View of A truncated bivariate inverted Dirichlet distribution

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A TRUNCATED BIVARIATE INVERTED DIRICHLET DISTRIBUTION Saralees Nadarajah

1. INTRODUCTION

The need for truncated bivariate distributions with heavy tails is frequently en-countered in many applied areas, see, for example, Lien (1985, 1986). A popular bivariate distribution with heavy tails is the bivariate inverted dirichlet distribu-tion. Because of the heavy tails, this distribution does not possess finite moments of all orders. In this note, we overcome this weakness by introducing a truncated version. The bivariate inverted dirichlet distribution is given by the joint probabil-ity densprobabil-ity function (pdf):

1 1 ( ) ( ) ( ) ( ) ( )(1 ) a b a b c a b c x y g x y a b c x y − − + + Γ Γ Γ Γ + + , = + + (1)

for x > 0, y > 0, a > 0, b > 0 and c > 0. The corresponding joint cumulative dis-tribution function (cdf) is:

( ) ( ) ( 1) ( 1) ( )(1 ) a b a b c a b c x y G x y a b c x y + + Γ Γ Γ Γ + + , = + + + + 2 1 1 1 1 1 1 y x F a b c a b x y x y ⎛ ⎞ × _⎜ + + ; , ; + , + ; , _⎟, + + + + ⎝ ⎠ (2)

where F denotes the Appell hypergeometric function of the second kind de-2

fined by 2 0 0 ( ) ( ) ( ) ( ) ( ) ( ) k l k l k l k l k l a b b z F a b b c c z c c k l η η ∞ ∞ + = = ′ ′ ′ , , ; , ; , = ′ ! !

∑ ∑

for | | | | 1z + η < , where ( )e k =e e( +1) ("e k+ −1) denotes the ascending factorial. The truncated version is given by the pdf:

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1 1 ( ) ( ) ( ) ( ) ( )(1 ) a b a b c a b c x y f x y a b c x y − − + + Γ ΩΓ Γ Γ + + , = + + (3)

for 0 B x< ≤ ≤A< ∞ and 0 D y C< ≤ ≤ < ∞ , where (A B C D a b c) G A C( ) G A D( ) G B C( ) G B D( )

Ω , , , , , , = , − , − , + , . The cdf

associ-ated with (3) is: 1

( ) { ( ) ( ) ( ) ( )}.

F x y G x y G x D G B y G B D

Ω

, = , − , − , + , (4)

The corresponding marginal cdfs and marginal pdfs are 1 ( ) { ( ) ( ) ( ) ( )}, X F x G x C G x D G B C G B D Ω = , − , − , + , 1 ( ) { ( ) ( ) ( ) ( )}, Y F y G A y G A D G B y G B D Ω = , − , − , + , 1 2 1 2 1 ( ) ( ) 1 1 ( ) ( 1) ( )(1 ) 1 1 a b X a b c b a b c x C f x C F b a b c b x a b c x D D F b a b c b x − + + Γ ΩΓ Γ Γ + + ⎧ ⎛ ⎞ = _⎨ _⎜ , + + ; + ;− _⎟ + + + ⎩ ⎝ ⎠ ⎫ ⎛ ⎞ − _⎜ , + + ; + ;− _⎟_⎬ + ⎝ ⎠⎭ and 1 2 1 2 1 ( ) ( ) 1 1 ( 1) ( ) ( )(1 ) 1 , 1 b a Y a b c a a b c y A f y A F a a b c a y a b c y B B F a a b c a y − + + Γ ΩΓ Γ Γ ⎧ ⎛ ⎞ + + ⎪ = _⎨ _⎜ , + + ; + ;− _⎟ + + + _⎪_⎩ _⎝ _⎠ ⎫ ⎛ ⎞⎪ − _⎜ , + + ; + ;− _⎟_⎬ + _⎪ ⎝ ⎠⎭

where 2 1F denotes the Gauss hypergeometric function defined by

2 1 0 ( ) ( ) ( , ; ; ) . ( ) k k k k k a b x F a b c x c k ∞ = = !

∑

We refer to (3) as the truncated bivariate inverted dirichlet distribution. Because (3) is defined over a finite interval, the truncated bivariate inverted dirichlet dis-tribution has all it moments. Thus, (3) may prove to be a better model for certain practical situations than one based on just the bivariate inverted dirichlet distribu-tion. Below, we discuss one such situadistribu-tion.

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The bivariate inverted dirichlet distribution has received applications in many areas, including biological analyses, clinical trials, stochastic modelling of decreas-ing failure rate life components, study of labor turnover, queuedecreas-ing theory, and re-liability (see, for example, Nayak (1987) and Lee and Gross (1991)). For data from these areas, there is no reason to believe that empirical moments of any or-der should be infinite. Thus, the choice of the bivariate inverted dirichlet distribu-tion as a model is unrealistic since its product moments (_{E X Y are not finite}m n) for all m and n . The alternative given by (3) will be a more appropriate model for the kind of data mentioned. The choice of the limits, A , B , C and D could be easily based on historical records. For example, to model the dependence be-tween the exchange rate data of the United Kingdom Pound and the Canadian Dollar (as compared to the United States Dollar) for last 100 years, one could choose D = , 100 C = , 0B = and A =10 (see http://www.gobalfindata.com).

The rest of this note is organized as follows. Explicit expressions for the mo-ments of (3) are derived in Sections 2 and 3. Maximum likelihood estimators and the associated Fisher information matrix are derived in Section 4. The calcula-tions use the special funccalcula-tions defined above as well as the hypergeometric func-tion defined by 3 2 0 ( ) ( ) ( ) ( , , ; , ; ) . ( ) ( ) k k k k k k k a b c x F a b c d e x d e k ∞ = = !

∑

The properties of the above special functions can be found in Prudnikov et al. (1986) and Gradshteyn and Ryzhik (2000).

2.MOMENTS

We derive two representations for the product moment (E X Y . Theorem m n) 1 expresses it as an infinite sum of incomplete beta functions while the expres-sion given by Theorem 2 is in terms of the hypergeometric functions.

Theorem 1 If (X Y, ) has the joint pdf (3) then

( ) { ( ) ( ) ( ) ( )} ( ) ( ) ( ) m n a b c E X Y H A C G A D G B C G B D a b c ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ Γ Γ Γ Γ + + = , − , − , + , (5)

for 1m ≥ and n ≥ , where 1

(1 ) 0 ( ) ( ) ( ) ( ) ( ). ( 1) k n b k k P P k k n b a b c Q Q H P Q B m a k b c m n b n b k + ∞ / + = + + + − , = + , + + − +

∑

+ + !

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Proof: Using (3), one can write 1 1 1 1 1 1 0 0 0 0 1 1 0 0 0 0 ( ) ( ) ( ) ( ) ( ) (1 ) ( ) ( ) ( ) ( ) (1 ) (1 ) (1 ) m a n b A C m n a b c B D m a n b m a n b A C A D a b c a b c m a n b m B C B D a b c a b c x y E X Y dydx a b c x y a b c x y x y dydx dydx a b c x y x y x y x dydx x y + − + − + + + − + − + − + − + + + + + − + − + + + Γ Γ Γ Γ Γ Γ Γ Γ + + = + + ⎧ + + = _⎨ − + + + + ⎩ − + + +

∫ ∫

₍₁ 1 ₎ 1 ( ) { ( ) ( ) ( ) ( )}, ( ) ( ) ( ) a n b a b c y dydx x y a b c H A C H A D H B C H B D a b c − + − + + Γ Γ Γ Γ ⎫ ⎬ + + _⎭ + + = , − , − , + , where 1 1 0 0 ( ) . (1 ) m a n b P Q a b c x y H P Q dydx x y + − + − + + , = + +

∫ ∫

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Application of equation (2.2.6.1) in Prudnikov et al. (1986, volume 1) shows that the double integral (H P Q, ) can be expressed as

1 1 0 0 1 2 1 0 1 0 0 0 ( ) (1 ) (1 ) 1 1 ( ) ( ) (1 ) ( 1) 1 ( ) ( ) ( b n P _{m a} Q a b c n b _P m a a b c k n b _P m a a b c k k k k n b k k k y H P Q x dydx x y Q _x _x _{F n b a b c n b} Q _dx n b x n b a b c Q _x _x Q _dx n b n b k x n b a b c Q n b + − + − + + + + − − − − + ∞ + − − − − = + ∞ = , = + + ⎛ ⎞ = + _⎜ + , + + ; + + ;− _⎟ + ⎝ + ⎠ + + + ⎛ ⎞ = + _⎜− _⎟ + + + ! ⎝ + ⎠ + + + = +

∫

∑

∫

∑

1 0 (1 ) ₁ ₁ 0 0 (1 ) 0 ) (1 ) ( 1) ( ) ( ) ( ) (1 ) ( 1) ( ) ( ) ( ) ( ), ( 1) k _P m a a b c k k k n b _P _P m a k b c m k k k k k n b k k P P k k Q x x dx n b k n b a b c Q Q y y dy n b n b k n b a b c Q Q _B _{m a k b c m} n b n b k θ + − − − − − + ∞ _{/ +} + − + + − − = + ∞ / + = − ₊ + + ! + + + − = − + + + ! + + + − = + , + + − + + + !

∫

∑

∫

∑

where we have set y x= / +(1 x) and used the definitions of the incomplete beta and the Gauss hypergeometric functions.

Theorem 2 If (X Y, ) has the joint pdf (3) and if a ≥ and 1 b ≥ are integers then 1 ( m n)

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1 1 1 0 0 1 ( 1) 1 ( ) { ( ) ( )}, 1 b n k b n b n k k l b n b n k H P Q k l k l k k a b c l α β + − − + − + − − = = + − + − − ⎛ ⎞ − ⎛ ⎞ , = _⎜ _⎟ _⎜ _⎟ , − , − − − + ⎝ ⎠ ⎝ ⎠

∑

1 2 1 1 3 2 ( 1) 1 1 1 if 1 ( ) log( 1) ( 1) 1 1 1 1 2 2 1 if 1 a m l k a b c a m l a m l P Q a m l P F a m l a b c k a m l Q k a b c k l P Q P Q a m l a m l P F a m l a m l Q k a b c α + + − − − + + + + + + ⎧ + ⎪ _{+ +} ⎪ ⎪ ⎛ ⎞ × + + , + + − − ; + + + ;− , ⎪ ⎜ ₊ ⎟ ⎝ ⎠ ⎪ ⎪ _{≠ + + − ,} ⎪ , = ⎨ + / + ⎪ ₊ ⎪ _{+ +} _{+ + +} ⎪ ⎛ ⎞ ⎪ _× _{+ + + , , ; , + + + ;−} _, ⎜ ⎟ ⎪ _⎝ ₊ _⎠ ⎪ = + + − ⎪⎩ and 2 1 1 3 2 ( 1 1 ) if 1 ( ) ( 1 1 1 2 2 ) 1 if 1 a m l a m l P F a m l a b c k a m l P a m l k a b c k l P F a m l a m l P a m l k a b c β + + + + + ⎧ + + , + + − − ; + + + ;− , ⎪ + + ⎪ ≠ + + − , ⎪ , = ⎨ ⎪ _{+ + + , , ; , + + + ;− ,} ⎪ _{+ + +} ⎪ _{= + + − .} ⎩

Proof: Setting z= + + in (6), the double integral (1 x y H P Q, ) can be expressed as

1 ₁ 1 0 1 1 ₁ 1 1 0 1 0 1 1 1 0 0 1 1 ( 1) ( ) 1 ( 1 ) 1 ( 1 ) (1 ) (1 ) 1 P _{a m} x Q _{a b c} _{b n} x b n P _{a m} _{b n} _k x Q _{k a b c} x k b n P _{a m} _{b n} _k k k a b c k a b c z x H P Q x z dzdx b n x x z dzdx k b n x x k x Q x _dx k a b c + + _{+ −} + − − − − + + − _{+ +} + − + − − − − − + = + − + − + − − = − − − + − − − + − − , = + − ⎛ ⎞ = _⎜ _⎟ − − ⎝ ⎠ + − ⎛ ⎞ = _⎜ _⎟ − − ⎝ ⎠ + + − + × − − − +

∫

∑

∫

∑

∫

{

}

1 1 1 0 0 1 1 1 0 1 1 1 0 0 1 ( 1) 1 1 (1 ) (1 ) 1 ( 1) 1 { ( ) ( )}, 1 b n k b n b n k k l P _{a m l} _{k a b c} _{k a b c} b n k b n b n k k l b n b n k k k a b c l x x Q x dx b n b n k k l k l k k a b c l α β + − − + − + − − = = + + − − − − + − − − + + − − + − + − − = = + − + − − ⎛ ⎞ − ⎛ ⎞ = _⎜ _⎟ _⎜ _⎟ − − − + ⎝ ⎠ ⎝ ⎠ × + + − + + − + − − ⎛ ⎞ − ⎛ ⎞ = _⎜ _⎟ _⎜ _⎟ , − , − − − + ⎝ ⎠ ⎝ ⎠

∑

∫

∑

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where 1 1 0 (_{k l}) P_xa m l (1 _x _Q)k a b c _dx α , =

_∫

+ + − + + − − − + and 1 1 0 (_{k l}) P_xa m l (1 _x)k a b c _dx. β _{, =} + + − ₊ − − − +

∫

If k a b c≠ + + − then a direct application of equation (2.2.6.1) in Prudnikov et 1 al. (1986, volume 1) shows that (α k l, and () β k l, reduce to the forms given in ) the theorem. If k a b c= + + − then a direct application of equation (2.6.10.31) 1 in Prudnikov et al. (1986, volume 1) shows that (α k l, and () β k l, reduce to the ) forms given in the theorem.

3. PARTICULAR CASES

Using special properties of the hypergeometric functions, one can obtain ele-mentary forms for (E X Y for given integer values of m n) a , b , m and n . This is illustrated in the corollaries below.

Corollary 1 If (X Y, ) has the joint pdf (3) with a b= = then ( )1 E X is given by (5) with 2 2 (1 ) (1 ) (1 ) (1 ) ( ) {1 (1 ) (1 ) (1 ) } { ( 1)} a a a a a a a A A B B A H A B a A a A B A B B B B A a a θ θ θ φ φ θ θ φ θ φ φ φ φ φθ θ − − − − − − − + + + + + , = − − − + + + + + + + − + / − .

Corollary 2 If (X Y, ) has the joint pdf (3) with a b= = then 1 E X is given by ( 2) (5) with 2 2 2 2 2 2 2 2 2 2 2 2 (1 ) (1 ) (1 ) (1 ) ( ) {2 2 2 (1 ) (1 ) (1 ) 2 2 (1 ) (1 ) (1 ) 4 4 2 (1 ) (1 ) 2 2 (1 ) 2(1 a a a a a a a a a a a a a a A A A A H A B A a A a a A B B A A a B A B B B A B B A a A B B B A a A B B A A a θ θ θ θ θ θ θ φ φ θ θ φ θ φ φ φ θ φ φ θ θ φ φ φ θ θ φ φ θ θ − − − − − − − − − − − − − − + + + + , = + − − − + + + + + − + + + + + + + − + + + + + − + + + − + 2 2 3 2 3 ) } { ( 2 2 )}. B A B a a a a φ θ φ φ θ + + / − − + +

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Corollary 3 If (X Y, ) has the joint pdf (3) with a b= = then (1 E XY is given by ) (5) with 2 2 2 2 2 2 2 2 2 2 2 (1 ) (1 ) (1 ) (1 ) ( ) {1 (1 ) (1 ) (1 ) (1 ) (1 ) (1 ) (1 ) (1 ) (1 ) (1 ) a a a a a a a a a a a a a a A A A A H A B a A A a A B A B B a B B A a A B B A a A B A B A a A B B a B B A A a B B A θ θ θ θ θ θ θ φ θ φ φ φ φ θ θ φ φ θ θ φ θ φ θ θ φ φ φ φ θ θ φ φ θ θ − − − − − − − − − − − − − − + + + + , = − − + − + + + + + − − + + + + + + + + + − + + + + − + − + + − 2 2 2 2 2 2 2 2 2 3 (1 ) (1 ) } { ( 2 2 )}. (1 ) a a a B A B A a B B a B a a a a B A φ θ φ φ φ φ φ θ φ θ − − − + + + + + + + + / − − + + 4. ESTIMATION

Here, we consider estimation of the seven parameters of (3) by the method of maximum likelihood and provide the associated Fisher information matrix. The log-likelihood for a random sample (x y … x y1, 1), ,( n, n) from (3) is:

1 1

1

log ( ) log ( ) log log ( ) log ( ) log ( ) ( 1) log ( 1) log ( ) log(1 ). n n j j j j n j j j L A B C D a b c n a b c n n a n b n c a x b y a b c x y = = = Γ Ω Γ Γ Γ , , , , , , = + + − + − − + − + − − + + + +

∑

It follows that maximum likelihood estimators of the seven parameters are the solutions of the simultaneous equations

1 1 ( ) ( ) n log 1 _j _j n log _j j j n n a b c n a x y x a ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ = = ∂Ω Ψ Ψ Ω ∂ + + − − =

∑

+ + −

∑

, 1 1 ( ) ( ) n log 1 j j n log j j j n n a b c n b x y y b ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ = = ∂Ω Ψ Ψ Ω ∂ + + − − =

∑

+ + −

∑

, 1 ( ) ( ) n log 1 j j j n n a b c n c x y c ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ = ∂Ω Ψ Ψ Ω ∂ + + − − =

∑

+ + , 0 n A ∂Ω Ω ∂ = , 0 n B ∂Ω Ω ∂ = ,

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0 n C ∂Ω Ω ∂ = , and 0 n D ∂Ω Ω ∂ = ,

where ( )Ψ x =dlog ( )Γ x dx/ is the digamma function. The second order deriva-tives of logL are all constants, so the elements of the associated Fisher informa-tion matrix are

2 2 2 2 2 2 log ( ) ( ) L n n E n a b c n a a a a ∂ ∂ Ω ∂Ω _′ _′ Ψ Ψ Ω ∂ ∂ ∂ Ω ⎛ ⎞ _⎛ _⎞ − = − − + + + , ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ 2 2 2 log ( ) L n n E n a b c a b a b a b ∂ ∂ Ω ∂Ω ∂Ω _′ Ψ ∂ ∂ Ω ∂ ∂ Ω ∂ ∂ ⎛ ⎞ − = − − + + , ⎜ ⎟ ⎝ ⎠ 2 2 2 log ( ) L n n E n a b c a c a c a c ∂ ∂ Ω ∂Ω ∂Ω _′ Ψ ∂ ∂ Ω ∂ ∂ Ω ∂ ∂ ⎛ ⎞ − = − − + + , ⎜ ⎟ ⎝ ⎠ 2 2 2 2 2 2 logL n n ₍ ₎ _{( )} E n a b c n b b b b ∂ ∂ Ω ∂Ω ′ ′ Ψ Ψ Ω ∂ ∂ ∂ Ω ⎛ ⎞ _⎛ _⎞ − = − − + + + , ⎜ ⎟ ⎜_⎝ ⎟_⎠ ⎝ ⎠ 2 2 2 logL n n ₍ ₎ E n a b c b c b c b c ∂ ∂ Ω ∂Ω ∂Ω ′ Ψ ∂ ∂ Ω ∂ ∂ Ω ∂ ∂ ⎛ ⎞ − = − − + + , ⎜ ⎟ ⎝ ⎠ and 2 2 2 2 2 2 log ( ) ( ) L n n E n a b c n c c c c ∂ ∂ Ω ∂Ω _′ _′ Ψ Ψ Ω ∂ ∂ ∂ Ω ⎛ ⎞ _⎛ _⎞ − = − − + + + , ⎜ ⎟ ⎜_⎝ ⎟_⎠ ⎝ ⎠

where ( )Ψ′x is the first derivative of ( )Ψ x . The remaining elements of the Fisher information matrix all take the form

2 2 2 1 2 1 2 1 2 logL n n E θ θ θ θ θ θ ∂ ∂ Ω ∂Ω ∂Ω ∂ ∂ Ω ∂ ∂ Ω ∂ ∂ ⎛ ⎞ − = − . ⎜ ⎟ ⎝ ⎠

School of Mathematics SARALEES NADARAJAH

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SUMMARY

A truncated bivariate inverted Dirichlet distribution

A truncated version of the bivariate inverted dirichlet distribution is introduced. Unlike the inverted dirichlet distribution, this possesses finite moments of all orders and could therefore be a better model for certain practical situations. One such situation is dis-cussed. The moments, maximum likelihood estimators and the Fisher information matrix for the truncated distribution are derived.