A. K. GUPTA AND D. G. KABE
Received 2 September 2002 and in revised form 9 July 2003
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 rela- tionship between zonal polynomial integrals and invariant polynomial integrals.
1. Introduction
Consider two p×p independent random matrices S1 and S2.The ma- trix S1, with noncentrality parameter matrixΩ and covariance matrix Σ, is Wishart withn1 degrees of freedom andS2is Wishart with covari- ance matrixΣandn2 degrees of freedom. Then the random matrixF= S−1/22 S1S−1/22 has the density
g(F) =K
|F|(1/2)(n1−p−1)S2(1/2)(n1+n2−p−1)
×exp
−1
2trΣ−12 S2−1
2trΣ−11 S1/22 FS1/22
×0F1
1 2n1;1
4ΩΣ−1/21 S1/22 FS1/22
dS2
=
δ,λ:φ
(−1)kθφδ,λCφ(Λ) k!!
n1/2 λCφ(I)
Cφδ,λ 1
2Σ−11 S2,1
2Σ−11 ΩΣ−1/21 S2
dS2,
(1.1)
Copyrightc2003 Hindawi Publishing Corporation Journal of Applied Mathematics 2003:11(2003)569–573 2000 Mathematics Subject Classification: 62H10, 62H12 URL:http://dx.doi.org/10.1155/S1110757X03209074
whereΛis thep×pdiagonal matrix of the roots ofFandKas a generic letter denotes the normalizing constants of density functions in this pa- per.
Mathai et al. [4, equations (5.5.11), (5.5.12), pages 274–275] do not evaluate the integral(1.1). However, they derive the density of the roots of F in terms of a series of invariant polynomials of two matrix argu- ments. This paper explicitly obtains the density ofF in terms of zonal polynomials, a result which has not been available in the literature so far, not even in terms of invariant polynomials. An expression for the density of the roots ofF is also derived in terms of invariant polynomi- als of two arguments, which is slightly different from the one presented by Mathai et al.[4, equation(5.5.11), page 274].
The next section states some useful results and the main results of the paper are derived inSection 3. Although it is not always possible to de- rive invariant polynomial results in terms of zonal polynomials, some- times it is as done in this paper. We assume that all matrices occurring in this paper are full-rank matrices, except the noncentrality parameterΩ which may have any rank.
2. Some useful results
If P is any p×p matrix, then it is known that (see Gupta and Nagar [2])
K
exp{trP V}|I−V V|(1/2)(n−2p−1)dV= 0F1
1 2n;1
4PP
. (2.1)
Now we can writeP= (PP)1/2Hfor somep×porthogonalH, see G. A.
Korn and T. M. Korn[3, page 412]. It follows that
K
exp{trQP V}|I−V V|(1/2)(n−2p−1)dV
=K
exp
trQ(PP)1/2HV
|I−V V|(1/2)(n−2p−1)dV
=K
exp
tr(PP)1/2QV
|I−V V|(1/2)(n−2p−1)dV
=0F1
1 2n;1
4PP QQ
=0F1
1 2n;1
4QQPP
=K
exp{trP QV}|I−V V|(1/2)(n−2p−1)dV.
(2.2)
Again it follows from(2.2)that
K
exp
trQ(PP)1/2HV
|I−V V|(1/2)(n−2p−1)dV
=K
exp tr
H(PP)1/2 Q
|I−V V|(1/2)(n−2p−1)dV
=K
exp tr
H(PP)1/2 (QQ)1/2R
|I−V V|(1/2)(n−2p−1)dV
= 0F1
1 2n;1
4RQQRHPP H
=0F1
1
2n;QQPP
,
(2.3)
whereR(p×p)andH(p×p)are any two arbitrary orthogonal matrices which may depend onQQandPP, respectively.
Now it follows from(2.3)that
0F1
1 2n;1
4PP QQTT
=0F1
1 2n;1
4HPP HRQQRUTTU
, (2.4) whereH,R, andUare anyp×porthogonal matrices, and note that
0F1
1 2n;1
4ABC
=0F1
1 2n;1
4∆
, (2.5)
where∆is any permutation ofABCand also holds for any permutation of four symmetric matrices.
Obviously,(2.4)in terms of zonal polynomials yields
Cθ(QQPP TT) =Cθ(RQQRHPP HUTTU). (2.6)
From(2.6), note that
exp{trPP QQRR}= 0F1exp{tr∆}=∞
t=0
1
t!Cθ(PP QQRR), (2.7) where∆is any permutation ofPP QQRR. Further, it is easy to verify that
0F1
1 2n;1
4PP QQTTGG
=0F1
1 2n;1
4PP QQGGTT
(2.8) because(2.8)is true for any permutation of the four distinct matrices in (2.8).
Now we proceed with the main result.
3. Main results
In view of(2.4),(2.5), and(2.7)and using(1.1), the density ofFis given by
g(F) =K|F|(1/2)(n1−p−1)
×
exp
−1
2trΣ−12 S−1
2trΣ−1/21 FΣ−1/21 S
|S|(1/2)(n1+n2−p−1)
×0F1
1 2n1;1
4Σ−1/21 ΩΣ−1/21 F1/2SF1/2
dS
=K|F|(1/2)(n1−p−1)Σ−12 + Σ−1/21 FΣ−1/21 −(1/2)(n1+n2)
×1F1
1 2
n1+n2 ,1 2n1; 1
2F1/2Σ−1/21 ΩΣ−1/21 F1/2
Σ−12 + Σ−1/21 FΣ−1/21 −1
=K|F|(1/2)(n1−p−1)Σ−12 + Σ−1/21 FΣ−1/21 −(1/2)(n1+n2)
×1F1
1 2
n1+n2 ,1 2n1;1
2
Σ−1/21 ΩΣ−1/21 1/2
×F
Σ−1/21 ΩΣ−1/21/2
Σ−12 + Σ−1/21 FΣ−1/21 −1 .
(3.1)
Now ifΣ1/21 Σ−12 Σ1/21 =θIfor any scalarθ, then note that
g(F) =K|F|(1/2)(n1−p−1)|θI+F|−(1/2)(n1+n2)
×1F1
1 2
n1+n2 ,1 2n1;1
2ΩF(θI+F)−1
. (3.2)
Thus the density ofΛ =diag(λ1, . . . , λp)of the roots ofFis
g(Λ) =K|Λ|(1/2)(n1−p−1)|θI+ Λ|−(1/2)(n1+n2)
i<j
λi−λj
×1F1
1 2
n1+n2 ,1 2n1;1
2ΩΛ(θI+ Λ)−1 .
(3.3)
The expression is not very suitable for obtaining the density ofΛfor the expression (3.1). For some Q (p×p) orthogonal, let QFQ be still denoted byF, letLbe thep×pdiagonal matrix of the roots ofΣ1/2ΩΣ1/2, and let∆ be the diagonal matrix of the roots ofΣ1/21 Σ−12 Σ1/21 . Then the
density of thisFis
g(F) =K|F|(1/2)(n1−p−1)|∆ +F|−(1/2)(n1+n2)
×1F1
1 2
n1+n2 ,1 2n1;1
2LF(∆ +F)−1
. (3.4)
Mathai et al.[4, equation(A.3.5), page 338]showed that
|∆ +F|−a1F1
a, u;LF(∆ +F)−1 =
δ,λ;φ
(a)φCφδ,λ
−∆−1F, LF kK!!(u)λ
. (3.5)
We have observed earlier thatCθ(QQPP)=Cθ(RQQRHPP H)and thus the invariant polynomial argument for matrices can also be replaced by their roots. Thus the density ofΛfor(3.4)is
g(Λ) =K|Λ|(1/2)(n1−p−1)
i<j
λi−λj
×
δ,λ;φ
(1/2) n1+n2
φCδ,λφ
−∆−1Λ, LΛ k!!
(1/2)n1 λ
.
(3.6)
However,(3.6)is not suitable for obtaining the density of(trF).
Acknowledgment
The authors are thankful to the referee for a critical reading of the paper.
References
[1] Y. Chikuse,Distributions of some matrix variates and latent roots in multivariate Behrens-Fisher discriminant analysis, Ann. Statist.9(1981), no. 2, 401–407.
[2] A. K. Gupta and D. K. Nagar, Matrix Variate Distributions, Chapman &
Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 104, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2000.
[3] G. A. Korn and T. M. Korn,Mathematical Handbook for Scientists and Engineers, McGraw-Hill, New York, 1961.
[4] A. M. Mathai, S. B. Provost, and T. Hayakawa,Bilinear Forms and Zonal Poly- nomials, Lecture Notes in Statistics, vol. 102, Springer-Verlag, New York, 1995.
A. K. Gupta: Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, OH 43403-0221, USA
E-mail address:[email protected]
D. G. Kabe: Department of Mathematics and Computer Science, Saint Mary’s University, Halefax, Nova Scotia, Canada B3H 3C3
