Consistent Estimators of the Population Covariance Matrix and Its Reparameterizations

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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Bibliographic Details
Main Authors:	Chia-Hsuan Tsai, Ming-Tien Tsai
Format:	Article
Language:	English
Published:	MDPI AG 2025-01-01
Series:	Mathematics
Subjects:	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
Online Access:	https://www.mdpi.com/2227-7390/13/2/191
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Summary:	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>.
ISSN:	2227-7390