Journal of Probability and Statistical Science 4(2), 233-244, Aug. 2006

Product Moments of Bivariate Wishart Distribution Anwar H. Joarder King Fahd University of Petroleum & Minerals ABSTRACT The moments of the multivariate Wishart distribution is known up to the fourth order. But the product moments of the elements, namely, the two sample variances and the correlation coefficient of the bivariate Wishart distribution, are not available in general. In this paper some product moments of an arbitrary order are derived for the three elements of the bivariate Wishart distribution. A theorem containing eight identities of infinite series involving product of two gamma functions has been established to facilitate the derivation. The general nature of the theorem indicates its use in other contexts. Keywords Bivariate Wishart distribution; Product moments, Correlation coefficient.

1. Introduction The moments of the multivariate Wishart distribution is known up to the fourth order. Some researchers have derived moments of some special type of functions, namely, sample trace and determinant of Wishart Matrix which have had applications in estimation theory. Only a few special cases of product moments are published in the literature which are mostly considered for applications in correlation analysis. This paper aims at deriving some product moments of the three elements, namely, two sample variances and the correlation coefficient of bivariate Wishart matrix. A theorem containing eight identities of infinite series involving product of two gamma functions has been established to facilitate the derivation. This will lead to generalization of some of the published results in the area. Let X 1 , X 2 ,L X N ( N > p) be a p-dimensional independent normal random vector with mean vector X so that the sums of squares and cross product matrix is given by N

∑(X j =1

j

− X )( X j − X )′ = A .

_______________________ □ Received June 2005, revised March 2006, in final form May 2006. □ Anwar H. Joarder is an Associate Professor in the Department of Mathematical Sciences at King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia; email: anwarj@kfupm. edu.sa. © 2006 Susan Rivers’ Cultural Institute, Hsinchu, Taiwan, Republic of China.

ISSN 1726-3328

JPSS Vol. 4 No.2 August 2006

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pp. 233-244

The random symmetric positive definite matrix A is said to have a Wishart distribution with parameters p, m = N − 1 > p and Σ( p × p) > 0 , written as A ~ W p (m , Σ) if its pdf (probability density function) is given by

⎛ 1 ⎞ | A |( m − p −1) / 2 exp ⎜ − trΣ −1 A ⎟ ⎝ 2 ⎠ f ( A) = , A > 0, m > p p ⎛1 ⎞ mp / 2 p ( p −1) / 4 m/2 2 π | Σ | ∏ Γ ⎜ (m + 1 − i ) ⎟ ⎝2 ⎠ i =1 (see, e.g., Anderson [1], p. 252). The symmetric matrix A for the bivariate case, i.e., for p = 2 , can be written as A = ( aik ), i = 1, 2; k = 1, 2 where N

aii = mS ii = mS i2 = ∑ (X ij − X i ) 2 , m = N − 1, (i = 1, 2) , j =1

N

a12 = ∑ (X 1 j − X 1 )(X 2 j − X 2 ) = mS 12 = mRS 1S 2 , j =1

and the quantity R = S12 /( S1 S 2 ) = a12 /( a11a 22 )1 / 2 is the sample product moment correlation coefficient. We want to derive the moments E ( S12 l1 S 22 l2 R l ) for finite l1 , l 2 , l . Some of them may not have closed forms. This paper may help estimate some parametric functions of the elements of Σ = (σ ik ), σ ii = σ i2 , σ ik = ρσ i σ k (i = 1, 2; k = 1, 2) where σ 1 > 0, σ 2 > 0 , and ρ , (−1 < ρ < 1) , is the product moment correlation coefficient between X 1 and X 2 . Fisher [3] derived the distribution of A for p = 2 in order to study the distribution of correlation coefficient from a normal sample. Wishart [8] obtained the distribution for arbitrary p as the joint distribution of sample variances and covariances from the multivariate normal population. Because of its important role in multivariate statistical analysis, various authors have derived it from different perspectives (see the references in Gupta and Nagar [5], pp. 87-88). The first moment of A, trA, det( A), A−1 and similar beautiful quantities are known (Muirhead [7]). A nice update of moments of the Wishart distribution is given by Gupta and Nagar [5]. But very few product moments can be deduced from the works published so far. Hence we derive some product moments of the bivariate Wishart distribution when the parent variables have the bivariate normal distribution.

2. The General Product Moments of Sample Variances and the Correlation Coefficient The moments of sample variances S i2 (i = 1, 2) are well known. The moments of product moment correlation coefficient R are also known though they do not have closed forms.

Product Moments of Bivariate Wishart Distribution

Anwar H. Joarder

235

Ghosh [4] expressed the moments of R in terms of hypergeometric functions. The first moment is the simplest one and is given by 2

2 ⎛ Γ((m + 1) / 2) ⎞ E ( R) = ⎜ ⎟ ρ m ⎝ Γ(m / 2) ⎠

2

⎛1 1 m+2 2⎞ F1 ⎜ , , ; ρ ⎟ , −1 < ρ < 1, m > 1 . ⎝2 2 2 ⎠

There are a number of representations of the moments of the correlation coefficient in Johnson et al. ([6], p.553). Understandably the derivation of the joint moments of S12l1 , S22l2 and R l will be formidably difficult. By differentiating the moment generating function of A ~ W p (m , Σ) , m > p , de Waal and Nel [2] derived the following results: (a )E (A 2 ) = m ( (m + 1)Σ + (tr Σ)I p ) Σ,

(

)

(b )E (A 3 ) = m (m 2 + 3m + 4)Σ 2 + 2(m + 1)(tr Σ)Σ + (m + 1)(tr Σ 2 )I p + (tr Σ) 2 I p Σ. Some product moments, e.g., E (S 1S 23R ), E (S 13S 2 R ) and E ( S12 S22 R 2 ) , can be deduced from the above identity in (a). Theorem 2.1 For finite l1 , l2 , and l, the product moments E ( S12 l1 S 22 l2 R l ) denoted by µ ′(l1 , l 2 , l ; ρ ) are given by

µ ′(l1 , l 2 , l ; ρ ) =

2l1 + l 2 1− ρ 2 l1 + l 2 m L (m , ρ )

(

)

l1 + l 2

σ 12 l σ 22 l 1

2

⎛ k +1+ l ⎞ Γ⎜ ⎟ 2 ⎛k +m ⎞ ⎛k +m ⎞ ⎝ ⎠ , k 2 ×∑ρ Γ⎜ + l1 ⎟ Γ ⎜ + l2 ⎟ k! ⎝ 2 ⎠ ⎝ 2 ⎠ Γ⎛ k + m + l ⎞ k =0 ⎜ ⎟ 2 ⎝ ⎠ ∞

k

where L (m , ρ ) = π Γ(m / 2) (1 − ρ 2 )

−m / 2

(2.1)

, m > 2, σ 1 > 0, σ 2 > 0, and −1 < ρ < 1 .

Proof. The pdf of the elements of A can be written as

(1 − ρ )

2 −m / 2

f 1 (a11 , a22 , a12 ) =

2

m

(σ 1σ 2 ) − m

π Γ(m / 2)Γ((m − 1) / 2)

(a a

11 22

− a122

)

( m − 3) / 2

⎛ 1 ⎞ ⎛ 1 ⎞ ⎛ ⎞ a11 a22 ρa12 × exp ⎜ − exp ⎜ − exp ⎜ ⎟ 2 2 ⎟ 2 2 ⎟ 2 ⎝ 2 (1 − ρ )σ 1 ⎠ ⎝ 2 (1 − ρ )σ 2 ⎠ ⎝ (1 − ρ )σ 1σ 2 ⎠ where a11 > 0, a22 > 0, −∞ < a12 < ∞, −1 < ρ < 1 , m > 2, σ 1 > 0, σ 2 > 0 . Under the transformation a11 = ms12 , a22 = ms22 , a12 = mrs1s2 with Jacobian m3 s1s2 , the joint pdf of S12 , S 22 and R is given by

JPSS Vol. 4 No.2 August 2006

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⎛ 1 ms12 ⎞ s1m − 2 exp ⎜ − 2 2 ⎟ 4πΓ(m − 1) ⎝ 2 (1 − ρ )σ 1 ⎠ ( m −3) / 2 ⎛ 1 ms 22 ⎞ ⎛ m ρ rs1s 2 ⎞ × s 2m − 2 exp ⎜ − 1− r 2 exp ⎜ ⎟ 2 2 ⎟ 2 ⎝ 2 (1 − ρ )σ 2 ⎠ ⎝ (1 − ρ )σ 1σ 2 ⎠

(2.2)

236

f 2 (s , s , r ) = 2 1

2 2

(

m m 1− ρ 2

)

−m / 2

(σ 1σ 2 ) − m

(

)

where we have also used the duplication formula of gamma function 2z −1 ⎛ z + 1 ⎞ ⎛ z ⎞ Γ(z ) = Γ⎜ ⎟Γ⎜ ⎟ π ⎝ 2 ⎠ ⎝2⎠

with z = m − 1 . By expanding the last term, we have

(

m m 1− ρ 2

f 3 (s , s , r ) = 2 1

2 2

)

−m / 2

(σ 1σ 2 ) − m

⎛ ⎛ s12 s 22 ⎞ ⎞ m exp ⎜ − + 2 ⎟⎟ 2 ⎜ 2 ⎜ ⎟ ⎝ 2(1 − ρ ) ⎝ σ 1 σ 2 ⎠ ⎠

4π Γ(m − 1)

(m ρ ) k (s1s 2 ) m − 2+ k k r 1− r 2 k 2 k k =0 (1 − ρ ) (σ 1σ 2 ) k ! ∞

(

×∑

)

( m − 3) / 2

.

Then the product moments µ ′(l1 , l 2 , l ; ρ ) are given by

µ ′(l1 , l 2 , l ; ρ ) =

(

m m 1− ρ 2

)

−m / 2

(σ 1σ 2 ) − m

4π Γ(m − 1)

(m ρ )k ×∑ 2 k k k =0 (1 − ρ ) (σ 1σ 2 ) k ! ∞

∫

∞

0

s

m + k + 2 l1 − 2 1

⎛ 1 ms12 ⎞ 2 exp ⎜ − ds1 2 2 ⎟ ⎝ 2 (1 − ρ )σ 1 ⎠

∞ ⎛ 1 ms 22 ⎞ 2 × ∫ s 2m + k + 2 l 2 − 2 exp ⎜ − ds 2 2 2 ⎟ 0 − ρ σ 2 (1 ) 2 ⎠ ⎝

∫

1

−1

(

r k +l 1 − r 2

)

( m −3) / 2

dr .

That is

µ ′(l1 , l 2 , l ; ρ ) =

(

m m 1− ρ 2

)

−m / 2

(σ 1σ 2 ) − m

4π Γ(m − 1)

∞ (m ρ )k ×∑ s12 2 k k ∫ 0 k =0 (1 − ρ ) (σ 1σ 2 ) k ! ∞

×∫

∞

0

( )

( ) s 22

( m + k ) / 2 + l 2 −1

which can further be evaluated as

( m + k ) / 2 + l1 −1

⎛ 1 ms12 ⎞ 2 exp ⎜ − ds1 2 2 ⎟ ⎝ 2 (1 − ρ )σ 1 ⎠

⎛ 1 ms 22 ⎞ 2 1 ( k + l −1) / 2 ( m − 3) / 2 ds 2 ∫ u du exp ⎜ − (1 − u ) 2 2 ⎟ 0 − ρ σ 2 (1 ) 2 ⎠ ⎝

Product Moments of Bivariate Wishart Distribution

µ ′(l1 , l 2 , l ; ρ ) =

l1 + l 2 + m

2 4π m l1 + l 2 ∞

×∑

Anwar H. Joarder

⎛ m −1 ⎞ Γ⎜ ⎟ ⎝ 2 ⎠ 1− ρ 2 Γ ( m − 1)

(

(2 ρ ) k ⎛ k + m k!

k =0

Γ⎜ ⎝

2

)

l1 + l 2 + m / 2

σ 12 l σ 22 l 1

237

2

⎛ k +1+ l ⎞ Γ⎜ ⎟ k m 2 ⎞ ⎛ + ⎞ ⎠ . + l1 ⎟ Γ ⎜ + l2 ⎟ ⎝ ⎠ ⎝ 2 ⎠ Γ⎛ k + m + l ⎞ ⎜ ⎟ 2 ⎝ ⎠

The pdf of the sample correlation coefficient originally derived by Fisher [3] has the following representation. Theorem 2.2 The probability density function of the correlation coefficient R is given by h( r ) =

2

m− 2

Γ ( m / 2)(1 − ρ ) π Γ( m − 1) 2

2 m/2

2 ( m−3) / 2

(1 − r )

E (e

ρr U 1U 2

), − 1 < r < 1

where m > 2, −1 < ρ < 1 and the expectation is over the variables U i ~ χ m2 (i = 1, 2) . Proof. By making the transformation

ms12 ms 22 u = , = u2 σ 12 (1 − ρ 2 ) 1 σ 22 (1 − ρ 2 ) 2 2 2 2 2 with Jacobian J ( s1 , s 2 → u1 , u 2 ) = (σ 1σ 2 / m) (1 − ρ ) in (2.2), we have

f 4 (u1 , u 2 , r ) =

(1 − ρ ) 2 ( m − 3 ) / 2 ρr (1 − r ) e 4πΓ( m − 1) 2 m/2

u1u2

(u1u 2 )

( m / 2 ) −1

e

− ( u1 + u2 ) / 2

.

Then the theorem is obvious by the integration over u1 and u 2 as the following: h( r ) =

∞∞

(1 − ρ ) ρr 2 ( m−3) / 2 e (1 − r ) ∫ ∫ 4πΓ( m − 1) 0 0 2 m/2

u1u2

(u1u 2 )

( m / 2 ) −1

e

− ( u1 + u2 ) / 2

du1du 2 .

Note that by expanding the term e ρ r u1u 2 and completing the integration over u1 and u 2 we have the following well known probability density function of R :

h (r ) =

(

2m − 2 1 − ρ 2

)

m /2

π Γ(m − 1)

(1 − r ) 2

(2 ρ r ) k 2 ⎛ m + k ⎞ Γ ⎜ ∑ ⎟ , −1 < r < 1 k! ⎝ 2 ⎠ k =0

( m −3) / 2 ∞

where m > 2, −1 < ρ < 1 (Muirhead [7], p. 154).

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3. Some Special Cases of Product Moments µ( l1 , l 2 , l ; ρ) To evaluate the product moments given by Theorem 2.1 for some special cases we need the following theorem. In what follows, for any nonnegative integer k, we define: c {k } = c (c − 1) L (c − k + 1), c {0} = 1 . 2k ⎛ k + 1 ⎞ ⎛ k + m ⎞ Γ⎜ ⎟Γ⎜ ⎟ , m > 2, k > 0. Then for L( m, ρ ), m > 2, k! ⎝ 2 ⎠ ⎝ 2 ⎠ − 1 < ρ < 1 defined in Theorem 2.1, we have

Theorem 3.1 Let bk ,m =

∞

∑ρ

(i)

b k ,m = L (m , ρ ),

k

k =0 ∞

(ii)

∑k ρ

k

k =0 ∞

(iii)

∑k

{2}

k =0 ∞

(iv)

∑k

{3}

k =0

(

b k ,m = m ρ 2 1 − ρ 2

)

−1

L (m , ρ ) = w 1 (m , ρ ) L (m , ρ ),

ρ k b k ,m = ( m (m + 1) ρ 4 + m ρ 2 )(1 − ρ 2 ) L (m , ρ ) = w {2} (m , ρ )L (m , ρ ), −2

ρ k b k ,m = w {3} (m , ρ )L (m , ρ ) where

( = ( m (m + 1)(m + 2) ρ

)(

w {3} (m , ρ ) = (m 3 + 3m 2 + 2m ) ρ 6 + (3m 2 + 6m ) ρ 4 1 − ρ 2

∞

(v)

∑k

{4}

k =0

6

)(

+ 3m (m + 2) ρ 4 1 − ρ 2

)

)

−3

−3

,

ρ k b k ,m = w {4} (m , ρ )L (m , ρ ) where

(

w {4} (m , ρ ) = m ⎡⎣(m 3 + 6m 2 + 11m + 6) ρ 8 + (6m 2 + 30m + 36) ρ 6 + (3m + 6) ρ 4 ⎤⎦ 1 − ρ 2

(

)

−4

= ⎡⎣ m (m + 1)(m + 2)(m + 3) ρ 8 + 6m (m + 2)(m + 3) ρ 6 + 3m (m + 2) ρ 4 ⎤⎦ 1 − ρ 2 ∞

(vi)

∑k

2

k =0

∑k

3

k =0

ρ k b k ,m = ( 4m ρ 2 + (6m 2 + 4m ) ρ 4 + m 3 ρ 6 )(1 − ρ 2 ) L (m , ρ ) = w 3 (m , ρ )L (m , ρ ),

∑k k =0

,

−2

−3

∞

(viii)

−4

ρ k b k ,m = ( m 2 ρ 4 + 2m ρ 2 )(1 − ρ 2 ) L (m , ρ ) = w 2 (m , ρ ) L (m , ρ ),

∞

(vii)

)

4

ρ k bk ,m = w4 (m, ρ ) L(m, ρ )

w4 ( m, ρ ) = [( m 4 + 18m 2 + 12m) ρ 8 + (12m 3

where

2 −4

− 20m2 + 8m) ρ + ( 46m − 4m) ρ + 32mρ ](1 − ρ ) . 6

2

4

2

Product Moments of Bivariate Wishart Distribution

Anwar H. Joarder

239

Proof. Since

µ ′(0, 0, 0; ρ ) =

∞ 1 ∑ ρ k b k ,m , L (m , ρ ) k =0

the proof of (i) is obvious by virtue of µ ′(0, 0, 0) = 1 where µ ′(l1 , l 2 , l ; ρ ) is defined in (2.1). The identity in (i) can be rewritten as ∞

∑ρ k =0

k

(

b k ,m = L (m , 0) 1 − ρ 2

)

−m / 2

.

(3.1)

Differentiating both sides of the identity in Theorem 3.1 (i) with respect to ρ we have ∞

∑kρ k =0

(

bk ,m = mL(m, 0) ρ 1 − ρ 2

k −1

)

− m / 2 −1

(3.2)

which yields the identity in (ii) above. Differentiating the identity in (3.2) again we have ∞

∑ k (k − 1) ρ

k −2

k =0

{

(

)

(

)

b k ,m = mL (m , 0) ⎡ ρ (−m / 2 − 1) 1 − ρ 2 ⎢⎣ = mL (m , 0) ⎡⎣1 + (m + 1) ρ 2 ⎤⎦ 1 − ρ 2

− m / 2− 2

− m / 2− 2

}

(

( −2 ρ ) + 1 − ρ 2

)

− m / 2 −1

⎤ ⎥⎦ (3.3)

.

which yields the identity in (iii) above. Differentiating the identity in (3.3) we have ∞

∑ k (k − 1)(k − 2) ρ k =0

k −3

bk ,m

){

(

(

= mL(m, 0) ⎡ 1 + (m + 1) ρ 2 (− m / 2 − 2) 1 − ρ 2 ⎢⎣

)

(

− m / 2 −3

= mL(m, 0) ⎡⎣ (m 2 + 3m + 2) ρ 3 + (3m + 6) ρ ⎤⎦ 1 − ρ 2

)

}

(

(−2 ρ ) + {(m + 1)2 ρ } 1 − ρ 2

)

− m / 2− 2

⎤ ⎥⎦

− m / 2 −3

(3.4) which yields the identity in (iv). Differentiating the identity in (3.4) we have ∞

∑ k (k − 1)(k − 2)(k − 3) ρ k =0

k −4

bk ,m

){

(

(

= mL(m, 0) ⎡ (m 2 + 3m + 2) ρ 3 + (3m + 6) ρ (− m / 2 − 3) 1 − ρ 2 ⎢⎣ − m / 2 −3 ⎤ + (m 2 + 3m + 2)3ρ 2 + (3m + 6) 1 − ρ 2 ⎥⎦

{

}(

)

− m / 2− 4

}

( −2 ρ )

)

(

= mL(m, 0) ⎡⎣ (m3 + 6m 2 + 11m + 6) ρ 4 + (6m 2 + 30m + 36) ρ 2 + (3m + 6) ⎤⎦ 1 − ρ 2 which yields the identity in (v). The identity in (vi) can be proved as follows:

)

− m / 2− 4

JPSS Vol. 4 No.2 August 2006

240 ∞

∑k

pp. 233-244

∞

2

k =0

ρ k b k ,m = ∑ ( k {2} + k )ρ k b k ,m k =0

(

)

= w {2} (m , ρ ) + w 1 (m , ρ ) L (m , ρ )

(

= ⎡w {2} (m , ρ ) 1 − ρ 2 ⎢⎣

)

2

(

(

)

(

2 + w 1 (m , ρ ) 1 − ρ 2 ⎤ L (m , ρ ) 1 − ρ 2 ⎥⎦

)

(

)

(

= ⎡⎣ (m 2 + m ) ρ 4 + m ρ 2 + m ρ 2 1 − ρ 2 ⎤⎦ L (m , ρ ) 1 − ρ 2

)

)

−2

−2

The identity in (vii) can be proved as follows: ∞

∞

k =0

k =0

(

)

∑ k 3 ρ k bk ,m = ∑ k {3} + 3k {2} + k ρ k bk ,m

(

)

= w {3} (m , ρ ) + 3w {2} (m , ρ ) + w 1 (m , ρ ) L (m , ρ )

(

= ⎡w {3} (m , ρ ) 1 − ρ 2 ⎣⎢

)

3

(

+ 3w {2} (m , ρ ) 1 − ρ 2

(

)

3

) (

(

)

(

3 + w 1 (m , ρ ) 1 − ρ 2 ⎤ L (m , ρ ) 1 − ρ 2 ⎦⎥

)(

= ⎡⎣ (m 3 + 3m 2 + 2m ) ρ 6 + (3m 2 + 6m ) ρ 4 + 3 (m 2 + m ) ρ 4 + m ρ 2 1 − ρ 2

(

)

(

2 + m ρ 2 1 − ρ 2 ⎤ L (m , ρ ) 1 − ρ 2 ⎥⎦

)

−3

)

.

The identity in (viii) can be proved as follows: ∞

∞

k =0

k =0

(

)

∑ k 4 ρ k bk ,m = ∑ k {4} + 6k {3} + 7k {2} + k ρ k bk ,m

(

)

= w {4} (m , ρ ) + 6w {3} (m , ρ ) + 7w {2} (m , ρ ) + w 1 (m , ρ ) L (m , ρ )

(

)

(

)

(

)

= ⎡w {4} (m , ρ ) 1 − ρ 2 + 6w {3} (m , ρ ) 1 − ρ 2 + 7w {2} (m , ρ ) 1 − ρ 2 ⎢⎣ 4 −4 + w 1 (m , ρ ) 1 − ρ 2 ⎤ L (m , ρ ) 1 − ρ 2 ⎥⎦ = ⎡⎣ (m 4 + 6m 3 + 11m 2 + 6m ) ρ 8 + (6m 3 + 30m 2 + 36m ) ρ 6 + (3m 2 + 6m ) ρ 4

(

4

)

(

4

)

( + 6 ( m + 3m + 2m ) ρ + (3m + 6m ) ρ )(1 − ρ ) + 7 ( (m + m ) ρ + m ρ )(1 − ρ ) + m ρ (1 − ρ ) ⎤ L (m , ρ ) (1 − ρ ) ⎥⎦ 3

2

2

6

4

2

2

2

4

4

2

2

)

2

2

3

2

−4

= w 4 (m , ρ )L (m , ρ ) where w {i } (m , ρ ) and wi (m, ρ ), (i = 1, 2,3, 4) are defined in the theorem. By Direct application of Theorem 3.1 to Theorem 2.1, special cases of moments µ ′(l1 , l 2 , l ; ρ ) of Wishart distribution having probability density function given by (2.2) can be calculated. Some are tabulated below where in general m > 2, σ 1 > 0, σ 2 > 0, −1 < ρ < 1 .

)

−2

Product Moments of Bivariate Wishart Distribution

Anwar H. Joarder

Case I: (One order is negative)

µ ′(−2, 0, 0; ρ ) =

m 2σ 1−4 , m > 4, (m − 2)(m − 4)

µ ′(−2,1, 0; ρ ) =

m (−4 ρ 2 + m )σ 1−4σ 22 (m − 2)(m − 4)

⎛ ⎞ −4 2 ρ2 m =m⎜ + ⎟ σ 1 σ 2 , m > 4, 2 (m − 2)(m − 4) ⎠ ⎝ (m − 2) ρ

µ ′(−2,1, 2; ρ ) = m

1 + (m − 5) ρ 2 −4 2 σ1 σ 2 , m > 4 , (m − 2)(m − 4)

µ ′(−2, 2, 0; ρ ) = ( 24 ρ 4 − 8(m + 2) ρ 2 + m (m + 2) )

(σ 2 / σ 1 ) 4 (m − 2)(m − 4)

⎛ 24(1 − ρ 2 ) 2 8(1 − ρ 2 ) ⎞ =⎜ + + 1⎟ (σ 2 / σ 1 ) 4 , m > 4, m −2 ⎝ (m − 2)(m − 4) ⎠

µ ′(−2, 2, 2; ρ ) = ( 6(5 − m ) ρ 4 + (m 2 − m − 24) ρ 2 + (m + 2) )

(σ 2 / σ 1 ) 4 (m − 2)(m − 4)

⎛ 6(1 − ρ 2 ) 2 ⎞ (1 + 6 ρ 2 )(1 − ρ 2 ) =⎜ + + ρ 2 ⎟ (σ 2 / σ 1 ) 4 , m > 4, m −2 ⎝ (m − 2)(m − 4) ⎠

µ ′(−2, 2, 4; ρ ) = ( (m 2 − 12m + 35) ρ 4 + 6(m − 5) ρ 2 + 3) )

(σ 2 / σ 1 ) 4 (m − 2)(m − 4)

⎛ 3(1 − ρ 2 ) 2 ⎞ 6 ρ 2 (1 − ρ 2 ) =⎜ + + ρ 4 ⎟ (σ 2 / σ 1 ) 4 , m > 4, m −2 ⎝ (m − 2)(m − 4) ⎠

µ ′(−1, 0, 0; ρ ) =

m σ 1−2 , m > 2, (m − 2) 2

2

m − 2 ρ 2 ⎛ σ 2 ⎞ ⎛ 2(1 − ρ 2 ) ⎞ ⎛ σ 2 ⎞ µ ′(−1,1, 0; ρ ) = + 1⎟ ⎜ ⎟ , m > 2 , ⎜ ⎟ =⎜ m − 2 ⎝ σ1 ⎠ ⎝ m − 2 ⎠ ⎝ σ1 ⎠ 2

(m − 3) ρ 2 + 1 ⎛ σ 2 ⎞ ⎛ 2 1 − ρ 2 ⎞ ⎛ σ 2 ⎞ µ ′(−1,1, 2; ρ ) = ⎜ ⎟ =⎜ρ + ⎟⎜ ⎟ m −2 m − 2 ⎠ ⎝ σ1 ⎠ ⎝ σ1 ⎠ ⎝

µ ′(−1, 2, 0; ρ ) = ( 8 ρ 4 − 2(m + 3) ρ 2 + m (m + 2) )

σ 1−2σ 24 m (m − 2)

2

,

m > 2,

, m > 2,

µ ′(−1, 2, 2; ρ ) = ( (−2m + 6) ρ 4 + (m 2 + m − 14) ρ 2 + (m + 2) )

σ 1−2σ 24 m (m − 2)

, m > 2,

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µ ′(0, −2, 0; ρ ) =

m2 σ 2−4 , m > 4 , (m − 2)(m − 4)

µ ′(0, −1, 0; ρ ) =

m σ 2−2 , m > 2, m −2

µ ′(1, −2, 2; ρ ) =

m ( (m − 5) ρ 2 + 1) (m − 2)(m − 4)

σ 12σ 2−4 , m > 4 ,

µ ′(2, −1, 0; ρ ) = ( 8ρ 4 − 4(m + 2) ρ 2 + m (m + 2) )

1 σ 14σ 2−2 , m > 2 , m (m − 2)

µ ′(2, −1, 2; ρ ) = ( −4(m − 3) ρ 4 + (m 2 + m − 14) ρ 2 + (m + 2) )

1 σ 14σ 2−2 , m > 2 . m (m − 2)

Case II: (Each order is nonnegative)

µ ′(0,1, 0; ρ ) = σ 22 , µ ′(0,1, 2; ρ ) = ( (m − 1) ρ 2 + 1)

σ 22 m

,

µ ′(0, 2, 2; ρ ) = ( −2(m − 1) ρ 4 + (m − 1)(m + 4) ρ 2 + (m + 2) ) µ ′(0, 2, 4; ρ ) = ( (m − 1)(m − 3) ρ 4 + 6(m − 1) ρ 2 + 3) µ ′(1,1, 0; ρ ) = ( m + 2 ρ 2 )

σ 12σ 22 m

σ 24 m2

σ 22 m2

,

,

,

µ ′(1, 0, 0; ρ ) = σ 12 , µ ′(1,1, 2; ρ ) = ( (m + 1) ρ 2 + 1) µ ′(2, 0, 0; ρ ) =

σ 12σ 22 m

,

m +2 4 σ1 , m

µ ′(2, 0, 2; ρ ) = ( −2(m − 1) ρ 4 + (m 2 + 3m − 4) ρ 2 + (m + 2) ) µ ′(2,1, 0; ρ ) =

m +2 (m + 3) ρ 2 + 1 σ 14σ 22 , m2

µ ′(2,1, 2; ρ ) =

m +2 (m + 3) ρ 2 + 1 σ 14σ 22 , m2

(

(

pp. 233-244

)

)

σ 14 m2

,

Product Moments of Bivariate Wishart Distribution

Anwar H. Joarder

µ ′(2, 2, 0; ρ ) =

m +2 8 ρ 4 + 8(m + 2) ρ 2 + m (m + 2) σ 14σ 24 , 3 m

µ ′(2, 2, 2; ρ ) =

m +2 4(m + 3) ρ 4 − (m 2 + 7 m + 4) ρ 2 + m σ 14σ 24 . m3

(

243

)

(

)

Corollary 3.1 The sample product moment correlation between S 12 and S 22 is given by ρ 2 . Proof. From the above we have, E (S 12 ) = µ ′(1, 0, 0) = σ 12 , E (S 22 ) = µ ′(0,1, 0) = σ 22 ,

m +2 4 σ1 , m m +2 4 σ2 , E (S 24 ) = µ ′(0, 2, 0) = m E (S 14 ) = µ ′(2, 0, 0) =

E (S S ) = µ ′(1,1, 0) = 2 1

2 2

m + 2ρ 2 (σ 1σ 2 ) 2 . m

The variances of S 12 and S 22 are given by 2σ 14 2σ 24 2 and V (S 2 ) = V (S ) = m m 2 1

respectively, while the covariance between them is given by Cov (S 12 , S 22 ) = E (S 12S 22 ) − E (S 12 )E (S 22 ) =

2ρ . m

Hence, the product moment correlation coefficient between S 12 and S 22 is given by

Corr (S 12 , S 22 ) =

Cov (S 12 , S 22 ) 2) 1/ 2 2

⎡⎣V (S ) V (S ⎤⎦ 2 1

= ρ2 .

Since the correlation between X 1 and X 2 is ρ , the above result is intuitively appealing. Note that if ρ = 0 , the pdf in (2.2) becomes the product of that of S 12 , S 22 and R.

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pp. 233-244

Acknowledgments The author acknowledges the excellent research support provided by King Fahd University of Petroleum & Minerals, Saudi Arabia especially through the Fast Track Research Project # FT 2004-22. The author is also thankful to the referees for making constructive suggestions which led to this improved version of the paper.

References [1] Anderson, T. W. (2003). An Introduction to Multivariate Statistical Analysis, John Wiley and Sons, New York. [2] de Waal and Nel, D. J. (1973). On some expectations with respect to Wishart matrices, South African Statistics Journal, 7, 61-67. [3] Fisher, R. A. (1915). Frequency distribution of the values of the correlation coefficient in samples from an indefinitely large population, Biometrika, 10, 507-521. [4] Ghosh, B. K. (1966). Asymptotic expansions for the moments of the distribution of correlation coefficient, Biometrika, 53, 258-262. [5] Gupta, A. K. and Nagar, D. K. (2000). Matrix Variate Distributions, Chapman and Hall, New York. [6] Johnson, N. L.; Kotz, S. and Balakrishnan, N. (1995). Continuous Univariate Distributions Volume 2, John Wiley and Sons, New York. [7] Muirhead, R. J. (1982). Aspects of Multivariate Statistical Theory, John Wiley and Sons, New York. [8] Wishart, J. (1928). The generalized product moment distribution in samples from a normal multivariate population, Biometrika, A20, 32-52.