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 F2 denotes the Appell hypergeometric function of the second kind de-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 ∞ ∞ +

= =

′ ′ ′

, , ; , ; , =

′ ! !

∑ ∑

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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

( )

( ) 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

( )

( ) 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

a b x F a b c x

c k

∞

=

!

∑

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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}m n)_{are not finite} 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=0, 10C = , 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

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 Ym n). Theorem 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≥1, where

(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

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≥1 and b≥1 are integers then ( 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= + +1 x y in (6), the double integral (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≠ + + −1 then a direct application of equation (2.2.6.1) in Prudnikov et al. (1986, volume 1) shows that (α k l, ) and (β k l, ) reduce to the forms given in the theorem. If k a b c= + + −1 then a direct application of equation (2.6.10.31) 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 Ym n) for given integer values of 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= =1 then ( )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= =1 then E X( 2) is given by (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

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= =1 then (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 )

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

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

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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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REFERENCES

I.S. GRADSHTEYN, I.M. RYZHIK,(2000), Table of Integrals, Series, and Products (sixth edition),

Aca-demic Press, San Diego.

M.L.T. LEE, A.J. GROSS, (1991), Lifetime distributions under unknown environment, “Journal of

Sta-tistical Planning and Inference”, 29, pp. 137-143.

D.D. LIEN, (1985), Moments of truncated bivariate log–normal distributions, “Economics Letters”,

19, pp. 243-247.

T.K. NAYAK, (1987), Multivariate lomax distribution - properties and usefulness in reliability theory,

“Journal of Applied Probability”, 24, 170-177.

A.P. PRUDNIKOV, Y.A. BRYCHKOV, O.I. MARICHEV, (1986), Integrals and Series (volumes 1, 2 and

3), Gordon and Breach Science Publishers, Amsterdam.

SUMMARY

A truncated bivariate inverted Dirichlet distribution