Family Of Dirichlet Distributions

Technical Report

The Dirichlet Family of Distributions

Gary M. Johnson Joint Warfare Analysis Center Dahlgren, Virginia [email protected]

November 7, 2007 Abstract The type 1 and type 2 forms of the Dirichlet distribution have been discussed for many years, as is evident in the work of Kotz, et al. [KBJ00]. Both types of distributions have diverse areas of application, ranging from biopharmaceuticals, genetics, forensic science, geology, pattern recognition, business, and economics. The type 3 distribution is becoming an area of interest for research, but has not received the same attention as the others. This report attempts to demonstrate the existence of a general distribution from which types 1,2 and 3 originate and from which a very broad class of Dirichlet distributions, as well as other well known multivariate distributions, are found. This work is based on the work of McDonald and Xu [MX95].

1

2

1

Introduction

The Dirichlet distribution is one of the most important multivariate distributions and appears in many applications. Areas of application include order statistics; probabilistic constrained programming models; project evaluation and review technique (PERT); biopharmaceuticals; genetics and evolution; forensic science; geology and geochemistry; pattern recognition; business and economics; political and social science; and artificial intelligence and machine learning. With such a diverse array of areas that this distribution is found, we wish to examine the common properties of the types of Dirichlet distributions used, to discuss how they are related, to discover a broader class of distributions within which such distributions exist, and to explore many of the fundamental properties of this class of distributions. To accomplish this, this report introduces a generalized Dirichlet distribution (GD) function which defines, as special cases, the Dirichlet type 1 (D1 ) and its generalization (GD1 ), the Dirichlet type 2 (D2 ) and its generalization (GD2 ), and the Dirichlet type 3 (D3 ) and its generalization (GD3 ). The generalized distribution is defined in Section 2 where it is shown to be an extension of the generalized beta (GB) distribution defined by McDonald and Xu [MX95]. General information on each of the classes of distributions listed here is provided in Section 3. It is shown in Section 3 that the distribution GD is derivable from either D1 or D2 , from Gamma distributions, and from beta distributions. Various properties of the distribution are established in Section 4. In this section it is shown, for instance, that the moment generating function and marginal distribution function properties are consistent in definition with those from D1 , D2 and D3 . Historically, the use of the expression “generalized Dirichlet distribution” has been used extensively in describing a class of Dirichlet distributions that satisfy various correlational properties among variables in the distribution (See, for example, the works of Connor and Mosimann [CM69] and Wong [Won98].) In a similar manner as McDonald and Xu, Section 5 lists several known multivariate distributions that are defined using special parameter setting in GD. A taxonomy of distributions is used to display the interrelationships of the distributions. A brief summary concludes this report. The following special notation will be used throughout this presentation: We will use the uppercase character to denote random vectors, such as X0 = (X1 , · · · , Xk ). This will commonly be a vector of length k where k ≥ 1. The vector x, which will denote an instance of the random vector X, will be defined as x0 = (x1 , · · · , xk ) where each k k P Q xi ≥ 0 and xi ≤ 1. For differentials of vector x, let dx = dxi . For a generic i=1

i=1

vector parameter α where α0 = (α1 , · · · , αk ), let −α0 = (−α1 , · · · , −αk ), and α10 = ³ ´ 1 1 0 α1 , · · · , αk . For any vector w, let w = (w1 , · · · , wk ), unless otherwise noted. In this presentation, we define a Dirichlet distribution to which we will assign a new type

3 designation; this will be the Dirichlet type c or generalized Dirichlet type c. The type c refers to a vector and is emboldened since the distribution that it is assigned to is defined distinctly by this vector. When the random vector X has the most general form of Dirichlet distribution we will symbolize this as X ∼ GD(a, b, c, ν).

2

Definition

The generalized Dirichlet distribution is of the following form:

Γ(ν+ ) GD(y | a, b, c, ν) =

k Q

·

k P

³ ´ai ¸νk+1 −1

1− (1 − ci ) ybii i=1 i=1 i=1 · ³ ´ai ¸ν+ k+1 k k Q Q P ai ν i Γ(νi ) bi 1+ ci ybii |ai |

i=1

0

k Q

k

yiai νi −1

i=1

(2.1)

i=1

0

where a = (a1 , · · · , ak ) ∈ < , b = (b1 , · · · , bk ) ∈ <+k , c0 = (c1 , · · · , ck ) ∈ [0, 1]k , k+1 P ν 0 = (ν1 , · · · , νk+1 ) ∈ <+k , with ν+ = νi . Throughout this presentation, we assume i=1

that, for all i, νi > 0. ¾ ½ ³ ´ ai k P yi ≤ 1 is the domain The simplex S = yi | yi ≥ 0 for i = 1, · · · , k , (1 − ci ) bi i=1

of integration for GD. Throughout this presentation, since parameters can modify the shape of the simplex S, the symbol S will be used without explicit reference to the vector parameters a, b, or c. Note 2.1. The generalized beta distribution, which is denoted as GB, is a special case of the generalized Dirichlet distribution when k = 1. McDonald and Xu [MX95] provide a complete examination of the properties of this distribution as well as its relationship to numerous well-known continuous probability distributions.

3

Derivation of GD

We will show that GD is a probability distribution function by first deriving this distribution from the Dirichlet type 1 distribution (or equivalently from the Dirichlet type 2). This will form a distribution that will be called GD(a, b, c, ν) defined at a point c = (c1 , · · · , ck ) where 0 ≤ ci ≤ 1 is a point in a unit hypercube. Using this distribution, and assuming a = b = 1 for simplicity, we derive GD(1, 1, c(2) , ν) from GD(1, 1, c(1) , ν), (1) (1) (2) (2) defined at points c(1) = (c1 , · · · , ck ) and c(2) = (c1 , · · · , ck ), respectively. Lastly, we derive this distribution from k + 1 independent gamma distributed random variables.

4

3.1

Derivation of GD from GD1

In this section, we will demonstrate that GD is derivable from the generalized Dirichlet type 1 distribution GD1 . This will establish GD as the most general Dirichlet distribution, defined for any c0 = (c1 , · · · , ck ) such that c ∈ [0, 1]k . In particular, this will include the Dirichlet type 1, defined at c0 = (0, · · · ,¡0), Dirichlet type 2, defined at ¢ c0 = (1, · · · , 1), and Dirichlet type 3, defined at c0 = 21 , · · · , 21 . Dirichlet types 1, 2, and 3 are discussed further in Sections 5. Assume that we have a Dirichlet type 1 distribution D1 (x | ν) or, equivalently, GD1 (x | 1, 1, ν). Let X ∼ D1 (ν) with X0 = (X1 , · · · , Xk ). Define the spaces X and Y with X ∈ X and Y ∈ Y, where Y is defined by the transformation T defined below. Our objective is to determine the distribution of Y. Thus, suppose we apply the transformation T such that T : Yi = 1−

Xi k P i=1

,

for i = 1, · · · , k

ci Xi

with inverse transformation T −1 : Xi = 1+

Yi k P i=1

,

for i = 1, · · · , k

ci Yi

To obtain the Jacobian of the transformation, we know that  µ ¶ k   −ci yi + 1 + P ci yi    i=1   , if i = j  µ ¶2  k  P   1+ ci yi   ∂xi i=1 =  ∂yj      −cj yi   if i = 6 j  µ ¶2 ,  k  P   1+ ci yi  i=1

Let y0 = (y1 , · · · , yk ) and Ik be the identity matrix of order k. The Jacobian is then

5 ¯ ¯ ¯ ∂(x1 , · · · , xk ) ¯ ¯ J = ¯¯ ∂(y1 , · · · , yk ) ¯ à =

1+

k X i=1

à =

1+

k X

¯ !−2k ¯ Ã ! k ¯ ¯ X ¯ ¯ ci yi ci yi Ik − c · y0 ¯ ¯ 1+ ¯ ¯ i=1

!−2k à ci yi

1+

i=1

à =

1+

k X

k X

!k−1 ci yi

i=1

!−(k+1) ci yi

.

i=1

using the method of expansion of determinants by diagonal elements, as defined by Aitken [See [Ait58], Section 37]. From this, the distribution of Y is

µ Γ GD(y | 1, 1, c, ν) =

k P

i=1 k Q i=1

¶ νi ·

Γ(νi )



k Y

 Ã !−1 νi −1 k X yi 1 +  ci yi

i=1

Ã

× 1 − yi

1+

i=1

k X

!−1 νk+1 −1 Ã !−(k+1) k X  ci yi 1+ ci yi

i=1

µ Γ =

k P

i=1 k Q i=1

i=1

¶ νi

Γ(νi )

·

k Y

à yiνi −1

1+

i=1

k X

!−ν+ " ci yi

1−

i=1

k X

#νk+1 −1 (1 − ci )yi

i=1

(3.1) ½ where S =

yi | yi ≥ 0 for i = 1, · · · , k ,

k P

¾ (1 − ci )yi ≤ 1

.

i=1

³ ∗ ´ ai y for i = 1, · · · , k, the probability distribuUsing the variable transformation yi = bii tion Equation 3.1 is written in the general Dirichlet distribution form GD(y∗ | a, b, c, ν) shown in Equation 2.1.

6

3.2

Derivation of GD(·|1, 1, c(2) , ν) from GD(·|1, 1, c(1) , ν)

Let Y ∼ GD(1, 1, c(1) , ν) with Y0 = (Y1 , · · · , Yk ). Assume the general Dirichlet dis(2) (1) tribution as defined in Equation 3.1, with c(1) = c. Let c∗i = ci − ci , where (1) (2) ci ∈ [0, 1] and ci ∈ [0, 1], for i = 1, · · · ¯, k. It is required that −1 ≤ c∗i ≤ 1, giv¯ ¯ (2) (2) (1) (1) ¯ ing us −1 ≤ ci − ci ≤ 1 or, equivalently, ¯ci − ci ¯ ≤ 1. Take the transformation Tc : Wi = 1−

Yi k P i=1

,

for i = 1, · · · , k

c∗i Yi

with inverse transformation Tc−1 : Yi =

Wi , k P 1+ c∗i Wi

for i = 1, · · · , k.

i=1

µ The Jacobian of this transformation is J = Dirichlet distribution is

1+

k P i=1

¶−(k+1) c∗i wi

. Using this, the general

7 GD(w | 1, 1,c(2) , ν)  νi −1   −ν+ µ k ¶ P Γ νi k  k     Y X wi wi     (1)  i=1 = k · 1 + c      i k k Q P P      i=1 Γ(νi ) i=1 1 + c∗i wi 1+ c∗i wi i=1

i=1





k ³  ´ X  (1)  × 1 − 1 − ci    i=1

νk+1 −1

1+

wi k P i=1

µ

k P

Γ =

i=1 k Q i=1

µ Γ =

i=1 k Q i=1

Ã

k X

1+

!−(k+1) c∗i wi

i=1

¶ νi ·

Γ(νi )

k P

c∗i wi

   

i=1

k Y

" wiνi −1

k ³ ´ X (1) 1+ ci + c∗i wi

i=1

#−ν+ (

k h ³ ´i X (1) 1+ 1 − ci + c∗i wi

i=1

i=1

¶ νi

Γ(νi )

·

k Y

à wiνi −1

1+

k X

!−ν+ " (2) ci wi

1−

i=1

i=1

k ³ X

1−

(2) ci

#νk+1 −1

´ wi

i=1

(3.2) ½ defined on the simplex S =

wi | wi ≥ 0 for i = 1, · · · , k ,

k ³ P i=1

1−

(2) ci

´

¾ wi ≤ 1 .

Consequently, we see that the general Dirichlet type c(2) distribution GD(·|1, 1, c(2) , ν) is derivable at all points c(2) ∈ [0, 1]k from any Dirichlet type c(1) distribution. Example 3.1. Let c(1) = 0 and c(2) = 1 so that c∗ = 1. Then our transformation is from the Dirichlet type 1 to Dirichlet type 2. Similarly, if we let c(1) = 1 and c(2) = 0 so that c∗ = −1, then our transformation is from the Dirichlet type 2 to Dirichlet type 1. Example 3.2. If we let c(1) = 0 and c(2) = c so that c∗ = c, then the transformation Tc allows a derivation of the distribution GD(·|1, 1, c, ν) from the distribution D1 , as was demonstrated in Section 3.1. A similar result follows when we let c(1) = 1 and c(2) = c so that c∗ = c − 1. In this case, the transformation allows a derivation of the distribution GD(·|1, 1, c, ν) from the inverse Dirichlet distribution D2 . The distribution D2 is discussed further in Section 5.6.

)νk+1 −1

8

Figure 1: Cube [0, 1]3 Containing Dirichlet Types Definition 3.1. To distinguish GD(· | a, b, c, ν) from GD(· | 1, 1, c, ν), we will call the latter form Dirichlet type c. Note that when Dirichlet type 1 is renamed Dirichlet type 0 where 00 = (0, · · · , 0) and Dirichlet type 2 is renamed Dirichlet type 1 where 10 = (1, · · · , 1), then this new vectordesignator is more descriptive in notation than the currently assigned notations of types 1 and 2. Likewise, the Dirichlet type 3 notation can now be renamed Dirichlet type 12 ¡ ¢ 0 where 12 = 21 · · · 12 . In general, the vector-designator for the general Dirichlet type c where c0 = (c1 , · · · , ck ) is used to define the general Dirichlet distribution.

3.3

Derivation of GD from the Gamma PDF

Suppose that the random variable Zi has a gamma distribution with parameter νi , or Zi ∼ Γ(νi ), for i = 1, · · · , k + 1 and that the transformation S is defined as  Zi  Xi = k+1 for i = 1, · · · , k  P   Zi   i=1 S:   k+1  P   Xk+1 = Zi i=1

Define the set Z with random variables Zi ∈ Z. Note that X0 = (X1 , · · · , Xk ) has the Dirichlet type 1 distribution, or X ∼ D1 (ν). ( Kotz, et al [KBJ00] Chapter 40, Section

9 1, for more information.) The transformation T ◦ S is defined using the transformations T from Section 3.1 and S:  Zi  k+1  P   Zj  Zi  j=1   =  for i = 1, · · · , k Y = i  k  P   k+1 Zk+1 + (1 − cj )Zj  P  Zi    1− ci  k+1 j=1  P i=1 Zj T ◦S : j=1          k+1  X    Y = Zi   k+1 i=1

The transformations S, T and the composite transformation T ◦S are shown in the figure below: X ~? @@@ S ~~~ @@T @@ ~~ ~ Â ~ /Y Z T ◦S

Figure 2: Transformations T and S

Special cases of the transformation T ◦ S are demonstrated in the following examples. Example 3.3. When c0 = (0, · · · , 0), Yi =

Zi

k+1 P

for i = 1, · · · , k, which corresponds

Zj

j=1

with the Dirichlet type 1 random variable (See Kotz, et al [KBJ00], Chapter 49, Section 1). i Example 3.4. When c0 = (1, · · · , 1), Yi = ZZk+1 for i = 1, · · · , k, which corresponds with the Dirichlet type 2 random variable (See Kotz, et al [KBJ00], Chapter 49, Section 2).

To determine the distribution of functions defined using T ◦ S, we must determine the inverse transformation S −1 ◦ T −1 which is equivalent to solving for Z1 , · · · , Zk in the k equations c1 Yj Z1 + · · · + (cj Yj + 1)Zj + · · · + ck Yj Zk − Yj Yk+1 = 0 for j = 1, · · · , k. This requires solving

10  c1 Y1 + 1  c1 Y2   ..  .

c2 Y1 c2 Y2 + 1 .. .

··· ··· .. .

ck Y1 ck Y2 .. .

c1 Yk

c2 Yk

···

ck Yk + 1



   Z1 Y1   Z2   Y2        ..  = Yk+1  ..  .  .  . Zk

Yk

The solution is of the form  Yi Yk+1   for i = 1, · · · , k Zi =  k  P   1+ cj Yj    j=1     Ã ! S −1 ◦ T −1 : k P   Yk+1 1 − (1 − cj )Yj  k  X  j=1   Zk+1 = Yk+1 − Zj = .   k  P  j=1  1 + c Y  j j j=1

The joint distribution of Z1 , · · · , Zk+1 is g(z1 , · · · , zk+1 ) = β ν+

1 k+1 Q i=1

k+1 Y

Γ(νi )

Pk+1

ziνi −1 e−

i=1 β

zi

(3.3)

i=1

where zi ≥ 0 for i = 1, · · · , k + 1. Substituting the solutions found in S −1 ◦ T −1 into Equation 3.3 we get  · ¸ νk+1 −1 k P   νi −1    Ã ! −1  y 1 − (1 − c )y k k i i    k+1 Y X 1 i=1 ∗   g (y1 , · · · , yk+1 ) = y y 1 + c y i k+1 i i k+1 k   Q P   i=1   β ν+ 1+ Γ(νi ) i=1 ci yi   i=1 −

×e

8 > < > :β

" yk+1 1+

k P i=1

c i yi

!

k P i=1

yi +1−

k P i=1

(1−ci )yi

i=1

9 #> = > ;

×J (3.4)

where the Jacobian J is

11 ¯ ¯ ¯ ∂(x1 , · · · , xk ) ¯ ¯ J = ¯¯ ∂(y1 , · · · , yk ) ¯ k yk+1

=µ ¶2k k P 1+ ci yi

¯Ã ¯ ! k ¯ ¯ X ¯ 0¯ ci yi Ik − c · y ¯ ¯ 1+ ¯ ¯ i=1

i=1

Ã

k yk+1

1+

=µ ¶2k k P 1+ ci yi

k X

!k−1 ci yi

i=1

i=1

k yk+1 =µ ¶k+1 . k P 1+ ci yi i=1

The function 3.4 can now be written and simplified to 1 k+1 Q

g ∗ (y1 , · · · , yk+1 ) = β ν+

i=1

× e−

k Y

Γ(νi )

yk+1 β

 ν −1

yi j

1 +

j=1

k X

−ν+  cj yj 

k X

1 −

j=1

νk+1 −1 (1 − cj ) yj 

j=1

ν −1

+ yk+1 .

(3.5) By integrating g ∗ with respect to the random variable Yk+1 we find g ∗∗ (y1 , · · · , yk ) = k+1 Q j=1



k Y

1

ν −1 yj j

1 +

Γ(νj ) j=1

Z∞ × 0

e−

yk+1 β

k Y

−ν+  cj yj 

1 −

j=1

k X

νk+1 −1 (1 − cj ) yj 

j=1

ν −1

+ yk+1

β ν+

k X

dyk+1



k X

−ν+ 

Γ (ν+ ) ν −1 yj j 1 + cj yj  = k+1 Q j=1 j=1 Γ (νj )

1 −

k X

νk+1 −1 (1 − cj ) yj 

.

j=1

j=1

(3.6) This is the generalized Dirichlet distribution function GD(y | 1, 1, c, ν).

12

4

Properties of GD

In order to characterize GD, we determine its moment generating function E (Y1r1 · · · Ykrk ). From this, we then demonstrate several well known special cases of the moment generating function. Following this, we derive the marginal distribution for GD along with several cases of special interest.

4.1

Moment Generating Function of GD

The moment generating function for GD is developed by the use of the Lauricella hypergeometric function type D and the Gauss hypergeometric function. See the monograph Exton [Ext76] for a thorough examination of hypergeometric functions used in this section. Using multivariate expected value operations with GD, we get the following: Z E

(Y1r1

· · · Ykrk )

=

···

Z Y k i=1

S

Z Γ(ν+ )

Z =

yiri GD(y | a, b, c, ν)dy

··· S

· ³ ´ai ¸νk+1 −1 k P yiai νi +ri −1 1 − (1 − ci ) ybii i=1 i=1 i=1 dy · ³ ´ai ¸ν+ k+1 k k Q Q P yi ai νi Γ(νi ) bi 1+ ci bi k Q

|ai |

k Q

i=1

i=1

i=1

(4.1) Using the transformation µ T : yi = bi

wi 1 − ci

¶ a1

i

with ci < 1 in Equation 4.1 for all i = 1, · · · , k, then E (Y1r1 · · · Ykrk ) becomes

(4.2)

13 · ³ ¶νk+1 −1 ´ a1 ¸ai νi +ri −1 µ k P i wi 1 |ai | bi 1−ci 1− wi Z Z k a −1 Y Γ(ν+ ) i=1 bi wi i i=1 i=1 · · · k+1 dw · ´ ¸ν+ k ³ k Q |ai | (1 − ci ) a1i P Q ai ν i −ci i=1 Γ(νi ) 1− bi S 1−ci wi k Q

k Q

i=1

i=1

i=1

Z Y Z  k k k r r νi + ai −1 Γ(ν+ ) Y ri Y − νi + ai i i ··· wi = k+1 bi (1 − ci ) Q i=1 i=1 i=1 Γ(νi ) S i=1

"

¶ k µ X −ci × 1− wi 1 − ci i=1

à 1−

k X

!νk+1 −1 wi

i=1

#−ν+ dw (4.3)

½ where S =

wi | wi ≥ 0, i = 1, · · · , k,

k P i=1

¾ wi ≤ 1 .

(k)

Using Lauricella function FD , then Equation 4.3 becomes Γ(ν+ ) k+1 Q i=1

Γ(νi )

k Y

bri i

k Y



i

i=1

i=1

à ×

r

− νi + ai

(1 − ci )

(k) FD

´ ³ Γ νi + arii i=1 ¶ µ k P ri Γ ν+ + ai

 Γ(νk+1 )

k Q

i=1

k X r1 rk ri −c1 −ck ν+ , ν1 + , · · · , νk + ; ν+ + ; ,··· , a1 ak a 1 − c 1 − ck 1 i=1 i

!

(4.4)

=

k Y i=1

i=1

Ã

×

(k) FD

´

³

Γ νi + arii µ ¶ k P ri Γ (νi ) Γ ν+ + ai

Γ(ν+ ) bri i k Q

k Q

i=1

i=1

k k X X ri r1 rk ri , ν1 + , · · · , νk + ; ν+ + ; c1 , · · · , ck a a a a 1 k i=1 i i=1 i

! .

14 Since

à (k) FD

k k X X ri ri r1 rk , ν1 + , · · · , νk + ; ν+ + ; c1 , · · · , ck a a a a 1 k i=1 i i=1 i

µ ¶ k P ri Γ ν+ + Z1 P ai k i=1 = µ k u i=1 ¶ P ri Γ Γ (ν+ ) 0 ai

ri ai

−1

Pk

ν+ −

(1 − u)

i=1

!





r

ci νi + ai −1

du

i

(4.5)

i=1

µ k P Γ ν+ + =

i=1

¶

·

ri ai

Γ (ν+ )

Γ ν+ − · Γ

k P i=1

k P i=1

³ ci νi +

³

(1 − ci ) νi +

ri ai

ri ai

´¸ ¸.

´ + νk+1

we have ³

k Q

E (Y1r1 , · · · , Ykrk ) =

k Y i=1

bri i

´ · ³ k P Γ ν+ − ci νi +

Γ νi + i=1 · k ³ P Γ (νi ) Γ (1 − ci ) νi +

i=1 k Q i=1

ri ai

i=1

ri ai

ri ai

´¸ ¸.

´

(4.6)

+ νk+1

Example 4.1. If we let ri = 0 for all i = 1, · · · , k, then from Exton ([Ext76], Equations 2.3.5 and 2.3.6), using Equation 4.4, we get Z

Z ···

(k)

GD(y | a, b, c, ν)dy = FD (0, ν1 , · · · , νk ; ν+ ; c1 , · · · , ck )

S

Γ(ν+ ) = k+1 Q Γ(νi )

Z ···

Z Y k

S

à ziνi −1

i=1

1−

k X

!νk+1 −1 zi

dz

i=1

i=1

k+1 Q

Γ(νi ) Γ(ν+ ) i=1 = k+1 Q Γ(ν+ ) Γ(νi ) i=1

=1 (4.7) k+1 Q

where the right-hand multiple integral has measure

i=1

Γ(νi )

Γ(ν+ ) probability distribution function, defined in Section 5.1.

, being a Dirichlet type 1

15 Example 4.2. If we set ci = 0 for all i = 1, · · · , k, then Equation 4.4 can be written as

E (Y1r1 · · · Ykrk ) =

k Y

Γ(ν+ ) bri i

k Q

k Q

³ Γ νi +

i=1 ·k+1 ³ P

ri ai

Γ(νi )Γ νi + i=1 ´  ³ Γ νi + arii k Q   Γ(νi ) k i=1 Y = bri i ·k+1 ³ ´¸ P ri i=1 Γ νi + ai i=1

i=1

´

ri ai

´¸

(4.8)

i=1

Γ(ν+ ) where rk+1 = 0. This is the moment generating function for the generalized Dirichlet type 1 probability distribution function, defined in Section 5.1. When ai = bi = 1 for all i = 1, · · · , k, we have the moment generating function for the Dirichlet Type 1 probability distribution function (See Kotz, et al. [KBJ00], p. 488). Example 4.3. If we set ci = 1 for all i = 1, · · · , k, then Equation 4.4 can be written as

E (Y1r1 · · · Ykrk ) =

k Y

Γ(ν+ ) bri i

i=1

k Q i=1

à ×

(k) FD

k Y

i=1

³ Γ νi +

·

Γ(νi )Γ ν+ +

bri i i=1

i=1

k+1 P³ i=1

³ Γ νi +

ri ai

i=1

bri i

ri ai

´¸

k+1 Q

´ µ k P Γ νk+1 −

i=1

Γ(νi )Γ(νk+1 )

´  ³ Γ νi + arii   Γ(νi ) i=1 k Q

=

´

!

i=1

k Y

ri ai

k k X X ri r1 rk ri , ν1 + , · · · , νk + ; ν+ + ; 1, · · · , 1 a a1 ak a i=1 i i=1 i

k Q

=

k Q

Γ(νk+1 ) µ k P Γ νk+1 −

i=1

¶ ri ai

¶ ri ai

16 where νk+1 −

k P i=1

ri ai

> 0. This is the moment generating function for the generalized

Dirichlet type 2 probability distribution function, defined in Section 5.2. In particular, if we set ai = bi = 1 for all i = 1, · · · , k, then we have the moment generating function for the Dirichlet type 2 probability distribution function. (See Kotz, et al. [KBJ00], p. 492 for more details). 1 2

Example 4.4. If we set ci =

E

(Y1r1

· · · Ykrk )

Γ(ν+ )

k Y

=

for all i = 1, · · · , k, then Equation 4.4 can be written as

Γ (νi + ri ) Γ(νk+1 ) · k ¸ P Γ(νi )Γ (νi + ri ) + νk+1

bri i k+1 Q

i=1

k Q

i=1

i=1

i=1

(4.9) Ã (k)

× FD

k k X X ri r1 rk ri 1 1 , ν1 + , · · · , νk + ; ν+ + ; ,··· , a a1 ak a 2 2 i=1 i i=1 i

!

This is defined as the moment generating function for the generalized Dirichlet type 3 distribution, defined in Section 5.3. Using Equation 4.9, if we set ai = 1 and bi = 12 , for i = 1, · · · , k, then

E

(Y1r1

· · · Ykrk )

=

k µ ¶ri Y 1

2

i=1

Γ(ν+ )

Γ (νi + ri ) Γ(νk+1 ) · k ¸ k+1 Q P Γ(νi )Γ (νi + ri ) + νk+1 i=1

à ×

(k) FD

−

k P

k Q

i=1

i=1

k X

k X

1 1 ri , ν1 + r1 , · · · , νk + rk ; ν+ + ri ; , · · · , 2 2 i=1 i=1 k Q

ri

!

(4.10)

Γ(ν+ ) Γ(νi + ri ) i=1 = k · k ¸ P Q Γ(νi )Γ (νi + ri ) + νk+1 2

i=1

i=1

× 2 F1

Ã

i=1

k X

k X

k X

1 ri , (νi + ri ); ν+ + ri ; 2 i=1 i=1 i=1

!

using results from Exton ([Ext76], p. 288, Equation A.2.10) and where 2 F1 is the Gauss hypergeometric function.

17 Equation 4.10 can be rewritten as k Q Ã ! 2−νk+1 Γ(ν+ ) Γ(νi + ri ) k X 1 i=1 rk r1 E (Y1 · · · Yk ) = k ri ; · k ¸ 2 F1 νk+1 , ν+ ; ν+ + Q P 2 i=1 Γ(νi )Γ (νi + ri ) + νk+1 i=1

i=1

corresponding to the moment generating function for D3 provided by Carden˜o, et al., [CNS05].

4.2

Marginal Distribution of GD

If we apply the transformation T defined in Equation 4.2 to GD, we get k k Y Γ(ν+ ) Y −ν f (w, c, ν) = k+1 (1 − ci ) i wiνi −1 Q i=1 Γ(νi ) i=1

à 1−

k X i=1

!νk+1 −1 " wi

¶ k µ X −ci 1− wi 1 − ci i=1

i=1

(4.11) Let c0i = −

ci (1) and ci = − 1 − ci

1−

c0i m P i=1

From Equation 4.11, the expression

where 1 ≤ m < k and ci < 1 for i = 1, · · · , m. c0i wi

µ ¶− ν+ k P 1− can be written as c0i wi i=1

#−ν+ .

18

 Ã 1−

m X

!− ν+

  1 − 

c0i wi

i=1

− ν +

k P i=m+1 m P

c0i wi

1−

i=1

à =

1−

m X

!− ν+ Ã c0i wi

=

1−

m X

·³

=

1−

X

c0i wi



´lm+1

lj , lk 

m+1 ≤ j < k

³ ´l ¸ lm+1 wlk (1) k wm+1 ··· k · · · −ck lm+1 ! lk !

!− ν+

X

c0i wi

i=1

X

(ν+ , lm+1 )(ν+ + lm+1 , lm+2 ) · · · ν+ +

lm+1 , ··· , lk (1)

m X



!− ν+

−cm+1

Ã

(1) ci wi

i=m+1

i=1

×

!− ν+

k X

1−

i=1

Ã

c0i wi

   

à ν+ ,

k X

!· ³ li

(1) −cm+1

´lm+1

³ ···

(1) −ck

´lk ¸ wlm+1 m+1

lm+1 !

i=m+1

lm+1 , ··· , lk

···

wklk lk !

P Γ(a + n) is the multiple , with a > 0 and n ∈ Z and where Γ(a) lm+1 , ··· , lk sum over lm+1 , · · · , lk , where 0 ≤ lj < ∞ for j = m + 1, · · · , k. Then the marginal distribution of GD, denoted GD(m) , for variables (w1 , · · · , wm ), is where (a, n) =

GD(m) (w | 1, 1, c, ν) Z Z Y k k Γ(ν+ ) Y −νi = k+1 (1 − ci ) ··· wiνi −1 Q i=1 Γ(νi ) i=1 S i=1

×

à 1−

m X

!− ν+ c0i wi

i=1

where dw = dwm+1 · · · dwk .

X lm+1 , ··· , lk

à ν+ ,

à 1−

k X i=m+1

k X

!νk+1 −1 wi

i=1

!· li

³

(1) −cm+1

´lm+1

³ ···

(1) −ck

´lk ¸ wlm+1 m+1

lm+1 !

···

wklk dw lk !

19 Since Γ(ν+ ) k+1 Q Γ(νi )

Z ···

Z Y m

S

wiνi −1

i=1

m Y

à wiνi +li −1

1−

k X

!νk+1 −1 wi

dw

i=1

i=1

i=1

k Q Pk à !Pk+1 Γ(ν+ ) Γ(νi + li ) i=m+1 li −1 i=m+1 νi + m m Y X i=m+1 νi −1 = k+1 w 1 − w µ k+1 ¶ i i k Q P P i=1 Γ(νi )Γ νi + li i=1 i=1

i=m+1

i=m+1

k Q

à ! (νi , li ) m m Y X νi −1 wi 1− = m wi µ k+1 ¶ µ k+1 ¶ k Q P P P i=1 Γ(νi )Γ νi Γ νi , li i=1 Γ(ν+ )

i=1

Pk+1 i=m+1

νi +

Pk

i=m+1 li −1

i=m+1

i=m+1

i=m+1

i=m+1

we get GD(m) (w | 1, 1, c, ν) k Y Γ(ν+ ) = m (1 − ci )−νi µ k+1 ¶ Q P Γ(νi )Γ νi i=1 i=1

×

lm+1 ,··· ,lk

1+

m X

k P

c0i wi

 ¶

k Q

ν+ , li (νi , li ) i=m+1 i=m+1 µ k+1 ¶ k P P νi , li i=m+1

!− ν+

i=1

i=m+1

µ X

Ã

k Y i=m+1

m Y

à wiνi −1

m X

1−

i=1



 − c0i 

1−

m P

m P

i=1

wi

i=1

wi

i=1

1+

!Pk+1 j=m+1 νj −1

li 

c0i wi

li !

i=m+1

(4.12) Applying the transformation T from Equation 4.2 we can write this expression as

20 GD(m) (y | a, b, c, ν) m Q Γ(ν+ ) |ai | i=1 = m µ k+1 ¶ m Q P Q ai νi Γ(νi )Γ νi bi i=1

i=m+1

i=1

P µ ¶ai # k+1 i=m+1 νi −1 m X y i νi −1 −νi × (1 − ci ) 1+ ci yi 1− (1 − ci ) bi i=m+1 i=1 i=1 i=1     ³ ´ ³ ´ ai   m m ai P P yi yi 1 − (1 − c ) 1 − (1 − c ) k+1 i i bi bi X     (k−m)  i=1 i=1 0 0 ν+ , νm+1 , · · · , νk ;  , · · · , − cm+1   × FD νi ; − ck  ³ ³ ´ ´ m m a a      i i P P i=m+1 1+ ci ybii 1+ ci ybii k Y

"

m X

µ

yi bi

"

¶ai #− ν+ Y m

i=1

i=1

(4.13) ½ Equation 4.13 is defined on the simplex S =

yi | yi ≥ 0 for i = 1, · · · , m ,

with c0 = (c1 , · · · , cm ) ∈ [0, 1]m and ci < 1, for m + 1 ≤ i ≤ k.

m P

(1 − ci )

i=1

³ ´ ai yi bi

Example 4.5. If we let ci = 0 for i = 1, · · · , m, then Γ(ν+ ) GD(m) (y | a, b, 0, ν) =

m Q i=1

|ai |

m Q

i=1 m Q

i=1

(m)

≡ GD1

· yiai νi −1

Γ(νi )Γ

µ

1−

m ³ ´ ai P yi bi

i=1 k+1 P

i=m+1

¶

νi

¸Pk+1 j=m+1 νj −1

m Q i=1

bai i νi

(4.14)

(y | a, b, 0, ν)

Equation 4.14 is defined on the simplex ( S=

yi | yi ≥ 0 for i = 1, · · · , m ,

¶a m µ X yi i i=1

bi

) ≤1

(4.15)

In this instance GD(m) is in the generalized Dirichlet type 1 family of distributions.

¾ ≤1 ,

21 Example 4.6. Let ci = 0 for i = 1, · · · , k, and m = 1. Then h ³ ´a1 iPk+1 j=2 νj −1 Γ (ν+ ) |a1 |y1a1 ν1 −1 1 − yb11 ! Ã GD(1) (y1 ; a, b, c, ν) = k+1 P Γ (ν1 ) Γ νj j=2

 ≡ GB1 y1 : a1 , b1 , ν1 ,

k+1 X



νj 

j=2

where ba1 1 > y1a1 ≥ 0. The function GB1 is the generalized beta type 1 distribution function, defined by McDonald and Xu [MX95], Equation 2.1.

Example 4.7. If we let bi = ci =

1 2

and ai = 1 for i = 1, · · · , m, then

µ ¶ 1 1 GD(m) y | 1, , , ν 2 2

µ ¶Pk+1 j=m+1 νj −1 m P Γ(ν+ ) 2 i=m+1 1− yi i=1 i=1 = µ k+1 ¶ m µ ¶ν+ m m Q P Q P Γ(νi )Γ νi 1+ yi Pk

i=1

(k−m)

× FD

νi



m Q

yiνi −1

i=m+1

i=1

 ν+ , νm+1 , · · · , νk ; 

i=1

k+1 X i=m+1



 νi ; −  

1− 1+

m P i=1 m P i=1

 yi yi



  ,··· ,−  

1− 1+

m P i=1 m P i=1

 yi yi

  

µ ¶Pk+1 j=m+1 νj −1 m P Γ(ν+ ) 2 i=m+1 νi yiνi −1 1 − yi i=1 i=1 = µ k+1 ¶ m µ ¶ν+ m m Q P Q P Γ(νi )Γ νi 1+ yi Pk

i=1

(k−m)

× 2 F1

m Q

  ν+ , 

i=m+1 k X

i=m+1

νi ;

i=1

k+1 X i=m+1

i=1



 νi ; −  

1− 1+

m P i=1 m P i=1

 yi yi

  . 

(4.16) This corresponds to the result of Carden˜o, et al., [CNS05] in which the marginal distribution GD(m) is shown to not be in Dirichlet type 3 family of distributions.

22

4.3

Mixed Type 1 - Type 2 Dirichlet Distribution Functions

Assume the transformation similarly defined as in Section 3.1, with a = b = 1 (without loss of generality) and c ∈ {0, 1}k . Also, define T = {i | ci = 1, i = 1, · · · , k} and T 0 = {i | ci = 0, i = 1, · · · , k}. Let |T | ≥ 1 and |T 0 | ≥ 1 with |T | + |T 0 | = k. Then, from Equation 2.1, we have µ ¶νk+1 −1 P yiνi −1 1 − yi i∈T ∪T 0 i∈T 0 µ ¶ν+ Q P Γ (νi ) Γ (νk+1 ) 1 + yi

Γ (ν+ ) GD(y | 1, 1, c, ν) =

Q

i∈T ∪T 0

i∈T

½ yi | yi ≥ 0 for i ∈ T ∪ T 0 and

defined over the simplex S =

P i∈T 0

¾ yi ≤ 1 . This proba-

bility distribution function will be called the mixed type 1 - type 2 Dirichlet distribution function since it combines properties of both Dirichlet type 1 and Dirichlet type 2. This is easily determined to be true through the following two examples: Example 4.8. The distribution for yT 0 = {yi | i ∈ T 0 } is the function f defined by Z∞

Z∞

dyi

i∈T

0

0

Y

GD(y | 1, 1, c, ν)

···

f (yT 0 ) =

¶ µ P Ã !νk+1 −1 νi Γ ν+ − X Y i∈T νi −1 1− yi = Q yi . Γ (νi ) Γ (νk+1 ) 0 0 i∈T

i∈T

i∈T 0

½ This is the Dirichlet type 1 defined on S1 =

0

yi | yi ≥ 0 for i ∈ T and

P i∈T 0

¾ yi ≤ 1 .

In a similar manner as just shown, we can find the distribution for yT = {yi | i ∈ T }: Example 4.9. The function g is defined by Z g(yT ) =

Z ···

GD(y | 1, 1, c, ν)

Y

dyi

i∈T 0

S1

Q

yiνi −1 Γ (ν+ ) µ ¶ µ i∈T ¶ν+ . = Q P P Γ (νi ) Γ ν+ − νi 1+ yi i∈T

i∈T

i∈T

This is the Dirichlet type 2 distribution defined on S2 = {yi | yi ≥ 0 for i ∈ T }.

23 From the prior two examples we observe that g(yT ) · f (yT 0 ) = GD(y|1, 1, c, ν), where c is suitably chosen from {0, 1}k .

5

Relationships Between GD and Other Multivariate PDFs

The multivariate probability distributions defined in this section are special cases of GD in Equation 2.1 when parameter values for a, b, ν and c from GD are selected and substituted into the function. The distribution functions that are considered include: • (Generalized) Dirichlet type 1; • (Generalized) Dirichlet type 2; • (Generalized) Dirichlet type 3; • (Generalized) Multivariate Lomax; • Multivariate f; • (Generalized) Multivariate Cauchy; • Multivariate Burr; • Multivariate log-logistic; and • Special cases of the multivariate gamma and the multivariate normal. For more information on many of the distributions listed, see Kotz, et al. [KBJ00].

5.1

Generalized Dirichlet Type 1

When ci = 0 for i = 1, · · · , k, then the generalized Dirichlet type 1 distribution (denoted GD1 ) is written as GD1 (y | a, b, ν) = GD(y | a, b, 0, ν) Γ(ν+ ) =

k Q i=1

|ai |

k Q

yiai νi −1

i=1 k+1 Q i=1

where yi > 0 for i = 1, · · · , k, and

k ³ ´ai P yi i=1

bi

Γ(νi )

≤ 1.

¸ · k ³ ´ai νk+1 −1 P yi 1− bi i=1

k Q i=1

bai i νi

(5.1)

24 1. Dirichlet type 1 Using the generalized Dirichlet type 1, Equation 5.1, when ai = bi = 1 for i = 1, · · · , k, then the Dirichlet type 1 distribution (denoted D1 ) is written D1 (y | ν) = GD1 (y | 1, 1, ν) Γ(ν+ ) =

k Q i=1

µ

yiνi −1

1−

i=1

k+1 Q i=1

where yi ≥ 0 for i = 1, · · · , k, and

k P

k P

¶νk+1 −1 yi

(5.2)

Γ(νi )

yi ≤ 1.

i=1

2. Inverse Dirichlet Type 1 Using Equation 5.1 with ai = −1, bi = 1 for i = 1, · · · , k, the inverse Dirichlet type 1 distribution (denote ID1 ) is defined as ID1 (y | ν) = GD1 (y | -1, 1, ν) µ ¶νk+1 −1 k k ³ ´νi −1 P Q 1 1 1− Γ(ν+ ) yi yi i=1 i=1 = k+1 Q Γ(νi )

(5.3)

i=1

where yi ≥ 0 for i = 1, · · · , k, and

k P

yi−1 ≤ 1.

i=1

3. Independent Generalized Gamma 1 a

i Let β = (β1 , · · · , βk ). Substituting bi = νk+1 βi for i = 1, · · · , k so that b∗ = ¶ µ 1 1 ak a1 βk in Equation 5.1, we get β1 , · · · , νk+1 νk+1

f (y | a, β, ν) = GD(y | a, b∗ , 0, ν) Γ(ν+ ) =

k Q i=1

|ai |

k Q

· yiai νi −1

i=1 k+1 Q i=1

where yi ≥ 0 for i = 1, · · · , k, and

Γ(νi )

k ³ P i=1

a

1−

i=1

k Q i=1

yi i a νk+1 βi i

k ³ P

a

yi i a νk+1 βi i

´¸νk+1 −1

νi νk+1 βi ai νi

´ ≤ 1.

Then the independent generalized Gamma is given by  a i   y k ai νi −1 − βii Y |a | y e i i   IGG(y | a, ν, β) = lim f (y | a, β, ν) = ai νi νk+1 →∞ Γ(ν )β i i i=1 where yi ≥ 0 .

(5.4)

25 4. Independent Normal (Special Case) Let β = σ = (σ1 , · · · , σk ). By setting ai = 2 and νi = 21 for i = 1, · · · , k and using the independent generalized gamma IGG we get the independent normal (denoted IN ), written as IN (y) =IGG(y | 2, 1/2, σ) ! Ã 2 2 k Y 2e−yi /σi √ = σi π i=1 k

=

2 k 2

0

π |Σ|

1 2

−1

e−y Σ

(5.5)

y

  where 0 < yi < ∞ for i = 1, · · · , k and where Σ = 

σ12

0 ..

0 this is defined only for positive variables.

5.2

.

   . Note that

σk2

Generalized Dirichlet Type 2

If we let ci = 1 for i = 1, · · · , k, then the generalized Dirichlet type 2 distribution (denoted GD2 ) is defined as GD2 (y | a, b, ν) = GD(y | a, b, 1, ν) k Q

k Q |ai | yiai νi −1 i=1 i=1 = ¸ · k+1 k k ³ ´ai ν+ Q Q P yi Γ(νi ) bai i νi 1 + bi

Γ(ν+ )

i=1

i=1

(5.6)

i=1

where 0 < yi < ∞ . 1. Dirichlet Type 2 When ai = bi = 1 for i = 1, · · · , k, then the Dirichlet type 2, more commonly referred to as the inverse Dirichlet distribution (denoted D2 ) is defined as D2 (y | ν) = GD2 (y | 1, 1, ν) Γ(ν+ ) =

k+1 Q i=1

where 0 < yi < ∞ .

Γ(νi )

k Q i=1

k Q i=1

yiνi −1

µ ¶ν+ k P 1+ yi i=1

(5.7)

26 2. Generalized Multivariate Cauchy If we let ai = 2, bi = 2, and νi = 12 for all i = 1, · · · , k and νk+1 = m − k2 , then we can say that ¡ ¢ ¢ ¡ Γ k+1 Γ k+1 Γ(ν+ ) 2 2 = k+1 = . k+1 k+1 Q Q ¡1¢ π( 2 ) Γ(νi ) Γ 2 i=1

i=1

Thus, the generalized multivariate Cauchy distribution (denoted GM C) is defined as µ µ µ ¶¶¶ 1 1 k GM C(y | m) = GD2 y | 2, 2, ,··· , , m − 2 2 2 Γ(m) h = k ¡ ¢ ¡ ¢2 ¡ ¢2 im π 2 Γ m − k2 1 + y21 + · · · + y2k

(5.8)

where 0 < yi < ∞ for i = 1, · · · , k. When we take m = k+1 in Equation 5.8, then we can define the multivariate 2 Cauchy distribution as Γ( k+1 2 )

M C(y) = π

k+1 2

£ ¤ k+1 1 + ( y21 )2 + · · · + ( y2k )2 2

(5.9)

where 0 < yi < ∞ . 3. Generalized Multivariate Lomax If we let ν = (`, a), where ` = (`´1 , · · · , `k ), a = 1, where 1 is a unit vector of ³ 1 1 length k, and b = θ = θ1 , · · · , θ1k where θ = (θ1 , · · · , θk ), then the generalized multivariate Lomax (denoted GM L) is defined as µ µ ¶ ¶ 1 1 GM L(y | a, θ, `) = GD y | 1, ,··· , , 1, (`1 , · · · , `k , a) θ1 θk µ Γ

k P

i=1

= Γ(a)

k Q i=1

¶ `i + a

k Q i=1

µ Γ(`i ) 1 +

θi`i

k Q i=1

¶ P `i +a k

k P i=1

θi y i

(5.10)

yi`i −1 .

i=1

where 0 < yi < ∞. When `i = 1 for i = 1, · · · , k, then the multivariate Lomax distribution is

27 defined similarly and we will write M L(y | a, θ) = GM L(y | a, θ, 1) k Q

Γ(k + a) =

i=1

µ

k P

Γ(a) 1 +

i=1

θi

(5.11)

¶k+a

θi y i

where 0 < yi < ∞ . Nayak [Nay87] studies the multivariate Lomax distribution with its generalization and demonstrates its relationship to multivariate f, multivariate Pareto Type 2, and multivariate Burr. 4. Multivariate f Using the generalized multivariate Lomax distribution, if we let θi = a`ii for all i = 1, · · · , k, then the multivariate f distribution (denoted M F ) is defined as M F (y | `, a, a) = GM L(y | a, θ, `) µ Γ

k P

i=1

=

¶ k ³ ´ k Q `i `i Q `i −1 yi `i + a ai i=1

i=1

(5.12)

· ¸ P `i +a k ³ ´ P i=1 `i Γ(`i ) 1 + ai yi k

Γ(a)

k Q i=1

i=1

where 0 < yi < ∞. For further information on the relationship between multivariate Lomax and multivariate f, see Nayak [Nay87], p.176. 5. Multivariate Log-Logistic Using the multivariate Lomax distribution Equation 5.11, when we set a = 1, then the multivariate log-logistic distribution (denote MLL) is defined as µ k ¶ Q θi k! i=1 M LL(y | θ) = µ (5.13) ¶k+1 k P 1+ θi y i i=1

where 0 < yi < ∞ for i = 1, · · · , k. Note that when k = 1, M LL(y) is the θ where 0 < y < ∞. log-logistic distribution f (y) = (1 + θy)2 6. Multivariate Burr Using GD2 , Equation 5.6, we let νi = 1 for i = 1, · · · , k, νk+1 = a and set bi = ³ ´ c1 i 1 and ai = ci for i = 1, · · · , k, we get the multivariate Burr distribution di

28 (denoted M B), defined as k Q

Γ(k + a) M B(y | a, c, d) = Γ(a)

k Q

· ³ ´ 1 ¸ ci c 1 di

i=1

i

i=1



 1 +

a(a + 1) · · · (a + k − 1) =

·

k P

1+

i=1

k Q

ci yi ci −1  k P i=1

ci a+k

 yi   ³ ´ 1   c 1 di

i

(5.14)

di ci yici −1

i=1 ¸a+k

di yici

where 0 < yi < ∞. This defines the multivariate Burr distribution discussed by Takahasi [Tak65]. For further information on the relationship between multivariate Lomax and multivariate Burr, see Nayak [Nay87], p.172. 7. Multivariate Pareto Type 2 Using the multivariate Lomax distribution Equation 5.11, when we set bi = θ1i = 1 for all i, then the multivariate Pareto Type 2 distribution (denoted M P2 ) is defined as à !−(a+k) k X (5.15) M P2 (y | a) = a(a + 1) · · · (a + k − 1) 1 + yi i=1

where 0 < yi < ∞ . The distribution M P2 is derivable from the inverted Dirichlet distribution Equation 5.7 when we set νi = 1 for i = 1, · · · , k and νk+1 = a, as well as from the multivariate Lomax distribution Equation 5.11 when θi = 1 for i = 1, · · · , k. For further information on the multivariate Pareto distribution see Mardia [Mar62].

5.3

Generalized Dirichlet Type 3

When we set ci = 12 for i = 1, · · · , k, by using Equation 2.1, we get the generalized Dirichlet type 3 (denoted GD3 ), defined as GD3 (y | a, b, ν) = GD(y | a, b, 1/2, ν) Γ(ν+ ) =

k Q

i=1 k+1 Q i=1

|ai |

k Q

i=1 k Q

Γ(νi )

·

yiai νi −1

i=1

1−

· bai i νi 1 +

k P

i=1 k P 1 2 i=1

1 2

³ ´ai ¸νk+1 −1 yi bi

³ ´ai ¸ν+ yi bi

(5.16)

29 where yi ≥ 0 for i = 1, · · · , k, and

k P i=1

1 2

³ ´ ai yi bi

≤ 1.

1. Dirichlet Type 3 In particular, if we set ai = 1 and bi = 12 , then the Dirichlet Type 3 distribution (denoted D3 ) is defined as D3 (y | ν) = GD3 (y | 1, 1/2, 1/2, ν) k Γ(ν+ ) Pki=1 νi Y νi −1 = k+1 2 yi Q i=1 Γ(νi )

à 1−

k X

!νk+1 −1 Ã yi

1+

i=1

k X

!−ν+ yi

i=1

i=1

(5.17) where 0 < yi , νi > 0 for all i,

k P i=1

yi < 1 and k ≥ 1. This distribution is the

multivariate generalization of the beta type 3 distribution, denoted B3 (See Carden˜o, et al. [CNS05]). In particular, when k = 1, D3 (y | ν) = B3 (y | ν1 , ν2 ). For additional information on B3 and D3 , see Carden˜o, et al., [CNS05]. 2. Inverse Dirichlet Type 3 When we set ai = −1, bi = 2, and ci = 12 for i = 1, · · · , k, then the inverse Dirichlet type 3 distribution (denoted ID3 ) is defined as ID3 (y | ν) = GD3 (y | -1, 2, ν) Γ(ν+ ) = k+1 Q Γ(νi )

2

−

i=1

where 0 < yi < ∞ for i = 1, · · · , k and

µ ¶νk+1 −1 k P 1 1− yi i=1 i=1 ¶ ν+ µ k k Q P 1 yiνi +1 1 + yi

Pk

i=1 νi

k Q

i=1 k P i=1

i=1

1 yi

≤ 1.

(5.18)

30

6

PDF Taxonomy

A taxonomy is provided to organize the classes of multivariate distributions discussed in Section 5 and illustrate the relationships between the three commonly used Dirichlet distributions and other common multivariate distributions. The general Dirichlet distribution (GD) is defined with the largest number of parameters, 4k + 1, and is depicted at the center. Distributions that are one step from GD have 3k + 1 parameters. Distributions that are two or more steps from GD have less than 3k + 1 parameters. D3 ( 5.17) O

ID3 ( 5.18) o7 o o oo o o oo ooo GD3 ( 5.16) O

GD1 ( 5.1) qq q q q qqq ² xqqq ID1 ( 5.3) D1 ( 5.2) ² IG( 5.4) ² IN ( 5.5)

ggg gggg g g g g gggg gggg gs ggg

GD( 2.1)

² GD2 ( 5.6) WW OOO WWWW OOO WWWWW WWWWW OOO WWWWW OOO WW+ ² ' GM L( 5.10) D2 ( 5.7) GM C( 5.8) nn n n nn nnn ² wnnn ² M F ( 5.12) M L( 5.11) M C( 5.9) OOO n OOO nnn n OOO n nn OOO n n ² wnn ' ² M LL( 5.13) M B( 5.14) M P2 ( 5.15)

Figure 3: PDF Taxonomy Although numerous distributions have been identified and located in this taxonomy, it is uncertain that it is complete.

7

Conclusion

By demonstrating that the Dirichlet distribution encompasses a much broader class of distributions than has been shown to date allows us an opportunity to extend our knowledge of this distribution. From what has been provided in this report, we have

31 seen that the generalized Dirichlet distribution GD includes a wide class of well-known distributions as special cases. We have also found that it is consistent with the other Dirichlet distributions when selected parameters are used. The down side of this work is that marginal distributions are found to not be in the same class of distributions as the original distribution. This may limit the possibility of achieving useful results in such areas as conditional distributions in the general case. This subject needs to be further investigated before making any further claims. The GD distribution has been shown to be derivable from gamma and beta distributions; a strong possibility exists for this distribution to be extended in several areas of investigation, including: 1. Developing methods for parameter estimation, including maximum likelihood or estimation-maximization. More elaborate methods are certain to be needed as the number of parameters increase in the distribution. 2. Based on the multitude of applications that have appeared in such areas as business, economics, social science, biological science, and others, it is essential that this new distribution be demonstrated in similar applications. 3. Developing further results that rely on the use of beta functions. This includes extending the results to concepts of neutrality with applications to the generalized Dirichlet distribution defined by Connor and Mosimann. 4. Since this work introduces us to a new type of Dirichlet distribution, we see the possibility for extending this work to include, at the minimum the following items: (a) Extending results in Dirichlet process theory; (b) Extending results in matrix variate theory. This would require extending results in matrix variate beta and gamma distributions and the Lauricella hypergeometric functions; (c) Extending results in Liouville distribution theory; and (d) Computing probability integral measures of the generalized Dirichlet distribution.

32

References [Ait58]

A. C. Aitken. Determinants and Matrices. Oliver and Boyd, 1958.

[CM69] R. J. Connor and J. E. Mosimann. Concepts of independence for propositions with a generalization of the Dirichlet distribution. Journal of the American Statistical Association, 64:194–206, 1969. [CNS05] Liliam Carden˜o, Daya K. Nagar, and Luz Estela S´anchez. Beta type 3 distribution and its multivariate generalization. Tamsui Oxford Journal of Mathematical Sciences, 2005. [Ext76] Herald Exton. Multiple Hypergeometric Functions and Applications. John Wiley and Sons Inc., New York, 1976. [KBJ00] Samuel Kotz, N. Balakrishnan, and Norman L. Johnson. Continuous Multivariate Distributions, Volume 1: Models and Applications. John Wiley and Sons, Inc., New York, 2nd edition, 2000. [Mar62] K. V. Mardia. Multivariate Pareto distributions. Annals of Mathematical Statistics, 33:1008–1015, 1962. [MX95] James B. McDonald and Yexiao J. Xu. A generalization of the beta distribution with applications. Journal of Econometrics, 66:133–152, 1995. [Nay87] T. K. Nayak. Multivariate Lomax distribution: Properties and usefulness in reliability theory. Journal of Applied Probability, 24:170–177, 1987. [Tak65] K. Takahasi. Note on the multivariate Burr’s distribution. The Annals of the Institute of Statistical Mathematics, 17:257–260, 1965. [Won98] T. T. Wong. Generalized Dirichlet distributions in Bayesian analysis. Applied Mathematics and Computation, 97:165–181, 1998.