Consistent Estimators of the Population Covariance Matrix and Its Reparameterizations
For the high-dimensional covariance estimation problem, when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo movablelimits="true" form="prefix">lim</mo><m...
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2025-01-01
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| author | Chia-Hsuan Tsai Ming-Tien Tsai |
| author_facet | Chia-Hsuan Tsai Ming-Tien Tsai |
| author_sort | Chia-Hsuan Tsai |
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| description | For the high-dimensional covariance estimation problem, when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo movablelimits="true" form="prefix">lim</mo><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></msub><mi>p</mi><mo>/</mo><mi>n</mi><mo>=</mo><mi>c</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, the orthogonally equivariant estimator of the population covariance matrix proposed by Tsai and Tsai exhibits certain optimal properties. Under some regularity conditions, the authors showed that their novel estimators of eigenvalues are consistent with the eigenvalues of the population covariance matrix. In this paper, under the multinormal setup, we show that they are consistent estimators of the population covariance matrix under a high-dimensional asymptotic setup. We also show that the novel estimator is the MLE of the population covariance matrix when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>. The novel estimator is used to establish that the optimal decomposite <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>T</mi><mrow><mi>T</mi></mrow><mn>2</mn></msubsup></semantics></math></inline-formula>-test has been retained. A high-dimensional statistical hypothesis testing problem is used to carry out statistical inference for high-dimensional principal component analysis-related problems without the sparsity assumption. In the final section, we discuss the situation in which <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>></mo><mi>n</mi></mrow></semantics></math></inline-formula>, especially for high-dimensional low-sample size categorical data models in which <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>></mo><mo>></mo><mi>n</mi></mrow></semantics></math></inline-formula>. |
| format | Article |
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| institution | Kabale University |
| issn | 2227-7390 |
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| spelling | doaj-art-ea8af4d7ab5e4c37983e277734b6014f2025-01-24T13:39:41ZengMDPI AGMathematics2227-73902025-01-0113219110.3390/math13020191Consistent Estimators of the Population Covariance Matrix and Its ReparameterizationsChia-Hsuan Tsai0Ming-Tien Tsai1Institute of Statistical Science, Academia Sinica, Taipei, TaiwanInstitute of Statistical Science, Academia Sinica, Taipei, TaiwanFor the high-dimensional covariance estimation problem, when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo movablelimits="true" form="prefix">lim</mo><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></msub><mi>p</mi><mo>/</mo><mi>n</mi><mo>=</mo><mi>c</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, the orthogonally equivariant estimator of the population covariance matrix proposed by Tsai and Tsai exhibits certain optimal properties. Under some regularity conditions, the authors showed that their novel estimators of eigenvalues are consistent with the eigenvalues of the population covariance matrix. In this paper, under the multinormal setup, we show that they are consistent estimators of the population covariance matrix under a high-dimensional asymptotic setup. We also show that the novel estimator is the MLE of the population covariance matrix when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>. The novel estimator is used to establish that the optimal decomposite <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>T</mi><mrow><mi>T</mi></mrow><mn>2</mn></msubsup></semantics></math></inline-formula>-test has been retained. A high-dimensional statistical hypothesis testing problem is used to carry out statistical inference for high-dimensional principal component analysis-related problems without the sparsity assumption. In the final section, we discuss the situation in which <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>></mo><mi>n</mi></mrow></semantics></math></inline-formula>, especially for high-dimensional low-sample size categorical data models in which <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>p</mi><mo>></mo><mo>></mo><mi>n</mi></mrow></semantics></math></inline-formula>.https://www.mdpi.com/2227-7390/13/2/191high-dimensional covariance matrixMLEsthe consistent estimatorthe decomposite <named-content content-type="inline-formula"><inline-formula> <mml:math id="mm3222"> <mml:semantics> <mml:msubsup> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi>T</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msubsup> </mml:semantics> </mml:math> </inline-formula></named-content>-test |
| spellingShingle | Chia-Hsuan Tsai Ming-Tien Tsai Consistent Estimators of the Population Covariance Matrix and Its Reparameterizations Mathematics high-dimensional covariance matrix MLEs the consistent estimator the decomposite <named-content content-type="inline-formula"><inline-formula> <mml:math id="mm3222"> <mml:semantics> <mml:msubsup> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi>T</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msubsup> </mml:semantics> </mml:math> </inline-formula></named-content>-test |
| title | Consistent Estimators of the Population Covariance Matrix and Its Reparameterizations |
| title_full | Consistent Estimators of the Population Covariance Matrix and Its Reparameterizations |
| title_fullStr | Consistent Estimators of the Population Covariance Matrix and Its Reparameterizations |
| title_full_unstemmed | Consistent Estimators of the Population Covariance Matrix and Its Reparameterizations |
| title_short | Consistent Estimators of the Population Covariance Matrix and Its Reparameterizations |
| title_sort | consistent estimators of the population covariance matrix and its reparameterizations |
| topic | high-dimensional covariance matrix MLEs the consistent estimator the decomposite <named-content content-type="inline-formula"><inline-formula> <mml:math id="mm3222"> <mml:semantics> <mml:msubsup> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi>T</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msubsup> </mml:semantics> </mml:math> </inline-formula></named-content>-test |
| url | https://www.mdpi.com/2227-7390/13/2/191 |
| work_keys_str_mv | AT chiahsuantsai consistentestimatorsofthepopulationcovariancematrixanditsreparameterizations AT mingtientsai consistentestimatorsofthepopulationcovariancematrixanditsreparameterizations |