DHAHRAN 31261 ● SAUDI ARABIA ● www.kfupm.edu.sa/math/ ● E-mail: mathdept@kfupm.edu.sa

King Fahd University of Petroleum & Minerals

DEPARTMENT OF MATHEMATICAL SCIENCES

Technical Report Series

TR 356

Sep 2006

Some Distributional Results in Two Correlated Chisquare Variables

Anwar H Joarder

Some Distributional Results in Two Correlated Chisquare Variables

Anwar H Joarder

Department of Mathematical Sciences King Fahd University of Petroleum & Minerals

Dhahran 31261 Saudi Arabia Email: anwarj@kfupm.edu.sa

Abstract Ratios of two independent chisquare variables are widely used in statistical tests of hypotheses. This paper ushers a horizon of statistical investigation where the assumption of independence is not met. Moments of the product and ratio of correlated chisquare variables are outlined. Distributions of the sum and product of two correlated chisquares are also derived. AMS Mathematics Subject Classification: 60E05, 60E10, 62E15 Key Words and Phrases: Chisquare distribution, Wishart distribution, product moments, Bivariate distribution, Correlation

1. Introduction Let 1 2, , ( 2)NX X X N >L be a two-dimensional independent normal random vectors with mean vector 1 2( , )X X X ′= so that the sums of squares and cross product matrix is given by

1( )( )

N

j jj

X X X X A=

′− − =∑ . Let the matrix A for the bivariate case be denoted by

( ), 1, 2; 1,2ikA a i k= = = where 2 2

1( ) , 1, ( 1, 2)

N

ii i ij ij

a ms X X m N i=

= = − = − =∑ and

12 1 1 2 2 1 21

( )( )N

j jj

a X X X X mrs s=

= − − =∑ . Also let the elements of the matrix ( )ikσΣ = ,

1, 2; 1,2i k= = where 2 211 1 22 2 12 1 2, ,σ σ σ σ σ ρσ σ= = = with 1 20, 0σ σ> > . The quantity ρ

( 1 1)ρ− < < is the product moment correlation coefficient between 1X and 2X . Fisher (1915) derived the distribution of the bivariate Wishart matrix in order to study the distribution of correlation coefficient for a bivariate normal sample. Wishart (1928) obtained the distribution of Wishart matrix as the joint distribution of sample variances and covariances from multivariate normal population. The bivariate matrix A is said to have a Wishart distribution with parameters 1m N= − and (2 2) 0Σ × > , written as 2~ ( , )A W m Σ if its probability density function is given by

( )

( 3) / 2 1

* 2/ 2

1

1| | exp2( )

2 | | ( 1 ) / 2

m

m m

i

A tr Af A

m iπ

− −

=

⎛ ⎞− Σ⎜ ⎟⎝ ⎠=

Σ Γ + −∏, 0, 2A m> >

(See e.g. Anderson, 2003, 252). The pdf of the elements of A can be written as

2

( )2 / 2 ( 3) / 221 2

11 22 12 11 22 12

11 22 122 2 2 2 2

1 2 1 2

(1 ) ( )( , , ) 2 ( / 2) (( 1) / 2)

exp2(1 ) 2(1 ) 2(1 )

m m m

mf a a a a a a

m m

a a a

ρ σ σπ

ρρ σ ρ σ ρ σ σ

− − −−= −

Γ Γ −

⎛ ⎞× − − +⎜ ⎟− − −⎝ ⎠

(1.1)

11 22 120, 0, , 2, 1 1a a a m ρ> > −∞ < < ∞ > − < < (Anderson, 2003, 123).

Because of its important role in multivariate statistical analysis, various authors have given different derivations. See the references in Gupta and Nagar (2000, 87-88) for a good update on the moments of Wishart distribution. In this paper we deduce some properties of a bivariate chisquare distribution of 2 2

1 1/U mS σ= and 2 2

2 2/V mS σ= introduced by Joarder (2005a). In particular, moments of the product and ratio of two correlated chisquare variables are also derived. Distributions of the sum and product of correlated chisqaures are also derived. Ratios of two independent chisquares are widely used in statistical tests of hypotheses. This paper ushers a horizon of statistical investigation where the assumption of independence is not met. In corollary 2.6, we have provided moments of the ratio of correlated chisquare variables. Further investigation is needed to utilize these moments to derive the distribution of the ratio of chisquares or many other interesting properties by the inverse Mellin transformation along Provost (1986). In case the correlation coefficient 0ρ = , the probability density function (pdf) of U and V becomes that of the product of two independent chisquare variables each with m degrees of freedom, and the product moments obtained for the correlated case are found to be in agreement with the special situation of independence. We refer to Kotz, Balakrishnan and Johnson (2000) for other type of bivariate chisquare distribution. For any nonnegative integer k , the following notations will be used in sequel:

{ }

{ }

( 1)( 2) ( 1),

( 1) ( 1).k

k

a a a a a k

a a a a k

= + + + −

= − − +

L

L

2. The Density Function of the Bivariate Chisquare Distribution The first three of the theorems are due to Joarder (2005a). Theorem 2.1 The random variables U and V are said to have a correlated bivariate chisquare distribution each with m degrees of freedom, if its probability density function is given by

( )( )

2( )

/ 2 1 2(1 )

22 / 2=0

( 1) / 2( ) ( , ) ,1 ! ( ) / 22 ( / 2)(1 )

u vkm

m mk

kuv e uvf u vk k mm

ρ ρρπ ρ

− +− − ∞ ⎛ ⎞ Γ +

= ⎜ ⎟⎜ ⎟− Γ +Γ − ⎝ ⎠∑ (2.1)

31 2m N= − > , 1 1ρ− < < . Note that the above pdf can also be written as

/ 2 12 / 2

2 2 2

(1 ) ( ) ( )( , ) exp exp2 ( / 2) 2(1 ) 1

mm

m

uv u v uv Yf u v Em

ρ ρρ ρ

−− ⎡ ⎤⎛ ⎞⎛ ⎞− − += ⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟Γ − −⎝ ⎠ ⎢ ⎥⎝ ⎠⎣ ⎦

where Y has a beta distribution ( , )B a b with parameters 1/ 2a = and ( 1) / 2b m= − . In case 0ρ = , the pdf of the joint probability distribution in Theorem 2.1, would be that of the product of two independent chisquare random variables given by

( )/ 2 1

( ) / 2

2 ( , ) , 0, 0.2 ( / 2)

mu v

m

uv ef u v u v

m

−− +

= > >Γ

Theorem 2.2 For max( , )m a b> and 1 1ρ− < < , the ( , )a b th product moment ( )a bE U V of the distribution of U and V , is given by

( )( )

2 / 2

0

( 1) / 22 (1 ) (2 )( , ; ) ! 2 2 ( ) / 2 ( / 2)

a b a b m k

k

kk m k ma b a bk k mm

ρ ρµ ρπ

+ + + ∞

=

Γ +− + +⎛ ⎞ ⎛ ⎞′ = Γ + Γ +⎜ ⎟ ⎜ ⎟ Γ +Γ ⎝ ⎠ ⎝ ⎠∑ .

The following theorem is due to Joarder (2006).

Theorem 2.3 Let ,2 1

! 2 2

k

k mk k mb

k+ +⎛ ⎞ ⎛ ⎞= Γ Γ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ and ( ) / 22( , ) ( / 2) 1

mL m mρ π ρ

−= Γ − .

Then for 2, 1 1m ρ> − < < and, we have

,0

( ) ( , )kk m

ki b L mρ ρ

∞

=

=∑

2 2 1, 1

0( ) (1 ) ( , ) ( , ) ( , )k m

kii kb m L m w m L mρ ρ ρ ρ ρ

∞−

=

= − =∑

( ){2} 4 2 2 2, (2)

0( ) ( 1) (1 ) ( , ) ( , ) ( , )k

k mk

iii k b m m m L m w m L mρ ρ ρ ρ ρ ρ ρ∞

−

=

= + + − =∑

{3}, (3)

0( ) ( , ) ( , )k

k mk

iv k b w m L mρ ρ ρ∞

=

=∑ where

( )( )

3 2 6 2 4 2 33

6 4 2 3(3)

( , ) ( 3 2 ) (3 6 ) (1 )

( 1)( 2) 3 ( 2) (1 ) ( , ) ( ) ( , )

w m m m m m m

m m m m m L m w m L m

ρ ρ ρ ρ

ρ ρ ρ ρ ρ

−

−

= + + + + −

= + + + + − =

{4}, (4)

0( ) ( , ) ( , )k

k mk

v k b w m L mρ ρ ρ∞

=

=∑ where

4 3 2 8 3 2 6 2 4 2 4(4)

8 6 4 2 4

( , ) ( 6 11 6 ) (6 30 36 ) (3 6 ) (1 )

( 1)( 2)( 3) 6 ( 2)( 3) 3 ( 2) (1 )

w m m m m m m m m m m

m m m m m m m m m

ρ ρ ρ ρ ρ

ρ ρ ρ ρ

−

−

⎡ ⎤= + + + + + + + + −⎣ ⎦⎡ ⎤= + + + + + + + + −⎣ ⎦

2 2 4 2 2 2, 2

0( ) ( 2 )(1 ) ( , ) ( , ) ( , )k

k mk

vi k b m m L m w m L mρ ρ ρ ρ ρ ρ ρ∞

−

=

= + − =∑

43 3 6 2 4 2 2 3

, 30

( ) ( (6 4 ) 4 )(1 ) ( , ) ( , ) ( , )kk m

kvii k b m m m m L m w m L mρ ρ ρ ρ ρ ρ ρ ρ

∞−

=

= + + + − =∑4

, 40

( ) ( , ) ( , )kk m

kviii k b w m L mρ ρ ρ

∞

=

=∑ where

4 8 3 2 6 2 4 2 2 44 ( , ) (12 16 8 ) (28 32 ) 8 (1 ) .w m m m m m m m mρ ρ ρ ρ ρ ρ −⎡ ⎤= + + + + + + −⎣ ⎦

For any nonnegative integer a , { }2 ( / 2 )( ,0; ) 2 ( / 2)

( / 2)

aa

am aa mm

µ ρ Γ +′ = =Γ

which is the a th

moment of usual chisquare distribution with m degrees of freedom. Similarly, { }(0, ; ) 2 ( / 2)bbb mµ ρ′ = . When both orders a and b are negative, it is difficult to get closed

form expressions for product moments ( , ; ) ( )a ba b E U Vµ ρ′ = . If 0ρ = , then ( , ;0) ( ,0;0) (0, ;0)a b a bµ µ µ′ ′ ′= . It is observed that if b is a nonnegative integer, then

2{ }(1, ; ) 2 ( / 2) ( 2 )bbb m m bµ ρ ρ′ = + .

The following theorem follows from Theorem 2.2. Theorem 2.4 Let U and V have the bivariate chisquare distribution with pdf given by Theorem 2.1. For (i) nonnegative integers a and b , with 2,m > or (ii) 0, 0a b> < with

0m b+ > , the ( , )a b th product moment of the distribution of U and V , is given by

2( ){ }

0{ } { }

( , ; ) 2 (1 )2 2

aa b a j

a jjb j

am ma b b a j a jj

µ ρ ρ+ −−

=

⎛ ⎞⎛ ⎞ ⎛ ⎞′ = + − + + −⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

∑

where 1 1ρ− < < . Corollary 2.1 Let ( )2 2

1 2( , , ; ) a b la b l E S S Rµ ρ′ = be the product moments of the bivariate Wishart distribution with pdf given by (1.1). Then the ( , )-tha b product moment of sample variances 2

1S and 22S can be calculated by

( )2 2

2 2 1 21 2 ( , ,0; ) ( , ; ).

a ba b

a bE S S a b a bmσ σµ ρ µ ρ+

′ ′= =

The following corollary is obvious from Theorem 2.2. Corollary 2.2 Let U and V have a bivariate chisquare distribution with pdf given by (2.1). Also let ( )( , ) ( ) ( )a ba b E U EU V EVµ = − − be the centered product moment between U and V . If a and b are of the same sign, then ( ) ( ) ( )a b b ai E U V E U V= ( ) ( , ) ( , ).ii a b b aµ µ= Corollary 2.3 For 2, 1 1m ρ> − < < , the a th moment of W UV= is given by

5( )( )

2 2

0 { }

( 1) / 24 (1 ) (2 )( ) ( , ) 2 ! 2 ( ) / 2

a a ka

k a

kk m k mE W aL m k k m

ρ ρρ

∞

=

Γ +− + +⎛ ⎞ ⎛ ⎞= Γ +⎜ ⎟ ⎜ ⎟ Γ +⎝ ⎠ ⎝ ⎠∑

where ( , )L m ρ is defined in Theorem 2.3. In case a is an integer,

2( ){ }

0{ } { }

( ) 4 (1 )2 2

aa a a j

a jja j

am mE W j a jj

ρ −−

=

⎛ ⎞⎛ ⎞ ⎛ ⎞= + + −⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

∑

(cf. Joarder, 2005a). In case 0ρ = , then W will be the product of two independent chisquare random variables each with m degrees of freedom and evidently the resulting moments are in agreement with that situation. In particular, if 0ρ = , the mean and variance of W will be ( )E W m= and

2( ) 4 ( 1)Var W m m= + respectively. Let /H U V= , the ratio of two correlated chisqaure variables U and V that have probability density function in Theorem 2.1. Then the following corollary follows from Theorem 2.2. Corollary 2.6 For 2 , 1 1m a ρ> − < < , the -tha moment of H is given by

( )1

0

1 (2 ) 1( , ) ! 2 2 2 2

ka

k

k m k m k k mE H a aL m k

ρρ

−∞

=

⎛ ⎞+ + + +⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= Γ + Γ − Γ Γ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠

∑

where ( , )L m ρ is defined in Theorem 2.3. In case a is a nonnegative integer, we have

( ) 2( ){ } { }

0

( / 2 )( ) ( 2 1) ( / 2) .( / 2)

aa a j

a j jj

am aE H a j m a jjm

ρ −−

=

⎛ ⎞Γ −= − + + + −⎜ ⎟Γ ⎝ ⎠

∑

In case 0ρ = , then H will be the ratio of two independent chisquare variables each m degrees of freedom, and the -tha moment will be simply ( ) { } { }( / 2) /( / 2 ) , 2a

a aE H m m a m a= − > which are evidently in agreement with the situation. In particular, if 0ρ = , the mean and variance of H will be

2

( ) , 2, and24 ( 1)( ) , 4

( 2) ( 4)

mE H mm

m mVar H mm m

= >−

−= >

− −

respectively.

6If X and Y are two random variables, then it is well known that the independence between X and Y implies uncorrelation, but the converse is sometimes true , e.g. in case X and Y have a bivarite normal distribution. The following corollary is another example along the line. Corollary 3.1 Let U and V have a bivariate chisquare distribution with pdf given by (2.1). If the product moment correlation between U and V vanishes, then they are independent. Proof. Since

2

2 2

( ) (1,1) ( 2 ), ( ) (1,0) , ( ) (0,1) ,( ) (2,0) ( 2), ( ) (0, 2) ( 2),

E UV m m E U m E V mE U m m E V m m

µ ρ µ µ

µ µ

′ ′ ′= = + = = = =

′ ′= = + = = +

it can be checked that

( ) ( ) ( )

2

2

2 2

( ) ( 2) 2 ,( ) ( 2) 2 ,( , ) ( 2 ) ( ) 2

Var U m m m mVar V m m m mCov U V E UV E U E V m m m m mρ ρ

= + − =

= + − =

= − = + − =

and hence the product moment correlation between U and V is given by

( ) ( )[ ]

21/ 2,

( ) ( )

Cov UVCorr U V

Var U Var Vρ= =

Note that if 0ρ = , the pdf in (2.1) becomes the product of that of the two independent chisquare distributions each with m degrees of freedom. In view of the above corollary we have the following: Corollary 3.2 Let X and Y have a bivariate normal distribution with correlation coefficient ρ . If 1 1 2( , , , )Nh X X XL and 2 1 2( , , , )Nh Y Y YL are two functions of ( 1 2, , , NX X XL )and ( 1 2, , , NY Y YL ) respectively, with correlation coefficient ( )h ρ such that ( ) 0h ρ = implies

0ρ = , then X and Y are independent. 4. On a Chisquare Distribution with a Nuisance Parameter

Theorem 4.1 A random variable U is said to have a chisquare distribution with a nuisance parameter ρ if it has the pdf given by

2/ 2 1 2(1 )

/ 2 2=0

2 (( 1) / 2)( ) !2 ( / 2) 1

u km

mk

u e u kg ukm

ρ ρπ ρ

−− − ∞ ⎛ ⎞ Γ +⎜ ⎟=

⎜ ⎟Γ −⎝ ⎠∑ , 0u > , 2, 1 1m ρ> − < < .

When 0ρ = , the above reduces to the pdf of chisquare distribution with m degrees of freedom. If 1/ 2ρ = , a special case of the chisquare distribution has the following pdf :

7/ 2 1 / 2

/ 2=0

(4 ) 1( ) ! 22 ( / 2)

m u k

mk

u e u kg ukm π

− − ∞ +⎛ ⎞= Γ⎜ ⎟Γ ⎝ ⎠∑ , 0u > , 2.m >

Theorem 4.2 The a th moment of the distribution of U is given by

(( / 2) )( ; ) 2( / 2)

a m aam

µ ρ Γ +′ =Γ

which is the a th moment of the usual chisquare random variable with m degrees of freedom. Proof. By the use of Theorem 2.3 in Theorem 2.2 , we have the following:

( ) ( ) ( )

( )

( )

( ) ( )

2

0

2

, 20

/ 22

/ 2 / 22 2

2 1 (2 )( ; ) ( 2 ) / 2 ( 1) / 2( , ) !

2 1

( , ) !

2 1 ( 2 , )

( , )

(( / 2) ) 2 1 1( / 2)

aa k

k

aa k

k m ak

a ma

a m a ma

a k m a kL m k

bL m k

L m aL m

m am

ρ ρµ ρρ

ρ ρρ

ρρ

ρ

ρ ρ

∞

=

∞

+=

+

+ − −

−′ = Γ + + Γ +

−=

−= +

⎡ ⎤Γ += − −⎢ ⎥Γ⎣ ⎦

∑

∑

since ( ) / 22( , ) ( / 2) 1m

L m mρ π ρ−

= Γ − , 2,m > and 1 1ρ− < < .

5. Some Functions of Correlated Chisquare Varibales Theorem 5.1 Let U and V be two correlated chisquare variables with pdf given by Theorem 2.1. Then the joint pdf of Y U V= + and W UV= is given by

( )( )

2/ 2 1 2 1/ 22(1 )

5 21 2 / 20

( 1) / 2 ( 4 )( , ) ,1 ! ( ) / 22 ( / 2)(1 )

ylm

m ml

lw e y w wf y wl l mm

ρ ρρπ ρ

−− −− ∞

−=

⎛ ⎞ Γ +−= ⎜ ⎟⎜ ⎟− Γ +Γ − ⎝ ⎠

∑

2 , 2, 1 1.y w m ρ> > − < < Proof. Let 1 2( , ) , ( , ) ,y h u v u v w h u v uv= = + = = 1 1

1 2( , ), ( , )u h y w v h y w− −= = in (2.1) so that the joint pdf of Y and W is given by

1 1 1 16 1 1 2 1 2 1 2 2( , ) ( ( , ), ( , )) | | ( ( , ), ( , )) | |f y w g h y w h y w J g h y w h y w J− − − −= +

where 1g and 2g are the pdf of Y and W in the domain 1 {( , ), }D u v u v= > and

2 {( , ) : }D u v u v= < respectively and | |, ( 1, 2)iJ i = is the Jacobians of transformations. In

1 {( , ) : }D u v u v= > we have 22 4 ,u y y w= + − 22 4v y y w= − − and

82 1/ 21 1 ( 4 ) ,

2 2u y y wy

−∂= + −

∂2 1/ 2( 4 ) ,u y w

w−∂

= − −∂

2 1/ 21 1 ( 4 ) ,

2 2v y y wy

−∂= − −

∂ 2 1/ 2( 4 )v y w

w−∂

= −∂

so that 2 1/ 2( 4 )u v u v y wy w w y

−∂ ∂ ∂ ∂− = −

∂ ∂ ∂ ∂

yielding ( ) 1/ 22( , , ) 4 , 2 .J u v y w y v y v−

→ = − > In 2 {( , ) : }D u v u v= < , we have 2 2 2 4 , 2 4u y y w v y y w= − − = + − and as above, it can be proved that

2 1/ 2( 4 )u v u v y w

y w w y−∂ ∂ ∂ ∂

− = − −∂ ∂ ∂ ∂

so that the Joacobian of the transformation is ( ) 1/ 22| ( , , ) | 4 , 2 .J u v y w y w y w−

→ = − > Then the probability density function of Y and V follows from Theorem 2.1. Corollary 5.1 If 0ρ = , then the joint distribution of sum and product of two chisquare variables would be

( ) 1/ 2/ 2 1 / 2 2

7 1 2

( ) 4 ( , )

2 ( / 2)

m y

m

yw e y wf y w

m

−− −

−

−=

Γ, 20 / 4,0 , 1 2.w y y m N< < < = − >

Theorem 5.2 Let U and V be two correlated chisquare variables with pdf given by Theorem 2.1. Then the pdf of Y U V= + is given by

( )( )( )2 21 /(2 2 )

8 / 22 1 2 0

/ 2 2 (( 1) / 2)( ) , 0

! (( 1) / 2)2 ( / 2) 1

lm y

mm l

y ly ef y yl l mm

ρ ρ ρ

ρ

− − − ∞

− =

− Γ += >

Γ + +Γ −∑

Proof. It follows from Theorem 2.1 that

( )

( )( ) ( )( ) ( )

2

2

/ 22/(2 2 )

9 1

2 / 41/ 2( 2) / 2 2

0 0

1 ( )

2 ( / 2)

/ 1 ( 1) / 2 4

! ( ) / 2

m

ym

ly

l m

l

f y em

lw y w dw

l l m

ρρ

π

ρ ρ

−

− −−

∞ −+ −

=

−=

Γ

− Γ +× −

Γ +∑ ∫

which can be simplified as

( )

( )( ) ( )( )

2/ 22 /(2 2 )

10 1

2 1

0

1( )

2 ( / 2)

/ 1 ( 1) / 2 (( ) / 2) (1/ 2) .! ( ) / 2 2 ( 1/ 2)

m y

m

ll m

m ll

ef y

m

l y l ml l m l m

ρρ

π

ρ ρ

− − −

−

+ −∞

+=

−=

Γ

− Γ + ⎡ ⎤Γ + Γ× ⎢ ⎥Γ + Γ + +⎣ ⎦∑

9 If 0ρ = , the pdf in Theorem 5.2 matches, as expected, with 2

2mχ . Theorem 5.3 Let U and V be two correlated chisquare variables with pdf given by Theorem 2.1. Then the pdf of W UV= is given by

( )( )

( )( ) ( )( )

2( 2) / 2 1/ 22 /(2 2 )

11 / 21 22

2

0

( ) 42 ( / 2) 1

/ 2 2 ( 1) / 2 , 0.

! ( ) / 2

my

mmw

l

l

wf w y w e dym

w lw

l l m

ρ

ρ π

ρ ρ

∞− − − −

−

∞

=

= −Γ −

− Γ +× >

Γ +

∫

∑

Proof. It follows from Theorem 2.1 that

( )( )

( )( ) ( )( )

2( 2) / 2 1/ 22 /(2 2 )

12 / 21 22

2

0

( ) 42 ( / 2) 1

/ 2 2 ( 1) / 2 .

! ( ) / 2

my

mmw

l

l

wf w y w e dym

w l

l l m

ρ

ρ π

ρ ρ

∞− − − −

−

∞

=

= −Γ −

− Γ +×

Γ +

∫

∑

If W is the product of two independent chisquare variables with d.f. 1m and 2m , then

1 2

1 21 2

( ) / 4 1

15 ( ) / 4/ 2 11 2

( ) ( ), 02 ( / 2) ( / 2)

m m

m mm m

wf w K w wm m

+ −

−+ −= >Γ Γ

(Springer, 1979, 365) where ( )K xα is the modified Bessel function of the third kind of order α which is neither zero nor a positive integer (Erdelyi, 1959, (13), p.5). Notice that the above density function does not allow the two chisquares having the same degrees of freedom. The following theorem provides a different form of the density of the product of two independent chisquare variables with the same degrees of freedom. Theorem 5.4 In case U and V are independent, the pdf of W UV= is given by

( 3) / 2

14 13/ 2 2

3 ( ) ( ) 2 ( / 2)

m

m

wf w K wmπ

−

−=

Γ

where 1/ 2

2 1/ 21 1

1

( ) ( 1) (0, , 2 ).2 ( 1/ 2) 2

xtxK x e t dt F xx

αα

α α

π π αα

∞− − ⎛ ⎞= − = ⎜ ⎟+ ⎝ ⎠∫

Proof. From Theorem 5.3 we have

10

( )/ 2 1 1/ 22 / 2

13 1 22

1/ 2/ 2 1 2/ 2

1 22

( ) 42 ( / 2)

4 1 . 2 ( / 2) 4

my

mw

my

mw

wf w y w e dym

w yw e dym w

∞− − −−

−∞−−

−

= −Γ

⎡ ⎤⎛ ⎞= −⎢ ⎥⎜ ⎟Γ ⎝ ⎠⎣ ⎦

∫

∫

Letting , 22

y x dy w dxw

= = , we have

( )/ 2 1 1/ 22

14 1 21

( 3) / 2

13/ 2 2

( ) 1 , 2 ( / 2)

3 ( ).2 ( / 2)

mw t

m

m

m

wf w t e dtm

w K wm π

∞− − −−

−

−

= −Γ

=Γ

∫

The moments of /H U V= , the ratio of two correlated chisquare variables, is outlined in Corollary 2.6. But further investigation is required to derive the distribution of /H U V= , along Provost (1986). 6. Estimation of 2ρ Theorem 5.1 Let 1X and 2X have a bivariate normal distribution with 2 2

1 2σ σ= . Then an unbiased estimators of 2ρ is given by

21

* 22

2 1 12

SmHS

⎛ ⎞−= − +⎜ ⎟

⎝ ⎠

with variance

2 2 2 4 2 2*

1( ) (1 ) (1 4 5 ) 8 (1 )4

Var H m mm

ρ ρ ρ ρ ρ⎡ ⎤= − − + − + −⎣ ⎦−

Proof. From Corollary 2.6 we have

22( ) .2

mE Hm

ρ−=

−

Then the unbiasedness of *H follows from

222

U mEV m

ρ−⎛ ⎞ =⎜ ⎟ −⎝ ⎠

where 2 2

1 1/U mS σ= and 2 22 2/V mS σ= . The variance of *H follows from Corollary 2.6 by

virtue of

112

*( 2)( ) ( ).

4mVar H Var H−

=

Acknowledgements The author gratefully acknowledges the research support provided by King Fahd University of Petroleum & Minerals through the project FT 2004-23. References 1. Anderson, T.W. (2003). An Introduction to Multivariate Statistical Analysis. John Wiley and

Sons. New York. 2. Erdelyi, A. (1959). Higher Transcendental Functions. Vol. 2. McGraw-Hill, New York. 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. Gupta, A.K. and Nagar, D.K. (2000). Matrix Variate Distributions. Chapman and Hall,

London, UK. 5. Joarder, A.H. (2005a). On a bivariate chisquare distribution. Technical Report No. 335.

Department of Mathematical Sciences, King Fahd University of Petroleum & Minerals, Saudi Arabia.

6. Joarder, A.H. (2005b). Moments of the product and quotient of two correlated chisquare random variables. Technical Report No. 341. Department of Mathematical Sciences, King Fahd University of Petroleum & Minerals, Saudi Arabia.

7. Joarder, A.H. (2006). Product moments of bivariate Wishart distribution. Journal of

Probability and Statistical Science. 4(2), 233-244. 8. Provost, S.B. (1986). The exact distribution of the ratio of a linear combination of a

chisquare variables over the root of a product of chi-square variables. 9. Kotz, S.; Balakrishnan, N. and Johnson, N.L. (2000). Continuous Multivariate

Distributions (volume 1). John Wiley and Sons, New York.

10. Springer, M.D. (1979). The Algebra of Random Variables. John Wiley and Sons. 11. Wishart, J. (1928). The generalized product moment distribution in samples from a

normal multivariate population. Biometrika, A20, 32-52.

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