Asymptotic Normality in Linear Regression with Approximately Sparse Structure
In this paper, we study the asymptotic normality in high-dimensional linear regression. We focus on the case where the covariance matrix of the regression variables has a KMS structure, in asymptotic settings where the number of predictors, <i>p</i>, is proportional to the number of obse...
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MDPI AG
2022-05-01
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| Online Access: | https://www.mdpi.com/2227-7390/10/10/1657 |
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| author | Saulius Jokubaitis Remigijus Leipus |
| author_facet | Saulius Jokubaitis Remigijus Leipus |
| author_sort | Saulius Jokubaitis |
| collection | DOAJ |
| description | In this paper, we study the asymptotic normality in high-dimensional linear regression. We focus on the case where the covariance matrix of the regression variables has a KMS structure, in asymptotic settings where the number of predictors, <i>p</i>, is proportional to the number of observations, <i>n</i>. The main result of the paper is the derivation of the exact asymptotic distribution for the suitably centered and normalized squared norm of the product between predictor matrix, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">X</mi></semantics></math></inline-formula>, and outcome variable, <i>Y</i>, i.e., the statistic <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>∥</mo></mrow><msup><mi mathvariant="double-struck">X</mi><mo>′</mo></msup><msubsup><mrow><mi>Y</mi><mo>∥</mo></mrow><mrow><mn>2</mn></mrow><mn>2</mn></msubsup></mrow></semantics></math></inline-formula>, under rather unrestrictive assumptions for the model parameters <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>β</mi><mi>j</mi></msub></semantics></math></inline-formula>. We employ variance-gamma distribution in order to derive the results, which, along with the asymptotic results, allows us to easily define the exact distribution of the statistic. Additionally, we consider a specific case of approximate sparsity of the model parameter vector <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> and perform a Monte Carlo simulation study. The simulation results suggest that the statistic approaches the limiting distribution fairly quickly even under high variable multi-correlation and relatively small number of observations, suggesting possible applications to the construction of statistical testing procedures for the real-world data and related problems. |
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| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T03:29:23Z |
| publishDate | 2022-05-01 |
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| spelling | doaj.art-e27a3993c9364956a0d3839c2c5d41f32023-11-23T12:00:31ZengMDPI AGMathematics2227-73902022-05-011010165710.3390/math10101657Asymptotic Normality in Linear Regression with Approximately Sparse StructureSaulius Jokubaitis0Remigijus Leipus1Faculty of Mathematics and Informatics, Institute of Applied Mathematics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, LithuaniaFaculty of Mathematics and Informatics, Institute of Applied Mathematics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, LithuaniaIn this paper, we study the asymptotic normality in high-dimensional linear regression. We focus on the case where the covariance matrix of the regression variables has a KMS structure, in asymptotic settings where the number of predictors, <i>p</i>, is proportional to the number of observations, <i>n</i>. The main result of the paper is the derivation of the exact asymptotic distribution for the suitably centered and normalized squared norm of the product between predictor matrix, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">X</mi></semantics></math></inline-formula>, and outcome variable, <i>Y</i>, i.e., the statistic <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>∥</mo></mrow><msup><mi mathvariant="double-struck">X</mi><mo>′</mo></msup><msubsup><mrow><mi>Y</mi><mo>∥</mo></mrow><mrow><mn>2</mn></mrow><mn>2</mn></msubsup></mrow></semantics></math></inline-formula>, under rather unrestrictive assumptions for the model parameters <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>β</mi><mi>j</mi></msub></semantics></math></inline-formula>. We employ variance-gamma distribution in order to derive the results, which, along with the asymptotic results, allows us to easily define the exact distribution of the statistic. Additionally, we consider a specific case of approximate sparsity of the model parameter vector <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> and perform a Monte Carlo simulation study. The simulation results suggest that the statistic approaches the limiting distribution fairly quickly even under high variable multi-correlation and relatively small number of observations, suggesting possible applications to the construction of statistical testing procedures for the real-world data and related problems.https://www.mdpi.com/2227-7390/10/10/1657linear regressionsparsityasymptotic normalityvariance-gamma distribution |
| spellingShingle | Saulius Jokubaitis Remigijus Leipus Asymptotic Normality in Linear Regression with Approximately Sparse Structure Mathematics linear regression sparsity asymptotic normality variance-gamma distribution |
| title | Asymptotic Normality in Linear Regression with Approximately Sparse Structure |
| title_full | Asymptotic Normality in Linear Regression with Approximately Sparse Structure |
| title_fullStr | Asymptotic Normality in Linear Regression with Approximately Sparse Structure |
| title_full_unstemmed | Asymptotic Normality in Linear Regression with Approximately Sparse Structure |
| title_short | Asymptotic Normality in Linear Regression with Approximately Sparse Structure |
| title_sort | asymptotic normality in linear regression with approximately sparse structure |
| topic | linear regression sparsity asymptotic normality variance-gamma distribution |
| url | https://www.mdpi.com/2227-7390/10/10/1657 |
| work_keys_str_mv | AT sauliusjokubaitis asymptoticnormalityinlinearregressionwithapproximatelysparsestructure AT remigijusleipus asymptoticnormalityinlinearregressionwithapproximatelysparsestructure |