1

Moments of the Bivariate T-Distribution

Anwar H. Joarder

Department of Mathematical Sciences King Fahd University of Petroleum and Minerals Dhahran 31261, Saudi Arabia Email: anwarj@ kfupm.edu.sa

Abstract The probability density function of the bivariate t-distribution can be represented by a scale mixture representation of the bivariate normal distribution with an 'inverted' chisquare distribution. The product moments of the bivariate t-distribution are derived by exploiting the scale mixture. Some standardized moments of the bivariate t-distribution are also derived.

AMS Mathematics Subject Classifications: 60E10, 60-02

Key Words: Bivariate normal distribution, bivariate t-distribution, product moments, standardized moments

1. Introduction

The bivariate t-distribution arises as a derived sampling distribution from the bivariate normal distribution and the chisquare distribution (Anderson, 2003, 289). In this paper we emphasize a scale mixture representation to calculate some product moments and standardized moments of the bivariate t-distribution.

Because of the increase in the use of the multivariate t-distribution in business, especially, in stock returns, the paper will stimulate research in business, econometrics and statistics. Interested readers may go through Lange, Little and Taylor (1989), Billah and Saleh (2000), Kibria and Saleh (2000) and Kotz and Nadarajah (2004).

(i) The Univariate T-Distribution

2 −22 Theorem 1.1 Let ZN~(0,)τ and νΤ dW ~χν where the symbol d means that both sides of it have the same distribution. Then

1/2 ()aWZt (/ν ) ~ν , ν /2 2(ν / 2) 2 ()bh ()ττ= −+(1)/(2)ν e −ντ, 0 <τ , (1.1) Τ Γ(/2)ν (c) the pdf of the univariate t-dsitribution with ν degrees of freedom has the following representation: ∞ 1 ⎛⎞z 2 f 1()zhd=− exp⎜⎟Τ ()τ τ . (1.2) ∫ 2τ 2 0 τπ2 ⎝⎠

Proof. The proof of the first two parts are well known while the third part follows by transformation ν /τ 2 =w with Jacobian Jww()τν→=−3/2 /2. 2

The representation in (1.2) can be expressed by (|ZNΤ=τ )~(0,)τ 2 . Thus EZ()=Τ== EEZ [( |)](0)0 E and Cov() Z=Τ+Τ E [ V ar ( Z |)] V ar [( E Z |)] =Τ+EVar (2 ) (0)

= γ 2 where γ 2 =−νν/( 2), ν >2 . In what follows, for any nonnegative integer a , we assume {}a {0} kkkkak{}a =+( 1)"" ( +−= 1), {0} 1; kkkkak =− ( 1) ( −+= 1), 1.

Corollary 1.1 The a-th moment of Τ is given by a /2 a (/2)ννΓ− (/2a /2) γνa =Τ=Ea() , > . Γ(/2)ν

Clearly γ 2 =−νν/( 2), ν >2 and a 2a (/2)ν γν2a =Τ=Ea() {}a , > 2 , (/21)ν − −−2aa γννν−2{}aa=ΤE ( ) =( / 2) ( / 2) , >2.

(ii) A Location Scale Bivariate T-Distribution

Let XXX′ = (,12 ) be the bivariate t - random vector with probability density function −ν /2− 1 −−11/2 − 1 f 2 ()xxx=Σ+−Σ− (2)πθνθ | |( 1( )'()( )) (1.3) where θ′ = (,θθ12 ) is an unknown vector of location parameters and Σ is the 22× unknown positive definite matrix of scale parameters while the scalar ν is assumed to be a known positive constant (Anderson, 2003, 289). The distribution will be denoted by T 2 (,;)θ Σ ν in contrast to the bivariate normal distribution denoted by N 2 (,θ Σ ). The pdf of the bivariate t- distribution in (1.3) can be written as a mixture of the bivariate normal distribution and an inverted chisquare distribution as follows:

∞ ⎛⎞−1 f ()xxxhd=Σ (2)π −−121/2 |τθντθττ | exp ( −Σ− )'( 21 )( − ) () (1.4) 3 ∫ ⎜⎟Τ 0 ⎝⎠2 which can be represented by 2 i.e. (|XNΤ=τθτ )~(,).2 Σ (1.5)

The pdf in (1.4) can be written as f 412(,xx ) 22 ∞ (1−−ρ 21/2 )− ⎡⎤ 1 ⎪⎪⎧⎫⎛⎞⎛⎞xx−−θθρθθ2( xx −− )( ) (1.6) =+−exp⎢⎥⎨⎬⎜⎟⎜⎟11 22 1122 hd (τ )τ . ∫ 22(1)πσ σ τ222⎢⎥ τ− ρ σ σ σ σ 0 12⎣⎦⎩⎭⎪⎪⎝⎠⎝⎠ 1 2 12 which can further be written as 21/2− −+(2)/2ν (1− ρ ) ⎛⎞qx(,12 x ) f 512(,xx )=+ ⎜⎟1 , (1.7) 2πσ12 σ⎝⎠ ν 3 where 22 2 ⎛⎞⎛⎞xx11−−θ 2θρθθ 22( xx 112 −− )( 2 ) (1−=+−ρ )qx (12 , x )⎜⎟⎜⎟ . (1.8) ⎝⎠⎝⎠σσ12 σσ 12 The above is the explicit form of the pdf of the bivariate t-distribution (cf. Anderson, 2003, 123). As ν →∞, the pdf of the bivariate t-distribution converges to the bivariate normal distribution with pdf 21/2− (1− ρ ) ⎛⎞−qx(,12 x ) f 612(,xx )= exp⎜⎟ , (1.9) 22πσ12 σ ⎝⎠ where qx(,12 x )is defined by (1.8). The following is a special case of the bivariate t- distribution with pdf (1.7): 21/2− 22 −+(2)/2ν (1− ρ ) ⎛⎞yy12+−2ρ yy 12 f 712(,yy )=+ ⎜⎟12 . (1.10) 2(1)πρν⎝⎠−

2. The Moments of the Bivariate T-Distribution

In the rest of the paper we mostly concentrate on the location scale representation of the bivariate t-distribution with equivalent probability density functions between (1.4) to (1.7). It follows from (1.5) that the expected value and the covariance matrix of the distribution are given by EX()=Τ== EEX [(|)]() Eθ θ and Cov()[(|)][(|)] X=Τ+Τ E Cov X Cov E X =ΤΣ+ECov()2 (θ )

=Σγ 2 , where

2 ⎛⎞µµ(2,0) (1,1) ⎛⎞σρσσ112 Σ=⎜⎟ =⎜⎟2 . ⎝⎠µµ(1,1) (0, 2) ⎝⎠ρσ22 σ σ 2

The raw product moments of X 1 and X 2 of order a and b are defined ab µ′(ab,; m) = E ( X12 X ) while the centered product moments of X 1 and X 2 are defined by ab µθθ(ab,; m) =− E [( X11 )( X 2 − 2 )], (2.1) where θ1122==EX(), θ EX().

For the bivariate normal distribution it is advisable to calculate the centered product moments ab instead of the raw product moments µ′(ab,()) = E X12 X . Clearly, µ(1, 0)==µ (0,1) 0. It is well known from the univariate normal distribution that

µ(2a += 1,0) 0 and µσµ(2aaa ,0)=−2 (2 1) (2 − 2,0) 1 2a =− 1(3)(5)" (2a 1)σ1 .

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3. Centered Product Moments of the Bivariate T-Distribution

The following theorem is due to Kendal and Stuart (1969, 91).

ab Theorem 3.1 The centered product moments µθθ(ab,[()()]) =− E X11 X 2 − 2of the bivariate normal distribution with pdf (1.9) are given by

ab µσσλ(,ab )= 12 (, ab ) where λρλρλ(,ab )=+− ( a b 1) ( a − 1, b −+− 1)( a 1)( b − 1)(1 −2 )( a − 2, b − 2), (2ab )!(2 )!min(ab , ) (2ρ )2 j λ(2ab ,2 )= , ab+ ∑ 2()!()!(2)!j =0 ajbj−− j (2ab++ 1)!(2 1)!min(ab , ) (2ρ )2 j λρ(2ab++= 1,2 1) , ab+ ∑ 2j =0 (ajbj−− )!( )!(2 j + 1)! λ(2ab ,2+= 1)λ (2 a + 1,2 b ) = 0.

The above can be rewritten as

222 µρσσµρσσµ(ab , )=+− ( a b 1)12 ( a − 1, b −+− 1) ( a 1)( b − 1)(1 − ) 1 2 ( a − 2, b − 2), (2ab )!(2 )!min(ab , ) (2ρ )2 j µσσ(2ab ,2 )= 22ab , 12 ab+ ∑ 2()!()!(2)!j =0 ajbj−− j (2ab++ 1)!(2 1)!min(ab , ) (2ρ )2 j µσσρ(2ab++= 1,2 1)2121ab++ , 12 ab+ ∑ 2j =0 (ajbj−− )!( )!(2 j + 1)! µ(2ab ,2+= 1)µ (2 a + 1,2 b ) = 0.

Theorem 3.2 The centered product moments of the bivariate t-distribution with pdf in (1.7) are given by

222 µ(ab , ;νρσσµγ )=+− ( a b 1)12 ( a − 1, b − 1) 2 +− ( a 1)( b − 1)(1 − ρσσµγ ) 1 2 ( a − 2, b − 2) 4 , (2ab )!(2 )!min(ab , ) (2ρ )2 j µνσσ(2ab ,2 ; )= 22ab γ, 12ab+ ∑ 22ab+ 2()!()!(2)!j =0 ajbj−− j (2ab++ 1)!(2 1)!min(ab , ) (2ρ )2 j µνσσρ(2ab++= 1,2 1; )2121ab++ γ, 12 ab+ ∑ 222ab++ 2j =0 (ajbj−− )!( )!(2 j + 1)! µ(2ab ,2+= 1;νµ ) (2 a + 1,2 b ; ν ) = 0 where γ a is the a -th moment of Τ .

Proof. By the scale mixture representation in (1.5), we have

ab µθθ(,;ab m )=−−Τ E [ E [( X11 )( X 2 2 )| ]]

=+−ΤΤ−−Eab [( 1)ρσ (12 )( σ ) µ ( a 1, b 1) 22 22 22 +− (ab 1)( − 1)(1 −ρυσ ) (12 Τ )( σ Τ ) µ ( ab − 2, − 2)] 2 =+− (ab 1)ρσ12 σ µ ( a − 1, b − 1) E ( Τ ) 222 4 + (ab−−−1)( 1)(1ρσσµ )12 ( abE −− 2, 2) ( Τ ). 5

Similarly we have (2ab )!(2 )!min(ab , ) (2ρ )2 j µνσσ(2ab ,2 ; )=Τ22ab E ( 22 ab+ ) 12 ab+ ∑ 2()!()!(2)!j =0 ajbj−− j (2ab++ 1)!(2 1)!min(ab , ) (2ρ )2 j µνσσρ(2ab++= 1,2 1; )2121ab++ E ( Τ222ab++ ) 12 ab+ ∑ 2j =0 (ajbj−− )!( )!(2 j + 1)! µνµ(2ab ,2+= 1; ) (2 ab ,2 +Τ 1) E (221ab++ ) = 0,

µνµ(2ab+=+Τ= 1,2 ; ) (2 abE 1,2 ) (221ab++ ) 0.

Some special cases are given below:

µ(aa ,1;νρσσµγ )=− ( 1)12 ( a − 1,0) 2

µ(1,bb ;νρσσµγ )=− ( 1)12 (0, b − 1) 2 222 µ(aa , 2;νρσσµγ )=+ ( 1)12 ( a − 1,1) 2 +− ( a 1)(1 − ρσσµγ ) 1 2 ( a − 2, 0) 4 222 µ(2,bb ;νρσσµγ )=+ ( 1)12 (1, b − 1) 2 +− ( b 1)(1 − ρσσµγ ) 1 2 (0, b − 2) 4 222 µ(,3;)(2)aaνρσσµγ=+12 (1,2)2(1)(1) a − 2 + a − − ρσσµγ 1 2 (2,1) a − 4 (2a )! µ(2a ,0;νσγ )= 2a 2a 12a (2b )! µ(0,2b ;νσγ )= 2b 2b 12b (2a )!(2) µνσσ(2a ,2; )=+22a (1 2 ργ ) 122!a+1a 22ab+ (2a + 1)! µ(2a += 1,1;νσσ )21a+ ργ 122ab+ 22a+ (2b + 1)! µνσσργ(1, 2b += 1; )21b + 122b 2b + 2 (2a + 1)! µνσσρ(2aaaa+= 1,3; )213a+ [3( −+ 1)! 2 ργ 2 !] , ≥1. 122!(1)!a aa− 24a+ 24 215a+ (2a + 1)!5!⎡⎤ 1 4ρρ 16 µνσσρ(2aa+= 1,5; )12 a+2 ⎢⎥ + + γ26a+ , ≥2. 2⎣⎦aa !(2) (−− 1)!(6) 120( a 2)!

Some special moments needed for calculating standardized moments (See Section 4) are given below:

µ(1, 0;ν )= 0,

µ(1,1;νσσργ )= 12 2 , µ(1, 2;ν )= 0, 3 µ(1, 3;νσσργ )= 312 4 , µ(1, 4;ν )= 0, 5 µ(1, 5;νσσργ )= 1512 6 ,

2 µ(2,0;νσγ )= 12 , µ(2,1;ν )= 0, 6

22 2 µ(2,2;νσσ )=+12 (1 2 ργ ) 4 , µ(2,3;ν )= 0, 24 2 µ(2,4;νσσργ )=+ 312 (1 4 ) 6 ,

µ(3,0;)νµ== (0,3;)0, ν 3 µ(3,1;νσσργ )= 312 4 , µ(3,2;ν )= 0, 33 2 µ(3,3;νσσρργ )=+ 312 (3 2 ) 6 ,

4 µ(4,0;νσγ )= 314 , µ(4,1;ν )= 0, 42 2 µ(4,2;νσσργ )=+ 312 (1 4 ) 6 ,

µ(5,0;ν )= 0, 5 µ(5,1;νσσργ )= 1512 6 ,

6 µ(6,0;νσγ )= 9016 .

ab By putting σ12==σ 1, we will have the moments E ()YY12 for the correlated bivarite t- distribution with pdf in (1.10).

Corollary 3.1 The product moment correlation between X and X is given by ρ = ρ . 1 2 XX12

Proof. By definition 1/2 ⎡⎤EX()()−−θ 22 EXθρ =−− EX ()(). θθ X ⎣⎦()()11 2 2XX12 , 11 2 2

The corollary then follows by virtue of 22 EX()(2,0;),11−=θµνσγν = 12 >2, 22 EX()(0,2;),22−=θµνσγν = 22 >2,

EX()()(1,1;),11−−==θθµνσσργν X 2 2 122 >2.

Note that if ρ = 0 in the pdf in (1.7), the components X 1 and X 2 do not become independent unless ν →∞.

Corollary 3.2 The product moments of the standard bivariate t-distribution

(i.e., θ12==θσσρ0, 1 = 2 = 1, = 0) are given by

µ(,ab ;νµγν )=− ( a 1)( b − 1)( a − 2, b − 2),4 >4, (2ab )!(2 )! µνσσγν(2ab ,2 ; )=>+22ab , 2 a 2 b , 122!!ab+ ab 22ab+ µ(2ab++= 1,2 1;ν ) 0, µ(2ab ,2+= 1;νµ ) (2 a + 1,2 b ; ν ) = 0, where γ a is the a-th moment of Τ . 7

Corollary 3.3 The product moments of the correlated bivariate t-distribution

(0,1,11)θ12==θσσ 1 = 2 =−<< ρwith pdf in (1.10) are given by

2 µν(,;ab )=+− ( a b 1) ρµ ( a − 1, b − 1) γ24 +− ( a 1)( b − 1)(1 − ρµ ) ( a − 2, b − 2), γν >4 (2ab )!(2 )!min(ab , ) (2ρ )2 j µν(2ab ,2 ; )=>+ γ, m2 a 2 b ab+ ∑ 22ab+ 2()!()!(2)!j =0 ajbj−− j (2ab++ 1)!(2 1)!min(ab , ) (2ρ )2 j µν(2ab+ 1,2 + 1; ) = ρ γν, >++2 ab 2 2 ab+ ∑ 222ab++ 2j =0 (ajbj−− )!( )!(2 j + 1)! µ(2ab ,2+= 1;νµ ) (2 a + 1,2 b ; ν ) = 0 where γ a is the a-th moment of Τ .

4. The Standardized Moments of the Bivariate T-Distribution

ab ab Let µθθ(ab,[()()]) =− E X11 X 2 − 2 and µν(ab,;) =− E [( X11 θ )( X 2 − θ 2 )]be the centered product moments of the bivarite normal distribution with pdf in (1.9) and the bivariate t- distribution with pdf in (1.7) respectively. Since the covariance matrix for the bivariate t- distribution is given by ⎛⎞⎛⎞µνµν(2,0;) (1,1;) µ (2,0) µ (1,1) Λ=⎜⎟⎜⎟ =γ 22 =γ Σ, ⎝⎠⎝⎠µνµ(1,1; ) (0,2; ν ) µ (1,1) µ (0,2) the quantity ()()||()||XX−Σθθ′′−−11/222 −=Σ XZZR − θ = = is not the standardized distance. The standardized distance for the bivariate t-distribution is defined by −1 QX=−()()()θ ′ γθ2 Σ X − (4.1)

a Letβa ==EQ ( ), ( a 1,2," ) be the a-th standardized moment of Q (Joarder, 2006). Some properties of standardized moments are discussed below.

−1 Theorem 4.1 Let QX=−()()()θ ′ γθ2 Σ X −be the standardized distance. Then a Γ+Γ(1)(/2)aaν − EQ()a =−()νν 2 , >2 a. Γ(/2)ν −−−1112 Proof. The quantity Q can be written as QX=−()()()θγ′′222 Σ X −= θ γ ZZR = γ where Σ−=−1/2 ()XZθ . Then from (1.3), the pdf of Z is −−ν /2 1 1 ⎛⎞zz′ f 8 ()z =+⎜⎟ 1 . (4.2) 2πν⎝⎠

By making the polar transformation zr12==cosθ , zr sinθ r ∈∞∈(0, ),θ (0,2π ) in (4.2) with Jacobian r , it follows that the probability density functions of R and Θ are given by

2(/21)−+ν gr1()=+ r (1 r /ν ) , r ∈∞(0, ) and −1 g 2 ()θ = (2)π ,(0,2)θ ∈ π

8 respectively. Then it can be checked easily that RF2 /2~ (2,ν ), and consequently

WQ=>γ 2 /2~ F (2,νν ), 2(cf. Muirhead, 1982, 37).

Corollary 4.1 The first three standardized moments of the bivariate t-distribution are given by

EQ()= 2, ν − 2 EQ()82 => , ν 4, ν − 4 48(ν − 2)2 EQ()3 => , ν 6. (4)(6)νν−−

Proof. The a-th moment of Q is given by aaaa− EQ()2= γ 2 EW ( ) Γ+Γ(1)(/2)aaν − = 2aaγν− ( / 2) a 2 Γ(/2)ν Γ+Γ(1)(/2)aaν − =− (νν 2)a , > 2a . Γ(/2)ν Notice that as ν →∞, the standardized moments of the bivariate t-distribution, as expected, coincide with that of the bivariate normal distribution.

Following Johnson, Kotz and Kemp (1993) we write moment ratios of the distribution of Τ by

−k /2 αµµkk()Τ= (2 ) , k =1,2," (4.3)

Note that α4 ()Τ is an index of kurtosis while α3 ()Τ is an index of skewness for the univariate random variable Τ. In the following theorem we calculate the standardized moments of the bivariate t-dsitribution from Theorem 2.1 of Joarder (2006) by the use of the centered product moments of Section 3 of the present paper.

Theorem 4.2 The the first three standardized moments of the bivariate t-distribution are given by

γ 4 γ 6 ββ12==2, 823 , β 3 =48 , γ 22γ where the moment ratio αk ()Τ is defined in ( 4.3).

Proof. Since the correlation coefficient between X 1 and X 2 is ρ (See Corollary 3.1), by the use of moments from Theorem 3.2 of the present paper in Theorem 2.1 of Joarder (2006), we have

22 2 ()iEQ (σγ12 )( σγ 22 )(1− ρ ) ( ) 22 =− 2[(σ12γσγ )( 22 ) 2 ρσγσγσσγρ ( 1 2 )( 2 2 )( 122 )] 222 2 =− 2σσγ122 (1 ρ ),

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2 ⎡⎤22 2 2 ()()()(1)()ii⎣⎦σγ12 σγ 22 − ρ E Q =+ 3()()4()()[(12)]σγσγ422222222 ρσγσγσσ + ργ 14 22 12 22 1 2 4 42221/223/23 +− 3()()4()()(3)σγ24 σγ 12 ρσγ 12 σγ 22 σσργ 1 2 4 22222 23/221/2 3 ++− 2(σγ12 )( σγ 22 )[ σσ 1 2 (1 2 ρ ) γ 4 ], 4 ρσγ ( 12 ) (σ 22γσσργ)(3 12 4 ), so that 22 2 4 2 2 2 2 (1−=+++−++−ρ )EQ ( )γρρρρρρργγ2 ( 34 (12)34(3)2(12)4(3)) 24 2222 =+() 6 (4ρρργγ + 2)(1 + 2 ) − 24 24 22 2 =−() 8(1ργγ )24 ,

3 ⎡⎤22 2 3 ()()()(1)()iii⎣⎦σγ12 σγ 22 − ρ E Q 6 2 3 3 2 3/2 2 3/2 3 3 2 =−(90σγ16 )( σγ 22 ) 8 ρ ( σγ 12 ) ( σγ 22 ) [3 σσρ 1 2 (3 + 2 ρ ) γ 6 ] +−()(90)6()()[15]σγ23 σγ 6 ρσγ 21/225/25 σγ σσργ 12 26 12 22 1 2 6 22 2 2 42 2 2 22 42 2 +12ρσγσγ (12 )( 22 ) [3 σσ 1 2 (1+ 4 ργ ) 6 ]+3( σγσ 12 )( 2γσσργ212)[3 (14+ ) 6 ] 2 22 24 2 22 22 24 2 ++3(σγ12 )( σγ 22 )[3 σσ 1 2 (14 ρ ) γ 6 ] +12 ρ ( σγ 22 )( σγ 22 )[3 σσ 1 2 (14 + ρ ) γ 6 ] 2 5/2 2 1/2 5 2 3/2 2 3/2 3 3 2 −−+6(ρ σγ12 )( σγ 22 )(15 σσργ 12 6 )12( ρσγ 12 )( σγ 22 )[3 σσρ 1 2 (32)] ρ γ 6 so that 23 6 3 (1− ργ )2 EQ ( ) =−(90 8ρρ32 [3 (3 + 2 ρ )] + (90) − 6 ρρ [15 ]+12 ρ 22 [3(1 + 4 ρ )]

2222 23 +3[3(1+++ 4ρ )] 3[3(1 4ρρ )] +12 [3(1 +− 4 ρρρρρργγ )] 6 (15 ) − 12 [3 (3 + 2 )]) 26 23 3 =−()48(1ργγ )26 . Hence, the second and the third standardized moments of the bivariate t-distribution are given by

γ 4 γ 6 βαβα2436==Τ=823 8 ( ), 48 = 48 ( Τ ), γγ22 which can be simplified by Corollary 1.1 as the following: νν−−2(2)2 βνβ=>=8 , 4; 48 , ν >6. 23ννν−−−4(4)(6)

Note that the coefficient of kurtosis β2 of the bivariate t-distribution is 8 times that of the invereted chisquare distribution with pdf in (1.1).

Corollary 4.1 Let Τ be an inverted chisqaure variable with pdf in (1.1), and

a µa ()Τ=EE ( Τ− ()) Τ . Then, as ν →∞, we have

2 (i) µµ42()Τ→() () Τ , 10

3 (ii) µµ62()Τ→( () Τ) . Proof. As ν →∞, the bivariate t-distribution converges to the bivariate normal distribution in which case it is known that β23→→8,β 48 ( Joarder, 2006). It is obvious from Theorem 4.2 that this happens if (i) and (ii) are true.

5. Shanon Entropy

The Shanon Entropy for any bivarite density function f (,xx12 ) is defined by

Hf()=− E (ln( f X12 , X )]. It follows from (1.9 ) that qx(, x ) lnfxx ( , )=−− ln 2πσ σ 1 ρ 2 12 612( 12 ) 2(1− ρ 2 ) so that EqX[( , X )] EfXX(ln ( , ))=− ln 2πσ σ 1 − ρ 2 − 12 612( 12 ) 2 2(1− ρ ) =− ln 2πσ σ 1 − ρ 2 − 1. ( 12 ) Hence, the Shanon Entropy for the bivariate normal distribution with pdf in (1.9) is given by

Hf()ln2=−+πσ σ 1 ρ 2 1. (5.1) 612( ) Theorem 5.1 Let the bivariate t-distribution have the pdf in (1.7). Then the Shanon entropy for the bivariate t-distribution has the following equivalent representations:

()iHf ( )=−+Τ+ ln2πσ σ 1 ρ 22 E (ln ) 1 512( ) ν + 2 (ii ) H ( f )=−++ ln 2πσ σ 1 ρ2 E⎡ln() 1 q ( X , X ) / ν ⎤ , 512( ) 2 ⎣ 12⎦

⎡⎤πνΓ+(/21) ν + 2 (iii ) H ( f )=−++Ψ−Ψ ln 2πσ σ 1 ρ2 ln ( ν / 2+1) ( ν / 2) , 512( ) ⎢⎥[] ⎣⎦Γ+((ν 1) / 2) 2

−22 where νΤ ~ χν and Ψ=()td ln()/ Γ tdt denotes the digamma function. (Nadarajah and Kotz, 2005).

Proof. By virtue of (1.5), it follows from (5.1) that

Hf()=ΤΤ−+ E ln2(⎡⎤πσ )()1 σ ρ2 1 512( ) ⎣⎦ =−+Τ+ ln 2πσ σ 1 ρ 22E (ln ) 1, ( 12 ) which is part (ii) of the theorem. It follows from (1.7) that 11

ν + 2 E[ln(,)]ln2 fXX=−πσ σ 1 − ρ2 − E⎡ln1(,)/,() + qXX ν ⎤ 512( 12) 2 ⎣ 12 ⎦ 22 which yields (ii) in the theorem. Note that EEWW(lnΤ= ) lnν − (ln ), ~χν . For part (iii), see Nadarajah and Kotz (2005).

Acknowledgements The author acknowledges the excellent research support provided by King Fahd University of Petroleum and Minerals, Saudi Arabia. The author is also grateful to Prof Saralees Nadarajah for providing me with preprints of some papers on t-distributions.

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9. Muirhead, R.J. (1982). Aspects of Multivariate Statistical Theory. John Wiley and Sons.

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