THE DENSITY OF A QUADRATIC FORM IN A VECTOR UNIFORMLY DISTRIBUTED ON THE n-SPHERE
AbstractThere are many instances in the statistical literature in which inference is based on a normalized quadratic form in a standard normal vector, normalized by the squared length of that vector. Examples include both test statistics (the Durbin Watson statistic) and estimators (serial correlation coefficients). Although much studied, no general closed-form expression for the density function of such a statistic is known. This paper gives general formulae for the density in each open interval between the characteristic roots of the matrix involved. Results are given for the case of distinct roots, which need not be assumed positive, and when the roots occur with multiplicities greater than one. Starting from a representation of the density as a surface integral over an (n 2)-dimensional hyperplane, the density is expressed in terms of top-order zonal polynomials involving difference quotients of the characteristic roots of the matrix in the numerator quadratic form.
Download InfoIf 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.
Bibliographic InfoArticle provided by Cambridge University Press in its journal Econometric Theory.
Volume (Year): 17 (2001)
Issue (Month): 01 (February)
Contact details of provider:
Postal: The Edinburgh Building, Shaftesbury Road, Cambridge CB2 2RU UK
Fax: +44 (0)1223 325150
Web page: http://journals.cambridge.org/jid_ECTProvider-Email:email@example.com
You can help add them by filling out this form.
CitEc Project, subscribe to its RSS feed for this item.
- Peter M Robinson & Francesca Rossi, 2013.
"Improved Tests for Spatial Correlation,"
STICERD - Econometrics Paper Series
/2013/565, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
- Grant Hillier & Raymond Kan & Xiaolu Wang, 2008.
"Computationally efficient recursions for top-order invariant polynomials with applications,"
CeMMAP working papers
CWP07/08, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
- Hillier, Grant & Kan, Raymond & Wang, Xiaolu, 2009. "Computationally Efficient Recursions For Top-Order Invariant Polynomials With Applications," Econometric Theory, Cambridge University Press, vol. 25(01), pages 211-242, February.
- Giovanni Forchini, . "The Distribution of a Ratio of Quadratic Forms in Noncentral Normal Variables," Discussion Papers 01/12, Department of Economics, University of York.
- Zeng-Hua Lu & Maxwell King, 2002. "Improving The Numerical Technique For Computing The Accumulated Distribution Of A Quadratic Form In Normal Variables," Econometric Reviews, Taylor & Francis Journals, vol. 21(2), pages 149-165.
- Grant Hillier & Federico Martellosio, 2004.
"Spatial design matrices and associated quadratic forms: structure and properties,"
CeMMAP working papers
CWP16/04, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
- Hillier, Grant & Martellosio, Federico, 2006. "Spatial design matrices and associated quadratic forms: structure and properties," Journal of Multivariate Analysis, Elsevier, vol. 97(1), pages 1-18, January.
- Hillier, Grant & Martellosio, Federico, 2006. "Spatial design matrices and associated quadratic forms: structure and properties," MPRA Paper 15807, University Library of Munich, Germany.
- Grant Hillier & Federico Martellosio, 2013. "Properties of the maximum likelihood estimator in spacial autoregressive models," CeMMAP working papers CWP44/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
- Lu, Zeng-Hua, 2006. "The numerical evaluation of the probability density function of a quadratic form in normal variables," Computational Statistics & Data Analysis, Elsevier, vol. 51(3), pages 1986-1996, December.
For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Keith Waters).
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.