The density of a quadratic form in a vector uniformly distributed on the nsphere
Hillier, G.H. (1999) The density of a quadratic form in a vector uniformly distributed on the nsphere , Southampton, UK University of Southampton (Discussion Papers in Economics and Econometrics, (doi:10.1017/S026646660117101X) , 9902).
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Description/Abstract
There are many instances in the statistical literature in which inference is based on a normalised quadratic form in a standard normal vector, normalised by the squared length of that vector. Examples include both test statistics (the DurbinWatson statistic, and many other diagnostic test statistics for linear models), and estimators (serial correlation coefficients).
Although the properties of such a statistic have been much studied – particularly for the special case of serial correlation coefficients – its density function remains unknown. Two of the earliest contributors to this literature, von Neuman (1941) and Koopmans (1942), provided what are still today almost the entire extent of our knowledge of the density.
This paper gives formulae for the density function of such a statistic in each of the open intervals between the characteristic roots of the matrix involved. We do not assume that these roots are positive, but do assume that they are distinct. The case of nondistinct roots can be dealt with by methods similar to those used here. Starting from a representation of the density as a surface integral over an (n2)dimensional hyperplane, the density is expressed in terms of toporder zonal polynomials involving differencequotients of the characteristic roots of the matrix in the numerator quadratic form
Item Type:  Monograph (Discussion Paper)  

Digital Object Identifier (DOI):  doi:10.1017/S026646660117101X  
Related URLs:  
Subjects:  H Social Sciences > HB Economic Theory Q Science > QA Mathematics H Social Sciences > HA Statistics 

ePrint ID:  33137  
Date : 


Date Deposited:  24 May 2007  
Last Modified:  16 Apr 2017 22:17  
Further Information:  Google Scholar  
URI:  http://eprints.soton.ac.uk/id/eprint/33137 
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