On the noncentral distribution of the ratio of the extreme roots of wishart matrix
The distribution of the ratio of the extreme latent roots of the Wishart matrix is useful in testing the sphericity hypothesis for a multivariate normal population. Let X be a p×n matrix whose columns are distributed independently as multivariate normal with zero mean vector and covariance matrix ∑....
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| Format: | Article |
| Language: | English |
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Wiley
1981-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171281000100 |
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| _version_ | 1832550207168970752 |
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| author | V. B. Waikar |
| author_facet | V. B. Waikar |
| author_sort | V. B. Waikar |
| collection | DOAJ |
| description | The distribution of the ratio of the extreme latent roots of the Wishart matrix is useful in testing the sphericity hypothesis for a multivariate normal population. Let X be a p×n matrix whose columns are distributed independently as multivariate normal with zero mean vector and covariance matrix ∑. Further, let S=XX′ and let 11>…>1p>0 be the characteristic roots of S. Thus S has a noncentral Wishart distribution. In this paper, the exact distribution of fp=1−1p/11 is derived. The density of fp is given in terms of zonal polynomials. These results have applications in nuclear physics also. |
| format | Article |
| id | doaj-art-3db5fcee03aa4942a296a6e0c15aeeb2 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1981-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-3db5fcee03aa4942a296a6e0c15aeeb22025-02-03T06:07:26ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251981-01-014114715410.1155/S0161171281000100On the noncentral distribution of the ratio of the extreme roots of wishart matrixV. B. Waikar0Department of Mathematics & Statistics, Miami University, Oxford, Ohio 45056, USAThe distribution of the ratio of the extreme latent roots of the Wishart matrix is useful in testing the sphericity hypothesis for a multivariate normal population. Let X be a p×n matrix whose columns are distributed independently as multivariate normal with zero mean vector and covariance matrix ∑. Further, let S=XX′ and let 11>…>1p>0 be the characteristic roots of S. Thus S has a noncentral Wishart distribution. In this paper, the exact distribution of fp=1−1p/11 is derived. The density of fp is given in terms of zonal polynomials. These results have applications in nuclear physics also.http://dx.doi.org/10.1155/S0161171281000100extreme rootsWishart distributionZonal polynomials. |
| spellingShingle | V. B. Waikar On the noncentral distribution of the ratio of the extreme roots of wishart matrix International Journal of Mathematics and Mathematical Sciences extreme roots Wishart distribution Zonal polynomials. |
| title | On the noncentral distribution of the ratio of the extreme roots of wishart matrix |
| title_full | On the noncentral distribution of the ratio of the extreme roots of wishart matrix |
| title_fullStr | On the noncentral distribution of the ratio of the extreme roots of wishart matrix |
| title_full_unstemmed | On the noncentral distribution of the ratio of the extreme roots of wishart matrix |
| title_short | On the noncentral distribution of the ratio of the extreme roots of wishart matrix |
| title_sort | on the noncentral distribution of the ratio of the extreme roots of wishart matrix |
| topic | extreme roots Wishart distribution Zonal polynomials. |
| url | http://dx.doi.org/10.1155/S0161171281000100 |
| work_keys_str_mv | AT vbwaikar onthenoncentraldistributionoftheratiooftheextremerootsofwishartmatrix |