J. Japan Statist. Soc. Vol.18 No.2 1988 123-130
EXACT MOMENTS OF THE MULTIVARIATE F AND BETA DISTRIBUTIONS
Yoshihiko Konno*
Based on integration by parts formula, for the multivariate F and Beta distribu-
tions, second order moments of the matrices and those of inverse are obtained.
1. Introduction
Closely related to the Wishart distribution are the multivariate F and Beta distribu- tions, which are generalization of usual F and Beta distributions in much the same way that the Wishart distribution generalizes the X2 distribution. Some of their properties are similar to those of the Wishart distribution, which were discussed in several papers such as Khatri [5], Olkin and Rubin [8], Tan [10], Mitra [6], De Waal [2], Perlman [9], and Dawid [1].
Let F and B be m×m positive definite random matrices. Then the multivariate F and Beta densities are defined by
(1.1) and
(1.2) respectively, where
n=n1+n2, n1>m+1, n2>m+1, and ⊿ is a positive definite matrix of parameters. We denote these distributions by F (m;n1, n2;⊿) and B (m;n1, n2;⊿) respectively.
In Section 2 of this paper, for a matrix V (F, ⊿) and a scalar h (F), an identity is obtained for EF [h (F) tr (⊿+F)-1V] from which second order moments EF [FAF] for an m×mmatrix A is derived. This identity is a generalization of an identity due to Muirhead and Verathaworn [7] which was introduced in the problem of estimating the latent roots of Σ1Σ-1 2 of two Wishart matrices. We also present alternative derivation (pointed out by Professor Sinha) of moments of F matrix by using Bayes prior approach.
In Section 3, similar type of identity for the multivariate Beta distribution is obtained from which we derive second order moments and those of inverse Beta distribution.
The F identity and Beta identity may be regarded as modification of the Wishart
Received December, 1987. Accepted February, 1988. * Institute of Mathematics , University of Tsukuba, Ibaraki 305, Japan. 124 J. JAPAN STATIST. SOC. Vol.18 No.2 1988
identity due to Haft [4], which is widely used in the problem of estimating a covariance matrix. The F and Beta moments include, as a special case, the Wishart and inverse
Wishart moments in Haff [4]. The results obtained in this paper will be useful for sta-
tistical inference on parameters of the multivariate F and Beta distributions.
2. An identity for the multivariate F distribution
For Qm×m=(qij), put ‖Q‖=(Σ Σq2 ij)1/2 and Q(1/2)=((1/2) qij/(1+δij)). Let Dm×m be a matrix of differential operators given by ((1/2)(1十 δij)∂/∂Fij). Then an identity for the
multivariate F distribution which is similar to the Wishart identity due to Haff [3,4] is
given in the following Theorem.
THEOREM 2.1. Let F follow F (m;n1,n2;⊿)distribution defined by (1.1). For a matrix Vm×m=(Vij (F, ⊿)) and a scalar h (F), assume that
(i) the function Vij (F, ⊿) h (F) satisfies the conditions of the Stoke's theorem on regions
(i) on b1(〓)={F;F〓0, ‖F‖=ρ1}
(ii) on b2(〓)={F;F〓0, ‖F‖=ρ2}
Then we have,for n=n1+n2,
(2.1)
OUTLET OF PROOF. Put
and
Let D* be an m×m matrix of differential operators with respect to Fij given by (∂/∂Fij).
From Lemma 2.2 in Haff [3], we have
(2.2)
when ηm×m=(cosηij), ηij=ηji, cosηij is the ij-th component of the outwardunit nomal on
b(〓)=b1 (〓)∪b2(〓), and dr is the differential surface area. Noting that ∂/∂F|F|=
(2-δij)Fij|F|where F-1=(Fij), the direct calculation shows that the left side of (2.2) becomes EXACT MOMENTS OF THE MULTIVARIATE F AND BETA DISTRIBUTIONS 125
and the second term of the right hand side except for the sign provides
Under the conditions (ii) and (iii),
then we obtain (2.1).
REMARK 1. When h (F)is a constant, (2.1) reduces to equation (5) in Muirhead and
Verathaworn [7].
By using the identity (2.1), we shall compute second order moments for the F matrix. We need the following Lemma.
LEMMA 2.2.
(i)
(ii)
where Qm×m is an arbitrary constant matrix.
PROOF. (i) is from Haff [4]. For (ii), put Q=eiet j where et is the i-th unit column vector. The direct calculation shows that
which completes the proof.
THEOREM 2.3. Let F follow F (m;n1, n2; ⊿) distribution and put⊿=(⊿ij), then
(i)
(ii)
PROOE (i) In the equation (2.1), set h (F)=l and V=(⊿+F)ejei tF. If n2-m-1 >0, each expectation in (2.1) exists and all the conditions in Theorem 2 .1 are satisfied. Then use (i) and (ii) of Lemma 2.2 to the first term in the right hand side of (2 .1).
(ii) Set h (F)=Fij and V=(⊿+F)etek tF. It is also seen that each expectation exists in (2.1) and h (F) and V (F, ⊿) satisfy all the conditions in Theorem 2 .1, if n2-m-3 126 J. JAPAN STATIST. SOC. Vol.18 No.2 1988
>0. The lefthand side of the equation (2.1)becomes E [FijFkl].Using (i)and (ii)of Lemma 2.2,the firstterm of the righthand sideprovides
Noting that ∂h (F) ∂F=(eiej t+ejei t)/(1+δij), the second term of the right hand side be-
comes
and
Combining these equationsand using (i)of Theorem 2.3 lead to
(2.3)
In similar way, from h(F)=Fik and V=(⊿+F)etej tF, we obtain
(2.4)
and from h (F)=Fu and V=(⊿+F)ejek tF,
(2.5)
Thus E [FijFkl]is determined by the linearequations of (2.3),(2.4), and (2.5).
COROLLARY 2.4. If F follows F (m; nt, n2; ⊿) distribution and n2-m-3>0, then
(i)
(ii)
for any m×m matrix A.
PROOF. From Theorem 2.3, direct calculation leads to the result.
The results of Theorem 2.3 and Corollary 2.4 include the Wishart moments in Haff [4] as a special case. Put F*=n2F in Corollary 2.4 and assume that A is symmetric, then we get EXACT MOMENTS OF THE MULTIVARIATE F AND BETA DISTRIBUTIONS 127
which is equal to E [WAW] when W has the Wishart distribution Wm (n1, ⊿), since F*
converges to W weakly.
We derived second order moments of the multivariate F distribution by using the
identity (2.1), but Professor Sinha pointed out that combining the Wishart moments by
Haff [4] with the inverted Wishart prior on a covariance matrix also gives the same result
without using the identity (2.1). Namely, assuming that
the joint density of W and Σ1 is
(2.6)
where n=n1+n2. By making a transformation φ=(W+⊿)1/2Σ-1(W+⊿)1/2, we have
the joint density of W andφ
(2.7)
Furthermore, by integrating out (2.7) with respect to φ, it is seen that the marginal density
of W becomes F (m; n1, n2; ⊿). It follows that second order moments of the multivariate
F distribution can be calculated by
(2.8)
where W=(wij). First, integrating (2.8) with respect to W having the Wishart distribu-
tion Wm (n1, Σ), gives
(2.9)
From Haff [4],we get
(2.10)
where Σ-1 follows Wm (n2. ⊿-1) distribution . By exchanging k with j or j with l, similar formula for E [ΣikΣjt] or E [ΣitΣkj] is obtained . This gives (ii) of Theorem 2.3.
If F has F (m; n1, n2; ⊿) distribution, then F-1 has F(m; n2, n1; ⊿-1) distribution. Immediately, Theorem 2.3 and Corollary 2 .4 give the following inverse moments .
THEOREM 2.5. If F follows F (m; n1, n2; ⊿) disEribution and n1-w-3>0, then
(i)
(ii) 128 J. JAPAN STATIST. SOC. Vol.18 No.2 1988
(iii)
(iv)
where A is any m×m matrix, F-1=(Fij) and ⊿-1=(⊿ij).
3. An Identity for the multivariate beta distribution
The multivariate Beta distribution may be regarded as an extension of the Wishart distribution. When B follows B (m; n1, n2; ⊿) distribution, we shall show an identity for
E [g (B) tr (⊿-B)-1 T (B, ⊿)], where T m×m is a matrix and g (B) is a scalar. Then we derive
second order moments of B.
THEOREM 3.1. Let B follow B (m; n1, n2; ⊿) distribution defined by (1.2). For a matrix T (B, ⊿) m×m=(Tij (B, ⊿)) and a scalar g (B), assume that
(i) the function Tij (B, ⊿) g (B) satisfies the conditions of the Stoke's theorem on regions
(i) on b1(〓)={B;0〓B〓 ⊿, |B|=ρ1}
(ii) on b2(〓)={B;0〓B〓 ⊿, ||⊿-B|=ρ2}
Then wee have
PROOF. Put
and
In (2.2), replace Fij by Bij. Then similar argument in the proof of Theorem 2.1 gives the identity.
Using this identity, we obtain moments of the Beta matrix in the same way as in
Section 2. We state the following results without proof. B*=n2B converges to the
Wishart distribution Wm (n1, ⊿) as n2 tends to infinity. Therefore the following results are give the moments of the Wishart distribution in Haff [4]. EXACT MOMENTS OF THE MULTIVARIATE F AND BETA DISTRIBUTIONS 129
THEOREM 3.2. If B follows B (m; n1, n2; ⊿) distribution, then
(i)
(ii)
COROLLARY 3.3. If B fbllows B (m; n1, n2; ⊿) distribution, then
(i)
(ii)
for any m×m matrix A.
It is pointed out in Tan [10] that, if F follows F (m; n1, n2; ⊿) distribution, then ⊿ (⊿+F)-1F follows B (m; n1, n2; ⊿) distribution and if B follows B (m; n1, n2; ⊿) distribu-
tion, then ⊿ (⊿-B)-1B follows F (m; n1, n2; ⊿) distribution. The distribution of F-1 is
equal to that of B-1 (⊿-B)⊿-1. Then, a simple calculation shows that F-1+⊿-1 and
B-1 have the same distribution. From the identity F-1+⊿-1=B-1, it can be seen that
Theorem 2.5 gives the following Theorem. Of course, the Beta identity in Theorem 3.1
provides the same results.
THEOREM 3.4. If B follows B (m; n1, n2; ⊿) distribution and n1-m-3>0, then
(i)
(ii)
(iii)
(iv)
for any m×m matrix A.
Mitra [6] shows that for any fixed m×l vector b(≠0), the ratio btBb/bt⊿b and bt⊿-1b/
btB-1b have standard Beta distributions with parameters n1/2 and n2/2 ,(n1-m-1)/2
and n2/2 respectively, which leads to (i) of Theorems 3 .2 and 3.4.
Acknowledgment
The auther would like to thank Professor Nariaki Sugiura for his many helpful com- ments and encouragement, and is also grateful to Professor B . K. Sinha for his valuable comments. 130 J. JAPAN STATIST. SOC. Vol.18 No.2 1988
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