variables in Multivariate Linear Regression Models with Covariance Structures
In this paper, we consider the high-dimensional consistencies of KOO methods for selecting response variables in multivariate linear regression with covariance structures. Here, the covariance structures are considered as (1) independent covariance structure with the same variance, (2) independent c...
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MDPI AG
2023-01-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/3/671 |
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| author | Yasunori Fujikoshi Tetsuro Sakurai |
| author_facet | Yasunori Fujikoshi Tetsuro Sakurai |
| author_sort | Yasunori Fujikoshi |
| collection | DOAJ |
| description | In this paper, we consider the high-dimensional consistencies of KOO methods for selecting response variables in multivariate linear regression with covariance structures. Here, the covariance structures are considered as (1) independent covariance structure with the same variance, (2) independent covariance structure with different variances, and (3) uniform covariance structure. A sufficient condition for model selection consistency is obtained using a KOO method under a high-dimensional asymptotic framework, such that sample size <i>n</i>, the number <i>p</i> of response variables, and the number <i>k</i> of explanatory variables are large, as in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>/</mo><mi>n</mi><mo>→</mo><msub><mi>c</mi><mn>1</mn></msub><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>/</mo><mi>n</mi><mo>→</mo><msub><mi>c</mi><mn>2</mn></msub><mo>∈</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>c</mi><mn>1</mn></msub><mo>+</mo><msub><mi>c</mi><mn>2</mn></msub><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula>. |
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| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T09:34:14Z |
| publishDate | 2023-01-01 |
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| series | Mathematics |
| spelling | doaj.art-503e0612fbc24bd08b8da965df9ca6bd2023-11-16T17:22:55ZengMDPI AGMathematics2227-73902023-01-0111367110.3390/math11030671High-Dimensional Consistencies of KOO Methods for the Selection of Variables in Multivariate Linear Regression Models with Covariance StructuresYasunori Fujikoshi0Tetsuro Sakurai1Department of Mathematics, Graduate School of Science, Hiroshima University, 1-3-2 Kagamiyama, Hiroshima 739-8626, JapanSchool of General and Management Studies, Suwa University of Science, 5000-1 Toyohira, Chino 391-0292, JapanIn this paper, we consider the high-dimensional consistencies of KOO methods for selecting response variables in multivariate linear regression with covariance structures. Here, the covariance structures are considered as (1) independent covariance structure with the same variance, (2) independent covariance structure with different variances, and (3) uniform covariance structure. A sufficient condition for model selection consistency is obtained using a KOO method under a high-dimensional asymptotic framework, such that sample size <i>n</i>, the number <i>p</i> of response variables, and the number <i>k</i> of explanatory variables are large, as in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>/</mo><mi>n</mi><mo>→</mo><msub><mi>c</mi><mn>1</mn></msub><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>/</mo><mi>n</mi><mo>→</mo><msub><mi>c</mi><mn>2</mn></msub><mo>∈</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>c</mi><mn>1</mn></msub><mo>+</mo><msub><mi>c</mi><mn>2</mn></msub><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula>.https://www.mdpi.com/2227-7390/11/3/671consistency propertycovariance structureshigh-dimensional asymptotic frameworkKOO methodsmultivariate linear regression |
| spellingShingle | Yasunori Fujikoshi Tetsuro Sakurai High-Dimensional Consistencies of KOO Methods for the Selection of Variables in Multivariate Linear Regression Models with Covariance Structures Mathematics consistency property covariance structures high-dimensional asymptotic framework KOO methods multivariate linear regression |
| title | High-Dimensional Consistencies of KOO Methods for the Selection of Variables in Multivariate Linear Regression Models with Covariance Structures |
| title_full | High-Dimensional Consistencies of KOO Methods for the Selection of Variables in Multivariate Linear Regression Models with Covariance Structures |
| title_fullStr | High-Dimensional Consistencies of KOO Methods for the Selection of Variables in Multivariate Linear Regression Models with Covariance Structures |
| title_full_unstemmed | High-Dimensional Consistencies of KOO Methods for the Selection of Variables in Multivariate Linear Regression Models with Covariance Structures |
| title_short | High-Dimensional Consistencies of KOO Methods for the Selection of Variables in Multivariate Linear Regression Models with Covariance Structures |
| title_sort | high dimensional consistencies of koo methods for the selection of variables in multivariate linear regression models with covariance structures |
| topic | consistency property covariance structures high-dimensional asymptotic framework KOO methods multivariate linear regression |
| url | https://www.mdpi.com/2227-7390/11/3/671 |
| work_keys_str_mv | AT yasunorifujikoshi highdimensionalconsistenciesofkoomethodsfortheselectionofvariablesinmultivariatelinearregressionmodelswithcovariancestructures AT tetsurosakurai highdimensionalconsistenciesofkoomethodsfortheselectionofvariablesinmultivariatelinearregressionmodelswithcovariancestructures |