Chi-Square Approximation for the Distribution of Individual Eigenvalues of a Singular Wishart Matrix
This paper discusses the approximate distributions of eigenvalues of a singular Wishart matrix. We give the approximate joint density of eigenvalues by Laplace approximation for the hypergeometric functions of matrix arguments. Furthermore, we show that the distribution of each eigenvalue can be app...
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| Format: | Article |
| Language: | English |
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MDPI AG
2024-03-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/12/6/921 |
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| author | Koki Shimizu Hiroki Hashiguchi |
| author_facet | Koki Shimizu Hiroki Hashiguchi |
| author_sort | Koki Shimizu |
| collection | DOAJ |
| description | This paper discusses the approximate distributions of eigenvalues of a singular Wishart matrix. We give the approximate joint density of eigenvalues by Laplace approximation for the hypergeometric functions of matrix arguments. Furthermore, we show that the distribution of each eigenvalue can be approximated by the chi-square distribution with varying degrees of freedom when the population eigenvalues are infinitely dispersed. The derived result is applied to testing the equality of eigenvalues in two populations. |
| first_indexed | 2024-04-24T18:02:23Z |
| format | Article |
| id | doaj.art-3c786a62233846fda3004f91bd62ecfc |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-04-24T18:02:23Z |
| publishDate | 2024-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-3c786a62233846fda3004f91bd62ecfc2024-03-27T13:53:19ZengMDPI AGMathematics2227-73902024-03-0112692110.3390/math12060921Chi-Square Approximation for the Distribution of Individual Eigenvalues of a Singular Wishart MatrixKoki Shimizu0Hiroki Hashiguchi1Department of Applied Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, JapanDepartment of Applied Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, JapanThis paper discusses the approximate distributions of eigenvalues of a singular Wishart matrix. We give the approximate joint density of eigenvalues by Laplace approximation for the hypergeometric functions of matrix arguments. Furthermore, we show that the distribution of each eigenvalue can be approximated by the chi-square distribution with varying degrees of freedom when the population eigenvalues are infinitely dispersed. The derived result is applied to testing the equality of eigenvalues in two populations.https://www.mdpi.com/2227-7390/12/6/921hypergeometric functionslaplace approximationspiked covariance model |
| spellingShingle | Koki Shimizu Hiroki Hashiguchi Chi-Square Approximation for the Distribution of Individual Eigenvalues of a Singular Wishart Matrix Mathematics hypergeometric functions laplace approximation spiked covariance model |
| title | Chi-Square Approximation for the Distribution of Individual Eigenvalues of a Singular Wishart Matrix |
| title_full | Chi-Square Approximation for the Distribution of Individual Eigenvalues of a Singular Wishart Matrix |
| title_fullStr | Chi-Square Approximation for the Distribution of Individual Eigenvalues of a Singular Wishart Matrix |
| title_full_unstemmed | Chi-Square Approximation for the Distribution of Individual Eigenvalues of a Singular Wishart Matrix |
| title_short | Chi-Square Approximation for the Distribution of Individual Eigenvalues of a Singular Wishart Matrix |
| title_sort | chi square approximation for the distribution of individual eigenvalues of a singular wishart matrix |
| topic | hypergeometric functions laplace approximation spiked covariance model |
| url | https://www.mdpi.com/2227-7390/12/6/921 |
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