Wishart and Chi-Square Distributions Associated with Matrix Quadratic Forms
✍ Scribed by Thomas Mathew; Kenneth Nordström
- Publisher
- Elsevier Science
- Year
- 1997
- Tongue
- English
- Weight
- 315 KB
- Volume
- 61
- Category
- Article
- ISSN
- 0047-259X
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✦ Synopsis
An explicit characterization is also obtained for the structure of 7 Y under which the distribution of Y$QY is Wishart. Assuming 7 Y positive definite, a necessary and sufficient condition is derived for every univariate quadratic from l $Y$QYl to be distributed as a multiple of a chi-square. For the case Q=I n , the corresponding structure of 7 Y is identified. An explicit counterexample is constructed showing that Wishartness of Y$Y need not follow when, for every vector l, l $Y$Yl is distributed as a multiple of a chi-square, complementing the well-known counterexample by Mitra (1969, SankhyaÄ A 31, 19 22). Application of the results to multivariate components of variance models is briefly indicated.