Matrix variate beta distribution

Vol. 24, No. 7 (2000) 449–459 S0161171200002398 © Hindawi Publishing Corp.

MATRIX-VARIATE BETA DISTRIBUTION

ARJUN K. GUPTA and DAYA K. NAGAR

(Received 24 September 1997)

Abstract.We propose matrix-variate beta type III distribution. Several properties of this distribution including Laplace transform, marginal distribution and its relationship with matrix-variate beta type I and type II distributions are also studied.

Keywords and phrases. Matrix variate, beta distribution, zonal polynomial, transforma-tion, Laplace transform.

2000 Mathematics Subject Classification. Primary 62H10; Secondary 62E15.

1. Introduction. The random variableuwith the probability density function (pdf)

β(a,b)−1ua−1_(1−u)b−1_{, 0< u <1, (1.1)} wherea >0 andb >0, is said to have a beta type I distribution with parameters(a,b). The random variablevwith pdf,

β(a,b)−1va−1_(1+v)−(a+b)_{, v >0, (1.2)} wherea >0 andb >0, is said to have beta type II distribution with parameters(a,b). Since (1.2) can be obtained from (1.1) by the transformation v =u/(1−u), some authors call the distribution ofvan “inverted beta distribution.”

The matrix variate generalizations of (1.1) and (1.2) are given as follows (see [1, 3, 4, 6, 11]).

Definition1.1. Ap×prandom symmetric positive definite matrixU is said to have a matrix-variate beta type I distribution with parameters(a,b), denoted asU∼

BI

p(a,b), if its pdf is given by

βp(a,b)−1det(U)a−(p+1)/2_detIp−Ub−(p+1)/2_{, 0< U < Ip, (1.3)} wherea > (p−1)/2, b > (p−1)/2, and βp(a,b) is the multivariate beta function given by

βp(a,b)=Γp(a)Γp(b)_Γp(a+b) , (1.4)

where

Γp(a)=πp(p−1)/4p j=1 Γ

a−j−₂1

Definition1.2. Ap×prandom symmetric positive definite matrixV is said to have a matrix-variate beta type II distribution with parameters(a,b), denoted asV∼

BII

p(a,b), if its pdf is given by

βp(a,b)−1det(V)a−(p+1)/2_detIp+V−(a+b)_{, V >0, (1.6)}

wherea > (p−1)/2,b > (p−1)/2, andβp(a,b)is the multivariate beta function.

As in the univariate case, the density (1.6) can be obtained from (1.3) by transforming U=(Ip+V)−1_{V, together with the JacobianJ(U→V)=det(Ip+V )}−(p+1)_.

The matrix-variate beta type I and type II distributions have been studied by many authors, e.g., see [7, 9, 10, 13, 14].

In this paper, a new matrix-variate beta distribution has been defined. We call it “Matrix-variate beta type III” distribution, which is then derived by using matrix trans-formation. Several properties of this distribution and its relationship with matrix-variate beta type I and type II distributions have also been studied.

2. Density function. First, we define the matrix-variate beta distribution of type III. Definition2.1. Ap×prandom symmetric positive definite matrixWis said to have a matrix-variate beta type III distribution with parameters (a,b), denoted as W∼BIII

p(a,b), if its pdf is given by

2pb_βp(a,b)−1_{det(W )}b−(p+1)/2

×detIp−Wa−(p+1)/2detIp+W−(a+b), 0< W < Ip, (2.1)

wherea > (p−1)/2,b > (p−1)/2, andβp(a,b)is the multivariate beta function. Forp=1, the beta type III density is given by

2b_β(a,b)−1_wb−1_(1−w)a−1_(1+w)−(a+b)_{, 0< w <1, (2.2)}

and in this case we writew∼BIII_(a,b).

By means of a bilinear transformation of the random matrixU, the matrix-variate beta type III distribution is generated as in the following theorem.

Theorem2.1. LetU∼B_p(a,b)I _{. DefineW=(Ip+U)}−1_{(Ip−U). ThenW∼B}III p(a,b).

Proof. Making the transformationW=(Ip+U)−1_{(Ip−U)with the Jacobian [12]} in the pdf (1.3) ofU, J(U→W )=2p(p+1)/2_{(Ip+W )}−(p+1)_{, we get the desired result.}

From the density of beta type III matrix it is apparent that

0<W <Ipdet(W )

b−(p+1)/2_detIp−Wa−(p+1)/2

The cumulative distribution function (cdf) ofWis obtained as

P(W <Ω)=2pb_βp(a,b)−1 0<W <Ω

det(W )b−(p+1)/2_detIp−Wa−(p+1)/2

×detIp+W−(a+b)dW .

(2.4)

For evaluating the above integral, we use the following results involving zonal poly-nomials [2, 5, 8]:

detIp−Wa−(p+1)/2= ∞

k=0

κ

−a+(p+1)/2κ

k! Cκ(W ), (2.5)

detIp+W−(a+b)= ∞

t=0

τ

(a+b)τ

t! (−1)tCτ(W ), (2.6)

Cκ(W )Cτ(W )=

δ gδ

κ,τCδ(W ), (2.7)

0<W <Ip

det(W )a−(p+1)/2_detIp−Wb−(p+1)/2_{Cκ(RW )dW}

=βp(a,b)_(a(a)κ_+b)κCκ(R), Re(a) >p₂−1,Re(b) >p−₂1,

(2.8)

whereκ=(k1,...,kp),k1≥ ··· ≥kp≥0,k1+ ··· +kp=k,τ=(t1,...,tp),t1≥ ··· ≥

tp≥0,t1+ ··· +tp=t,δ=(d1,...,dp),d1≥ ··· ≥dp≥0,d1+ ··· +dp=d=k+t,

andgδ

κ,τis the coefficient ofCδ(W )inCκ(W )Cτ(W ). Substituting (2.5) and (2.6) in (2.4) and subsequently using (2.7), we obtain

P(W <Ω)=2pb_βp(a,b)−1∞ k=0

∞

t=0

κ

τ

−a+(p+1)/2κ

k!

×(a+_t!b)τ(−1)t

δ gδ

κ,τ

0<W <Ω

det(W )b−(p+1)/2_{Cδ(W )dW .} (2.9)

Now substitutingX=Ω−1/2_WΩ−1/2_{with the JacobianJ(W →X)=det(Ω)}(p+1)/2_in the above integral, we get

P(W <Ω)=2pb_βp(a,b)−1_det(Ω)b∞ k=0

∞

t=0

κ

τ

−a+(p+1)/2κ

k!

×(a+b)τ

t! (−1)t

δ gδ κ,τ 0<X<Ip

det(X)b−(p+1)/2_Cδ(ΩX)dX

=2pbΓp(a+b)Γp

(p+1)/2

Γp(a)Γpb+(p+1)/2 det(Ω)b ∞

k=0 ∞

t=0

κ

τ

−a+(p+1)/2κ

k!

×(a+_t!b)τ(−1)t

δ gδ

κ,τ_b+(p(b)δ_+1)/2

δCδ(Ω),

(2.10)

The Laplace transform of the density ofWis

L(Z)=2pb_βp(a,b)−1 0<W <Ip

etr(−ZW )det(W )b−(p+1)/2

×detIp−Wa−(p+1)/2detIp+W−(a+b)dW ,

(2.11)

whereZ(p×p)=((1+δij)zij/2). Now, using the expansions

detIp+W−(a+b)=2−p(a+b)_detIp−Ip−W 2

−(a+b)

=2−p(a+b)∞

t=0

τ

(a+b)τ t!

₁

2

t

CτIp−W,

etr(−ZW )=etr(−Z)etrZIp−W=etr(−Z) ∞

k=0

κ 1 k!Cκ

ZIp−W, (2.12)

in (2.11), we obtain

L(Z)=2−pa_βp(a,b)−1_etr(−Z)∞ k=0

∞

t=0

κ

τ 1 k!

(a+b)τ

2t_t! Φ(Z), (2.13)

where

Φ(Z)=

0<W <Ip

det(W )b−(p+1)/2_detIp−Wa−(p+1)/2_{CκZIp−WCτIp−WdW}

=

0<X<Ip

detIp−Xb−(p+1)/2det(X)a−(p+1)/2_{Cκ(ZX)Cτ(X)dX.}

(2.14)

SinceΦ(Z)=Φ(H_{ZH), H∈O(p), integrating outH inΦ(HZH)using [5,} equa-tion 23], we have

Φ(Z)= Cκ(Z)

CκIp

0<X<Ip

detIp−Xb−(p+1)/2det(X)a−(p+1)/2_{Cκ(X)Cτ(X)dX. (2.15)}

Now using (2.7), and integratingXusing (2.8), we obtain

Φ(Z)=_CκCκ(Z)

Ip

δ gδ

κ,τ_(a(a)δ_+b)δΓp(a)Γp(b)_Γp(a+b) Cδ

Ip. (2.16)

Substituting (2.16) in (2.13), we finally obtain

L(Z)=2−pa_etr(−Z)∞ k=0

∞

t=0

κ

τ 1 k!

(a+b)τ 2t_t!

Cκ(Z) CκIp

δ gδ

κ,τ_(a(a)δ_+b)δCδIp. (2.17)

Theorem3.1. LetW ∼BIII

p(a,b)andA(p×p)be a constant nonsingular matrix.

Then the density ofX=AW A_is

2pb_βp(a,b)−1_det(AA)(p+1)/2_det(X)b−(p+1)/2

×det(AA_−X)a−(p+1)/2_det(AA+X)−(a+b)_{, 0< X < AA.} (3.1) Proof. Making the transformation X = AW A _{with the Jacobian J(W → X)=} det(A)−(p+1)_{in the pdf (2.1) ofW, the density ofXis obtained as}

2pb_βp(a,b)−1_det(AA)(p+1)/2_det(X)b−(p+1)/2

×det(AA_−X)a−(p+1)/2_det(AA+X)−(a+b)_{, 0< X < AA.} (3.2) which is the desired result.

We will writeX∼BIII

p(a,b;AA). In the next theorem, it is shown that the matrix-variate beta distribution of type III is orthogonally invariant.

Theorem3.2. LetW∼BIII

p(a,b)andH (p×p)be an orthogonal matrix, whose

ele-ments are either constants or random variables distributed independently ofW. Then, the distribution ofW is invariant under the transformationW→HW H_{, and does not}

depend onH.

Proof. First, let H be a constant orthogonal matrix. Then, from Theorem 3.1, HW H_∼BIII

p(a,b)sinceHH=Ip. If, however,His a random orthogonal matrix, then the conditional distribution ofHUH_|H∼BIII

p(a,b). Since this distribution does not depend onH,HW H_∼BIII

p(a,b).

The relationship between beta type I, type II, and type III matrices is now exhibited. First, we derive the density ofW−1_.

Theorem3.3. LetW∼BIII

p(a,b). Then the density ofY=W−1is

2pb_βp(a,b)−1_detY−Ipa−(p+1)/2_detIp+Y−(a+b)_{, Y > Ip. (3.3)} Proof. Making the transformation Y = W−1 _{with the Jacobian J(W → Y ) =} det(Y )−(p+1)_{, in the density ofWthe result follows.}

The density derived above may be called the inverse beta type III. From Theorem 3.3, it is clear that ifW∼BIII

p(a,b), thenW−1does not follow beta type III distribution. Theorem3.4. (i) LetU∼BI

p(a,b)andW=(Ip+U)−1(Ip−U), thenW∼BpIII(a,b). (ii) Similarly, ifW∼BIII

p(a,b)andU=(Ip+W )−1(Ip−W ), thenU∼BpI(a,b). Proof. Part (i) is proved in Theorem 2.1. SinceW=(Ip+U)−1_{(Ip−U), the} Jaco-bian of transformation isJ(W →U)=2p(p+1)/2_det(Ip+U)−(p+1)_{. Now, making the} substitution in the density ofWgiven by (2.1) the result (ii) follows.

Theorem3.5. (i) LetV∼BII_{p(a,b)andW=(Ip+2V)}−1_{, thenW∼B}III p(a,b). (ii) Similarly, ifW∼BIII

Theorem3.6. LetW = W11W12 W21W22

, W11(q×q), and W22·1=W22−W21W11−1W12. If W∼BIII

p(a,b), thenW22·1∼BpIII−q(a,b−q/2). Proof. From the partition ofW, we have

det(W )=det(W11)detW22·1, (3.4)

detIp−W=detIq−W11detIp−q−W22·1−W21W11−1Iq−W11−1W12, (3.5) detIp+W=detIq+W11detIp−q+W22·1+W21W11−1Iq+W11−1W12. (3.6)

Now making the transformation W11 = W11, X = W21W11−1/2, and W22·1 = W22− W21 W11−1 W12 = W22−XX with Jacobian J(W11, W22, W21 → W11, W22·1, X) = det(W11)(p−q)/2_{and substituting (3.4), (3.5), and (3.6) in the density ofW, we get the} joint density ofW11,W22·1, andXas follows:

2pb_βp(a,b)−1_det(W11)b−(q+1)/2_detIq−W11a−(p+1)/2_detIq+W11−(a+b)

×detW22·1b−(p+1)/2detIp−q−W22·1a−(p+1)/2detIp−q+W22·1−(a+b)

×detIp−q−Ip−q−W22·1−1XIq−W11−1Xa−(p+1)/2

×detIp−q+Ip−q+W22·1−1XIq+W11−1X−(a+b).

(3.7)

Substituting Y = (Ip−q−W22·1)−1/2X(Iq−W11)−1/2 with Jacobian J(X → Y ) = det(Ip−q−W22·1)q/2det(Iq−W11)(p−q)/2 and integratingY, we get the joint density ofW11andW22·1as

2pb_βp(a,b)−1_det(W11)b−(q+1)/2_detIq−W11a−(q+1)/2_detIq+W11−(a+b)

×det(W22·1)b−(p+1)/2detIp−q−W22·1a−(p−q+1)/2detIp−q+W22·1−(a+b)g(A,B), (3.8)

where

g(A,B)=

Ip−q−Y Y_>0

detIp−q−Y Ya−(p+1)/2detIp−q+AY BY−(a+b)dY

=

0<Z<Ip−q

Y Y_=Z

detIp−q−Y Ya−(p+1)/21F0(q)(a+b;−YAY B)dY dZ, forp−q≤q,

=

0<Z<Iq

Y_Y=Z

detIq−Y_Ya−(p+1)/2

1F0(p−q)(a+b;−AY BY)dY dZ, forp−q > q, (3.9)

g(A,B)=

0<Z<Ip−q

Y Y_=Z

detIp−q−Y Ya−(p+1)/21F0(q)(a+b;−AY Y,B)dY dZ

=_Γpπq(p−q)/2 −q(q/2)

0<Z<Ip−q

det(Z)(q−p+q−1)/2_detIp

−q−Za−(p+1)/21F0(q)(a+b;−AZ,B)dZ

= πq(p−q)/2

Γp−q(q/2)·

Γp−q(q/2)Γp−qa−q/2 Γp−q(a) 2F

(q) 1

_q

2,a+b;a;−A,B

=πq(p−q)/2Γp−q

a−q/2 Γp−q(a) 2F

(q) 1

_q

2,a+b;a;−A,B

. (3.10)

Substitutingg(A,B)in (3.8), we get the joint density ofW11andW22·1as 2qb_βq(a,b)−1_det(W11)b−(q+1)/2_detIq−W11a−(q+1)/2

×detIq+W11−(a+b)2(p−q)(b−q/2)_βp

−q

a,b−q₂ −1

detW22·1b−(p+1)/2

×detIp−q−W22·1a−(p−q+1)/2detIp−q+W22·1−(a+b)2(p−q)q/2

×2F1(q)

_q

2,a+b;a;−A,B

.

(3.11)

ClearlyW11andW22·1are not independent. IntegratingW11, using 2qb

0<W11<Iq

det(W11)b−(q+1)/2_detIq−W11a−(q+1)/2_detIq+W11−(a+b)

×2F1(q)

_q

2,a+b;a;−A,B

dW11

=

0<B<Iq

det(B)a−(q+1)/2_detIq−Bb−(q+1)/2 2F1

_q

2,a+b;a;−A,B

dB

=βq(a,b)3F2(q)

a,q₂,a+b;a,a+b;−A

=βq(a,b)detIp−q+A−q/2

=2−(p−q)q/2_{βq(a,b)detIp}

−q+W22·1q/2, (3.12)

we get the density ofW22·1. Forq < p−q, using (3.9) and following similar steps we get the same result.

Alternately, Theorem 3.6 can be proved using the relationship between type II and type III beta matrices. LetV∼BII

p(a,b)andV=

_V 11V12 V21V22

,V11(q×q)andV11·2=V11− V12V22−1V21. It is well known thatV11·2andV22are distributed independently (see [4]), V11·2∼BqII(a−(p−q)/2,b)andV22∼BpII−q(a,b−q/2). According to Theorem 3.5(i), if V∼BII

p(a,b), thenW=(Ip+2V )−1∼BIIIp(a,b). Furthermore, W−1₌

W11 _W12 W21 _W22

=

Iq+2V11 2V12 2V21 Ip−q+2V22

. (3.13)

That is,W22_{((p−q)×(p−q))=W}−1

Now, sinceV22∼BII

p−q(a,b−q/2), thenW22·1=(Ip−q+2V22)−1∼BpIII−q(a,b−q/2). The distribution of(AW−1_A)−1_{whereA(q×p)is a constant matrix of rankq (≤p),} is now derived.

Theorem3.7. LetA(q×p)be a constant matrix of rankq (≤p). IfW∼BIII p(a,b),

then(AW−1_A)−1_∼BIII

q(a,b−(p−q)/2;(AA)−1).

Proof. We writeA=M(Iq 0)G, whereM (q×q)is nonsingular andG (p×p)is orthogonal. Now,

(AW−1_A)−1_{=MIq 0GW}−1_{GIq 0M}−1

=(M₎−1

Iq 0Y−1

Iq 0

−1 M−1

=(M₎−1_(Y11₎−1_M−1_,

(3.14)

whereY =Y11Y12 Y21Y22

=GW G_∼BIII

p(a,b),Y11(q×q)andY11=(Y11−Y12Y22−1Y21)−1= Y−1

11·2. From Theorem 3.6, Y11·2 ∼ BqIII(a,b−(p−q)/2) and from Theorem 3.1, (M₎−1_Y22

·1M−1∼BqIII(a,b−(p−q)/2;(MM)−1). The proof is now completed by ob-serving thatMM_=AA.

Corollary3.8. LetW∼BIII

p(a,b)anda∈ p,a≠0, thenaa/aW−1a∼BIII(a,b− (p−1)/2).

In Corollary 3.8, the distribution ofa_a/aW−1_{a does not depend ona. Thus, if} y(p×1)is a random vector, independent ofW, andP(y≠0)=1, then it follows that y_y/yW−1_y∼BIII_{(a,b−(p−1)/2).}

Theorem3.9. LetW∼BIII

p(a,b), then

(i) E

det(W )h detIp+Wh

=₂Γp(a_phΓp(b)(a+b)Γp(b_+b+₊h)_h), Re(h) >−b+p−₂1,

(ii) Edet(W )h

=2−paΓp(a+b)Γp(b+h)

Γp(b)Γp(a+b+h) 2F1(a,a+b;a+b+h, 1

2Ip), Re(h) >−b+p−₂1, (iii) EdetIp−Wh

=2−_Γp(a)Γp(apaΓp(a+b)Γp(a_+b+h)+h)2F1(a+h,a+b;a+b+h,1₂Ip), Re(h)>−a+p−₂1,

where2F1is the hypergeometric function of matrix argument.

Proof. (i) From the density ofW, we have

E

det(W )h detIp+Wh

=2pb_βp(a,b)−1 0<W <Ip

det(W )b+h−(p+1)/2_detIp−Wa−(p+1)/2

×detIp+W−(a+b+h)dW

=2pb_βp(a,b)−1βp(a,b+h)

2p(b+h) , Re(h) >−b+ p−1

Simplifying this last expression using (1.4), we get the desired result. (ii) From the density ofW, we have

Edet(W )h₌₂pb_βp(a,b)−1

×

0<W <Ip

det(W )b+h−(p+1)/2_detIp−Wa−(p+1)/2_detIp+W−(a+b)_{dW .} (3.16)

Writing det(Ip+W )−(a+b)_{in series involving zonal polynomials using (2.12), we} ob-tain

Edet(W )h

=2−pa_βp(a,b)−1∞ t=0

τ

(a+b)τ 2t_t!

×

0<W <Ip

det(W )b+h−(p+1)/2_detIp−Wa−(p+1)/2_{Cτ(Ip−W )dW}

=2−paβp(a,b+h) βp(a,b)

∞

t=0

τ

(a+b)τ st_t! ·

aτ (a+b+h)τCτ

Ip, Re(h) >−b+p−₂1, (3.17)

where the integral has been evaluated using (2.8). Finally, simplifying the expression using results on hypergeometric functions [2, 5], we get the desired result.

SimilarlyE[det(Ip−W )h_{]can be derived.} From the density ofW, we have

ECκ(W ) =2pb_βp(a,b)−1 0<W <Ip

Cκ(W )det(W )b−(p+1)/2

×detIp−Wa−(p+1)/2detIp+W−(a+b)dW .

(3.18)

Writing det(Ip+W )−(a+b)_{in series involving zonal polynomials using (2.6), we obtain}

ECκ(W ) =2pb_βp(a,b)−1∞ t=0

τ

(a+b)τ t! (−1)t

×

0<W <Ip

Cκ(W )Cτ(W )det(W )b−(p+1)/2_detIp−Wa−(p+1)/2_dW

=2pb_βp(a,b)−1∞ t=0

τ

(a+b)τ t! (−1)t

δ gδ

κ,τ

×

0<W <Ip

Cδ(W )det(W )b−(p+1)/2_detIp−Wa−(p+1)/2_dW

=2pb∞

t=0

τ

(a+b)τ t! (−1)t

δ gδ

κ,τ_(a(b)δ_+b)δCδIp,

(3.19)

Theorem3.10. (i) LetU∼BI_{p(a,b)andW∼B}III

p(c,a+b)be independent. Then Y=W1/2_UW1/2_{,0< Y < Iphas the density}

2p(a+b)_βp(a,b+c)−1_{det(Y )}a−(p+1)/2_detIp−Yb+c−(p+1)/2

×detIp+Y−(a+b+c)2F1b,a+b+c;b+c;−Ip+Y−1Ip−Y, 0< Y < Ip. (3.20)

Further, pdf ofX=(Ip+Y )−1_{(Ip−Y )=(Ip+W}1/2_UW1/2₎−1_(Ip−W1/2_UW1/2_)is

2pb_βp(a,b+c)−1_det(X)b+c−(p+1)/2_detIp−Xa−(p+1)/2

×2F1(b,a+b+c;b+c;−X), 0< X < Ip. (3.21)

(ii) LetU∼BI

p(a,b)andV∼BpI(a+b,c)be independent. ThenY=(Ip+V1/2UV1/2)−1

×(Ip−V1/2_UV1/2_)∼BIII

p(a,b+c).

Proof. The joint density ofUandVis

2p(a+b)_{βp(a,b)βp(c,a+b)}−1_det(U)a−(p+1)/2_detIp−Ub−(p+1)/2

×det(W )a+b−(p+1)/2_detIp−Wc−(p+1)/2

×detIp+W−(a+b+c), 0< U < Ip,0< W < Ip.

(3.22)

Making the transformationY=W1/2_UW1/2_{with JacobianJ(U→Y )=det(W )}−(p+1)/2 in above, we get the joint density ofY andWas

2p(a+b)_{βp(a,b)βp(c,a+b)}−1_{det(Y )}a−(p+1)/2_{det(W−Y )}b−(p+1)/2

×detIp−Wc−(p+1)/2detIp+W−(a+b+c), 0< Y < W < Ip. (3.23)

Substituting Z =(Ip−Y )−1/2_{(W−Y )(Ip−Y )}−1/2 _{with the Jacobian J(W →Z)=} det(Ip−Y )(p+1)/2_{and integratingZ, we get the density ofY as}

2p(a+b)_{βp(a,b)βp(c,a+b)}−1_{det(Y )}a−(p+1)/2

×detIp−Yb+c−(p+1)/2detIp+Y−(a+b+c)

×

0<Z<Ip

det(Z)b−(p+1)/2_detIp−Zc−(p+1)/2

×detIp+Ip+Y−1Ip−YZ−(a+b+c)dZ.

(3.24)

Integration ofZusing [5, equation 48] completes the proof.

(ii) From [6], we haveV1/2_UV1/2_∼BI_{p(a,b+c)and from Theorem 2.1, we have} (Ip+V1/2_UV1/2₎−1_(Ip−V1/2_UV1/2_)∼BIII

Acknowledgement. This research was supported by the Comité para el Desar-rollo de la Investigación, Universidad de Antioquia research grant no. IN91CE.

References

[1] Y. Asoo,On theΓ-distribution of matrix argument and its related distribution, Natural Science2(1969), 1–13, Memoirs of Faculty of Literature and Sciences, Shimane University. Zbl 184.22304.

[2] A. G. Constantine,Some non-central distribution problems in multivariate analysis, Ann. Math. Statist.34(1963), 1270–1285. MR 31#5285. Zbl 123.36801.

[3] A. P. Dawid, Some matrix-variate distribution theory: notational considerations and a Bayesian application, Biometrika 68 (1981), no. 1, 265–274. MR 83m:62083. Zbl 464.62039.

[4] A. K. Gupta and D. K. Nagar,Matrix Variate Distributions, Chapman & Hall/CRC, Boca Raton, FL, 2000. CMP 1 738933.

[5] A. T. James,Distributions of matrix variates and latent roots derived from normal samples, Ann. Math. Statist.35(1964), 475–501. MR 31#5286. Zbl 121.36605.

[6] W. R. Javier and A. K. Gupta,On generalized matric variate beta distributions, Statistics 16(1985), no. 4, 549–558. MR 86m:62023. Zbl 602.62039.

[7] C. G. Khatri,A note on Mitra’s paper “A density-free approach to the matrix variate beta distribution”, Sankhy¯a Ser. A32(1970), 311–318. Zbl 225.62016.

[8] C. G. Khatri and K. C. S. Pillai,On the non-central distributions of two test criteria in multi-variate analysis of variance, Ann. Math. Statist.39(1968), 215–226. MR 36#2265. Zbl 155.27102.

[9] R. Khattree and R. D. Gupta,Some probability distributions connected with beta and gamma matrices, Comm. Statist. Theory Methods 21 (1992), no. 2, 369–390. MR 93b:62081. Zbl 800.62266.

[10] Y. Konno,Exact moments of the multivariateF and beta distributions, J. Japan Statist. Soc.18(1988), no. 2, 123–130. MR 90f:62166. Zbl 669.62031.

[11] S. K. Mitra,A density-free approach to the matrix variate beta distribution, Sankhy¯a Ser. A32(1970), 81–88. MR 45#2832. Zbl 203.51906.

[12] I. Olkin,Note on ‘the Jacobians of certain matrix transformations useful in multivariate analysis’, Biometrika40(1953), 43–46. MR 15,94b. Zbl 052.14904.

[13] I. Olkin and H. Rubin,Multivariate beta distributions and independence properties of the Wishart distribution, Ann. Math. Statist.35(1964), 261–269. MR 28#3511. [14] W. Y. Tan,Note on the multivariate and the generalized multivariate beta distributions, J.

Amer. Statist. Assoc.64(1969), 230–241. MR 39#2244. Zbl 175.17307.

Arjun K. Gupta: Department of Mathematics and statistics, Bowling Green State University, Bowling Green, Ohio43403-0221, USA