Invariant polynomials with two matrix arguments extending the zonal polynomials: Applications to multivariate distribution theory

1979 ◽  
Vol 31 (3) ◽  
pp. 465-485 ◽  
Author(s):  
A. W. Davis
Keyword(s):  
Distribution Theory ◽  
Multivariate Distribution ◽  
Invariant Polynomials ◽  
Zonal Polynomials

Methods for Constructing Top Order Invariant Polynomials

1987 ◽  
Vol 3 (2) ◽  
pp. 195-207 ◽  
Author(s):  
Yasuko Chikuse
Keyword(s):  
Symmetric Matrix ◽  
Distribution Theory ◽  
Multivariate Distribution ◽  
Invariant Polynomials ◽  
Zonal Polynomials ◽  
Explicit Formulae ◽  
Order Invariant

The invariant polynomials (Davis [8] and Chikuse [2] with r(r ≥ 2) symmetric matrix arguments have been defined, extending the zonal polynomials, and applied in multivariate distribution theory. The usefulness of the polynomials has attracted the attention of econometricians, and some recent papers have applied the methods to distribution theory in econometrics (e.g., Hillier [14] and Phillips [22]).The ‘top order’ invariant polynomials , in which each of the partitions of ki 1 = 1,…,r, and has only one part, occur frequently in multivariate distribution theory (e.g., Hillier and Satchell [17] and Phillips [27]). In this paper we give three methods of constructing these polynomials, extending those of Ruben [28] for the top order zonal polynomials. The first two methods yield explicit formulae for the polynomials and then we give a recurrence procedure. It is shown that some of the expansions presented in Chikuse and Davis [4] are simplified for the top order invariant polynomials. A brief discussion is given on the ‘lowest order’ invariant polynomials.

A Survey on the Invariant Polynomials with Matrix Arguments in Relation to Econometric Distribution Theory

1986 ◽  
Vol 2 (2) ◽  
pp. 232-248 ◽  
Author(s):  
Yasuko Chikuse ◽  
A. W. Davis
Keyword(s):  
Power Series ◽  
Distribution Theory ◽  
Multivariate Distribution ◽  
Group Representations ◽  
Invariant Polynomials ◽  
Zonal Polynomials

Invariant polynomials with matrix arguments have been defined by the theory of group representations, generalizing the zonal polynomials. They have developed as a useful tool to evaluate certain integrals arising in multivariate distribution theory, which were expanded as power series in terms of the invariant polynomials. Some interest in the polynomials has been shown by people working in the field of econometric theory. In this paper, we shall survey the properties of the invariant polynomials and their applications in multivariate distribution theory including related developments in econometrics.

scholarly journals COMPUTATIONALLY EFFICIENT RECURSIONS FOR TOP-ORDER INVARIANT POLYNOMIALS WITH APPLICATIONS

2009 ◽  
Vol 25 (1) ◽  
pp. 211-242 ◽  
Author(s):  
Grant Hillier ◽  
Raymond Kan ◽  
Xiaolu Wang
Keyword(s):  
International Economic ◽  
Quadratic Forms ◽  
Numerical Evaluation ◽  
Distribution Theory ◽  
Recursive Algorithms ◽  
Computationally Efficient ◽  
Invariant Polynomials ◽  
Zonal Polynomials ◽  
Theoretical Results ◽  
Order Invariant

The top-order zonal polynomials Ck(A), and top-order invariant polynomials Ck1,…,kr (A1, …, Ar) in which each of the partitions of ki, i = 1, …, r, has only one part, occur frequently in multivariate distribution theory, and econometrics — see, for example, Phillips (1980, Econometrica 48, 861–878; 1984, Journal of Econometrics 26, 387–398; 1985, International Economic Review 26, 21–36; 1986, Econometrica 54, 881–896), Hillier (1985, Econometric Theory 1, 53–72; 2001, Econometric Theory 17, 1–28), Hillier and Satchell (1986, Econometric Theory 2, 66–74), and Smith (1989, Journal of Multivariate Analysis 31, 244–257; 1993, Australian Journal of Statistics 35, 271–282). However, even with the recursive algorithms of Ruben (1962, Annals of Mathematical Statistics 33, 542–570) and Chikuse (1987, Econometric Theory 3, 195–207), numerical evaluation of these invariant polynomials is extremely time consuming. As a result, the value of invariant polynomials has been largely confined to analytic work on distribution theory. In this paper we present new, very much more efficient, algorithms for computing both the top-order zonal and invariant polynomials. These results should make the theoretical results involving these functions much more valuable for direct practical study. We demonstrate the value of our results by providing fast and accurate algorithms for computing the moments of a ratio of quadratic forms in normal random variables.

scholarly journals On a zonal polynomial integral

2003 ◽  
Vol 2003 (11) ◽  
pp. 569-573
Author(s):  
A. K. Gupta ◽  
D. G. Kabe
Keyword(s):  
Multiple Integral ◽  
Invariant Polynomial ◽  
Zonal Polynomial ◽  
Invariant Polynomials ◽  
Zonal Polynomials

A certain multiple integral occurring in the studies of Beherens-Fisher multivariate problem has been evaluated by Mathai et al. (1995) in terms of invariant polynomials. However, this paper explicitly evaluates the context integral in terms of zonal polynomials, thus establishing a relationship between zonal polynomial integrals and invariant polynomial integrals.

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

2013 ◽  
Vol 30 (2) ◽  
pp. 436-473 ◽  
Author(s):  
Grant Hillier ◽  
Raymond Kan ◽  
Xiaolu Wang
Keyword(s):  
Generating Functions ◽  
Quadratic Forms ◽  
Symmetric Functions ◽  
Statistical Computing ◽  
Invariant Polynomials ◽  
Zonal Polynomials ◽  
Siam Journal ◽  
Computing Power ◽  
Normal Vectors ◽  
Recursive Relations

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.

scholarly journals On Multivariate Distribution Theory

1954 ◽  
Vol 25 (2) ◽  
pp. 329-339 ◽  
Author(s):  
I. Olkin ◽  
S. N. Roy
Keyword(s):  
Distribution Theory ◽  
Multivariate Distribution
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