Wishart and Chi-Square Distributions Associated with Matrix Quadratic Forms
For a normally distributed random matrixYwith a general variance-covariance matrix[Sigma]Y, and for a nonnegative definite matrixQ, necessary and sufficient conditions are derived for the Wishartness ofY'QY. The conditions resemble those obtained by Wong, Masaro, and Wang (1991,J. Multivariate Anal.39, 154-174) and Wong and Wang (1993,J. Multivariate Anal.44, 146-159), but are verifiable and are obtained by elementary means. An explicit characterization is also obtained for the structure of[Sigma]Yunder which the distribution ofY'QYis Wishart. Assuming[Sigma]Ypositive definite, a necessary and sufficient condition is derived for every univariate quadratic fromlY'QYlto be distributed as a multiple of a chi-square. For the caseQ=In, the corresponding structure of[Sigma]Yis identified. An explicit counterexample is constructed showing that Wishartness ofY'Yneed not follow when, for every vectorl,Â l'Y'Ylis distributed as a multiple of a chi-square, complementing the well-known counterexample by Mitra (1969,Sankhya31, 19-22). Application of the results to multivariate components of variance models is briefly indicated.
If you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.
As the access to this document is restricted, you may want to look for a different version under "Related research" (further below) or search for a different version of it.
Volume (Year): 61 (1997)
Issue (Month): 1 (April)
|Contact details of provider:|| Web page: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description|
|Order Information:|| Postal: http://www.elsevier.com/wps/find/supportfaq.cws_home/regional|
References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
- Robert Boik, 1988. "The mixed model for multivariate repeated measures: validity conditions and an approximate test," Psychometrika, Springer, vol. 53(4), pages 469-486, December.
- Wong, Chi Song & Masaro, Joe & Wang, Tonghui, 1991. "Multivariate versions of Cochran's theorems," Journal of Multivariate Analysis, Elsevier, vol. 39(1), pages 154-174, October.
- Wong, C. S. & Wang, T. H., 1993. "Multivariate Versions of Cochran's Theorems II," Journal of Multivariate Analysis, Elsevier, vol. 44(1), pages 146-159, January.
- Mathew, Thomas, 1989. "MANOVA in the multivariate components of variance model," Journal of Multivariate Analysis, Elsevier, vol. 29(1), pages 30-38, April.
When requesting a correction, please mention this item's handle: RePEc:eee:jmvana:v:61:y:1997:i:1:p:129-143. See general information about how to correct material in RePEc.
For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Zhang, Lei)
If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.
If references are entirely missing, you can add them using this form.
If the full references list an item that is present in RePEc, but the system did not link to it, you can help with this form.
If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your profile, as there may be some citations waiting for confirmation.
Please note that corrections may take a couple of weeks to filter through the various RePEc services.