Generating functions and short recursions, with applications to the moments of quadratic forms in noncentral normal vectors
Using generating functions, the top-order zonal polynomials that occur in much distribution theory under normality can be recursively related to other symmetric functions (power-sum and elementary symmetric functions, Ruben (1962), Hillier, Kan, and Wang (2009)). Typically, in a recursion of this type the k-th object of interest, dk say, is expressed in terms of all lower-order dj â€™s. In Hillier, Kan, and Wang (2009) we pointed out that, in the case of top-order zonal polynomials (and generalizations of them), a shorter (i.e., fixed length) recursion can be deduced. The present paper shows that the argument in Hillier, Kan, and Wang (2009) generalizes to a large class of objects/generating functions. The results thus obtained are then applied to various problems involving quadratic forms in noncentral normal vectors Keywords; generating functions, invariant polynomials, non-central normal distribution, recursions, symmetric functions, zonal polynomials
|Date of creation:||01 Jan 2009|
|Date of revision:|
|Contact details of provider:|| Postal: |
Phone: (+44) 23 80592537
Fax: (+44) 23 80593858
Web page: http://www.economics.soton.ac.uk/
More information through EDIRC
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.:
- Bock, M.E. & Judge, G.G. & Yancey, T.A., 1984. "A simple form for the inverse moments of non-central [chi]2 andF random variables and certain confluent hypergeometric functions," Journal of Econometrics, Elsevier, vol. 25(1-2), pages 217-234.
- Ghazal, G. A., 1994. "Moments of the ratio of two dependent quadratic forms," Statistics & Probability Letters, Elsevier, vol. 20(4), pages 313-319, July.
- Hillier, Grant & Kan, Raymond & Wang, Xiaolu, 2009.
"Computationally Efficient Recursions For Top-Order Invariant Polynomials With Applications,"
Cambridge University Press, vol. 25(01), pages 211-242, February.
- 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.
- Magnus, J.R., 1986. "The exact moments of a ratio of quadratic forms in normal variables," Other publications TiSEM c6725407-ac3c-44fd-b6d1-5, Tilburg University, School of Economics and Management.
- repec:ner:tilbur:urn:nbn:nl:ui:12-153219 is not listed on IDEAS
- Kan, Raymond, 2008. "From moments of sum to moments of product," Journal of Multivariate Analysis, Elsevier, vol. 99(3), pages 542-554, March.
- Chikuse, Yasuko, 1987. "Methods for Constructing Top Order Invariant Polynomials," Econometric Theory, Cambridge University Press, vol. 3(02), pages 195-207, April.
- Sawa, Takamitsu, 1972. "Finite-Sample Properties of the k-Class Estimators," Econometrica, Econometric Society, vol. 40(4), pages 653-80, July.
When requesting a correction, please mention this item's handle: RePEc:stn:sotoec:0918. 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: (Chris Thorn)
If references are entirely missing, you can add them using this form.