Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-20T00:48:00.416Z Has data issue: false hasContentIssue false
  • English
  • Français

GENERATING FUNCTIONS AND SHORT RECURSIONS, WITH APPLICATIONS TO THE MOMENTS OF QUADRATIC FORMS IN NONCENTRAL NORMAL VECTORS

Published online by Cambridge University Press:  17 October 2013

Grant Hillier ,
Raymond Kan  and
Xiaolu Wang
Grant Hillier
Affiliation:
CeMMAP University of Southampton
Raymond Kan*
Affiliation:
University of Toronto
Xiaolu Wang
Affiliation:
Iowa State University
*
*Address correspondence to Raymond Kan, 105 St. George Street, Toronto, Ontario, Canada M5S 3E6, e-mail: [email protected].
Get access Rights & Permissions [Opens in a new window]

Abstract

Recursive relations for objects of statistical interest have long been important for computation, and they remain so even with hugely improved computing power. Such recursions are frequently derived by exploiting relations between generating functions. For example, the top-order zonal polynomials that occur in much distribution theory under normality can be recursively related to other (easily computed) symmetric functions (power-sum and elementary symmetric functions; Ruben, 1962, Annals of Mathematical Statistics 33, 542–570; Hillier, Kan, and Wang, 2009, Econometric Theory 25, 211–242). Typically, in a recursion of this type the kth object of interest, dk, say, is expressed in terms of all lower order dj’s. In Hillier et al. (2009) we pointed out that, in the case of top-order zonal polynomials and other invariant polynomials of multiple matrix argument, a fixed length recursion can be deduced. We refer to this as a short recursion. The present paper shows that the main results in Hillier et al. (2009) can be generalized and that short recursions can be obtained for a much larger class of objects/generating functions. As applications, we show that short recursions can be obtained for various problems involving quadratic forms in noncentral normal vectors, including moments, product moments, and expectations of ratios of powers of quadratic forms. For this class of problems, we also show that the length of the recursion can be further reduced by an application of a generalization of Horner’s method (cf. Brown, 1986, SIAM Journal on Scientific and Statistical Computing 7, 689–695), producing a super-short recursion that is significantly more efficient than even the short recursion.

Type
ARTICLES
Information
Econometric Theory , Volume 30 , Issue 2 , April 2014 , pp. 436 - 473
Copyright
Copyright © Cambridge University Press 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Bao, Y. & Ullah, A. (2010) Expectation of quadratic forms in normal and nonnormal variables with applications. Journal of Statistical Planning and Inference 140, 1193–1205.CrossRefGoogle Scholar
Brown, R.H. (1986) The distribution function of positive definite quadratic forms in normal random variables. SIAM Journal on Scientific and Statistical Computing 7, 689–695.CrossRefGoogle Scholar
Chikuse, Y. (1987) Methods for constructing top order invariant polynomials. Econometric Theory 3, 195–207.CrossRefGoogle Scholar
Cressie, N., Davis, A.S., Folks, J.L., & Policello, G.E. II (1981) The moment-generating function and negative integer moments. American Statistican 35, 148–150.Google Scholar
Davis, A.W. (1979) Invariant polynomials with two matrix arguments, extending the zonal polynomials: Applications to multivariate distribution theory. Annals of the Institute of Statistical Mathematics A31, 465–485.CrossRefGoogle Scholar
Davis, A.W. (1981) On the construction of a class of invariant polynomials in several matrices, extending the zonal polynomials. Annals of the Institute of Statistical Mathematics A33, 297–313.CrossRefGoogle Scholar
Ghazal, G.A. (1994) Moments of the ratio of two dependent quadratic forms. Statistics and Probability Letters 20, 313–319.CrossRefGoogle Scholar
Hillier, G.H., Kan, R., & Wang, X. (2009) Computationally efficient recursions for top-order invariant polynomials with applications. Econometric Theory 25, 211–242.CrossRefGoogle Scholar
James, A.T. (1961) Zonal polynomials of the real positive definite symmetric matrices. Annals of Mathematics 74, 456–469.CrossRefGoogle Scholar
Kan, R. (2008) From moments of sum to moments of product. Journal of Multivariate Analysis 99, 542–554.CrossRefGoogle Scholar
Kotz, S., Johnson, N.L., & Boyd, D.W. (1967) Series representations of distributions of quadratic forms in normal random variables II: Non-central case. Annals of Mathematical Statistics 38, 838–848.CrossRefGoogle Scholar
Krishnan, M. (1967) The moments of a doubly noncentral t-distribution. Journal of the American Statistical Association 62, 278–287.Google Scholar
Magnus, J. (1986) The exact moments of a ratio of quadratic forms in normal variables. Annales d’Économie et de Statistique 4, 95–109.CrossRefGoogle Scholar
Mathai, A.M. & Provost, S.B. (1992) Quadratic Forms in Random Variables: Thoery and Applications, Marcel Dekker.Google Scholar
Meng, X.L. (2005) From unit root to Stein’s estimator to Fisher’s k statistics: If you have a moment, I can tell you more. Statistical Science 20, 141–162.CrossRefGoogle Scholar
Phillips, P.C.B. (1980) Finite sample theory and the distribution of alternative estimators of the marginal propensity to consume. Review of Economic Studies 47, 183–224.CrossRefGoogle Scholar
Phillips, P.C.B. (1986) The exact distribution of the Wald statistic. Econometrica 54, 881–896.CrossRefGoogle Scholar
Press, S.J. (1966) Linear combinations of non-central chi-square variates. Annals of Mathematical Statistics 37, 480–487.CrossRefGoogle Scholar
Ruben, H. (1962) Probability content of regions under spherical normal distributions, IV: The distribution of homogeneous and non-homogeneous quadratic functions of normal variables. Annals of Mathematical Statistics 33, 542–570.CrossRefGoogle Scholar
Sawa, T. (1972) Finite sample properties of the k-class estimators. Econometrica 40, 653–680.CrossRefGoogle Scholar
Shah, B.K. & Khatri, C.G. (1961) Distribution of a definite quadratic forms in non-central normal variates. Annals of Mathematical Statistics 32, 883–887.CrossRefGoogle Scholar
Shah, B.K. & Khatri, C.G. (1963) Correction to “Distribution of a definite quadratic forms in non-central normal variates.” Annals of Mathematical Statistics 34, 673.CrossRefGoogle Scholar
Shenton, L.R. & Johnson, W.L. (1965) Moments of a serial correlation coefficient. Journal of the Royal Statistical Society, Series B 27, 308–320.Google Scholar
Smith, M. (1989) On the expectation of a ratio of quadratic forms in normal random variables. Journal of Multivariate Analysis 31, 244–257.CrossRefGoogle Scholar
Smith, M. (1993) Expectations of ratios of quadratic forms in normal variables: Evaluating some top-order invariant polynomials. Australian Journal of Statistics 35, 271–282.CrossRefGoogle Scholar
Smith, P.J. (1995) A recursive formulation of the old problem of obtaining moments from cumulants and vice versa. American Statistican 49, 217–218.Google Scholar
White, J.S. (1961) Asymptotic expansions for the mean and variance and of the serial correlation coefficient. Biometrika 48, 85–94.CrossRefGoogle Scholar
Wilf, H.S. (2005) Generatingfunctionology. 3rd ed. A.K. Peters.CrossRefGoogle Scholar
Williams, J.D. (1941) Moments of the ratio of the mean square successive difference to the mean square difference in samples from a normal universe. Annals of Mathematical Statistics 12, 239–241.CrossRefGoogle Scholar
Withers, C.S. & Nadarajah, S. (2007) A recurrence relation for moments of the noncentral chi square. American Statistician 61, 337–338.CrossRefGoogle Scholar