Technical Report

The Dirichlet Family of Distributions

Gary M. Johnson

Joint Warfare Analysis CenterDahlgren, Virginiagjohnson@jwac.mil

November 7, 2007

Abstract

The type 1 and type 2 forms of the Dirichlet distribution have been discussedfor many years, as is evident in the work of Kotz, et al. [KBJ00]. Both types ofdistributions 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 notreceived the same attention as the others. This report attempts to demonstrate theexistence of a general distribution from which types 1,2 and 3 originate and fromwhich a very broad class of Dirichlet distributions, as well as other well knownmultivariate distributions, are found. This work is based on the work of McDonaldand Xu [MX95].

1

2

1 Introduction

The Dirichlet distribution is one of the most important multivariate distributions andappears in many applications. Areas of application include order statistics; probabilis-tic 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 artificialintelligence and machine learning. With such a diverse array of areas that this distri-bution is found, we wish to examine the common properties of the types of Dirichletdistributions used, to discuss how they are related, to discover a broader class of distri-butions within which such distributions exist, and to explore many of the fundamentalproperties 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 2where it is shown to be an extension of the generalized beta (GB) distribution defined byMcDonald and Xu [MX95]. General information on each of the classes of distributionslisted 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 dis-tribution are established in Section 4. In this section it is shown, for instance, that themoment generating function and marginal distribution function properties are consistentin definition with those from D1, D2 and D3.

Historically, the use of the expression “generalized Dirichlet distribution” has been usedextensively in describing a class of Dirichlet distributions that satisfy various correlationalproperties among variables in the distribution (See, for example, the works of Connorand Mosimann [CM69] and Wong [Won98].)

In a similar manner as McDonald and Xu, Section 5 lists several known multivariatedistributions that are defined using special parameter setting in GD. A taxonomy ofdistributions is used to display the interrelationships of the distributions. A brief sum-mary concludes this report.

The following special notation will be used throughout this presentation: We will usethe uppercase character to denote random vectors, such as X′ = (X1, · · · , Xk). Thiswill commonly be a vector of length k where k ≥ 1. The vector x, which will denotean instance of the random vector X, will be defined as x′ = (x1, · · · , xk) where each

xi ≥ 0 andk∑

i=1

xi ≤ 1. For differentials of vector x, let dx =k∏

i=1

dxi. For a generic

vector parameter α where α′ = (α1, · · · , αk), let −α′ = (−α1, · · · , −αk), and 1α′ =(

1α1

, · · · , 1αk

). 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 typec refers to a vector and is emboldened since the distribution that it is assigned to isdefined distinctly by this vector. When the random vector X has the most general formof Dirichlet distribution we will symbolize this as X ∼ GD(a,b, c, ν).

2 Definition

The generalized Dirichlet distribution is of the following form:

GD(y | a,b, c, ν) =Γ(ν+)

k∏i=1

|ai|k∏

i=1

yaiνi−1i

[1−

k∑i=1

(1− ci)(

yi

bi

)ai]νk+1−1

k+1∏i=1

Γ(νi)k∏

i=1

baiνii

[1 +

k∑i=1

ci

(yi

bi

)ai]ν+ (2.1)

where a′ = (a1, · · · , ak) ∈ <k, b′ = (b1, · · · , bk) ∈ <+k, c′ = (c1, · · · , ck) ∈ [0, 1]k,

ν′ = (ν1, · · · , νk+1) ∈ <+k, with ν+ =k+1∑i=1

νi. Throughout this presentation, we assume

that, for all i, νi > 0.

The simplex S ={

yi | yi ≥ 0 for i = 1, · · · , k ,k∑

i=1

(1− ci)(

yi

bi

)ai ≤ 1}

is the domain

of integration for GD. Throughout this presentation, since parameters can modify theshape of the simplex S, the symbol S will be used without explicit reference to the vectorparameters a, b, or c.

Note 2.1. The generalized beta distribution, which is denoted as GB, is a special caseof the generalized Dirichlet distribution when k = 1. McDonald and Xu [MX95] providea complete examination of the properties of this distribution as well as its relationshipto numerous well-known continuous probability distributions.

3 Derivation of GD

We will show that GD is a probability distribution function by first deriving this dis-tribution from the Dirichlet type 1 distribution (or equivalently from the Dirichlet type2). This will form a distribution that will be called GD(a,b, c, ν) defined at a pointc = (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),defined at points c(1) = (c(1)

1 , · · · , c(1)k ) and c(2) = (c(2)

1 , · · · , c(2)k ), 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 Dirichlettype 1 distribution GD1. This will establish GD as the most general Dirichlet distri-bution, defined for any c′ = (c1, · · · , ck) such that c ∈ [0, 1]k. In particular, this willinclude the Dirichlet type 1, defined at c′ = (0, · · · , 0), Dirichlet type 2, defined atc′ = (1, · · · , 1), and Dirichlet type 3, defined at c′ =

(12 , · · · , 1

2

). 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 X′ = (X1, · · · , Xk). Define the spaces X and Y with X ∈ X andY ∈ Y, where Y is defined by the transformation T defined below. Our objective isto determine the distribution of Y. Thus, suppose we apply the transformation T suchthat

T : Yi =Xi

1−k∑

i=1

ciXi

, for i = 1, · · · , k

with inverse transformation

T−1 : Xi =Yi

1 +k∑

i=1

ciYi

, for i = 1, · · · , k

To obtain the Jacobian of the transformation, we know that

∂xi

∂yj=

−ciyi +(

1 +k∑

i=1

ciyi

)

(1 +

k∑i=1

ciyi

)2 , if i = j

−cjyi(1 +

k∑i=1

ciyi

)2 , if i 6= j

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

5

J =∣∣∣∣∂(x1, · · · , xk)∂(y1, · · · , yk)

∣∣∣∣

=

(1 +

k∑

i=1

ciyi

)−2k ∣∣∣∣∣

(1 +

k∑

i=1

ciyi

)Ik − c · y′

∣∣∣∣∣

=

(1 +

k∑

i=1

ciyi

)−2k (1 +

k∑

i=1

ciyi

)k−1

=

(1 +

k∑

i=1

ciyi

)−(k+1)

.

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

GD(y |1,1, c, ν) =Γ

(k∑

i=1

νi

)

k∏i=1

Γ(νi)·

k∏

i=1

yi

(1 +

k∑

i=1

ciyi

)−1

νi−1

×1− yi

(1 +

k∑

i=1

ciyi

)−1

νk+1−1 (1 +

k∑

i=1

ciyi

)−(k+1)

=Γ

(k∑

i=1

νi

)

k∏i=1

Γ(νi)·

k∏

i=1

yνi−1i

(1 +

k∑

i=1

ciyi

)−ν+ [1−

k∑

i=1

(1− ci)yi

]νk+1−1

(3.1)

where S ={

yi | yi ≥ 0 for i = 1, · · · , k ,k∑

i=1

(1− ci)yi ≤ 1}

.

Using the variable transformation yi =(

y∗ibi

)ai

for i = 1, · · · , k, the probability distribu-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 Y′ = (Y1, · · · , Yk). Assume the general Dirichlet dis-tribution as defined in Equation 3.1, with c(1) = c. Let c∗i = c

(2)i − c

(1)i , where

c(1)i ∈ [0, 1] and c

(2)i ∈ [0, 1], for i = 1, · · · , k. It is required that −1 ≤ c∗i ≤ 1, giv-

ing us −1 ≤ c(2)i − c

(1)i ≤ 1 or, equivalently,

∣∣∣c(2)i − c

(1)i

∣∣∣ ≤ 1.

Take the transformation

Tc : Wi =Yi

1−k∑

i=1

c∗i Yi

, for i = 1, · · · , k

with inverse transformation

T−1c : Yi =

Wi

1 +k∑

i=1

c∗i Wi

, for i = 1, · · · , k.

The Jacobian of this transformation is J =(

1 +k∑

i=1

c∗i wi

)−(k+1)

. Using this, the general

Dirichlet distribution is

7

GD(w |1,1,c(2), ν)

=Γ

(k∑

i=1

νi

)

k∏i=1

Γ(νi)·

k∏

i=1

wi

1 +k∑

i=1

c∗i wi

νi−1 1 +

k∑

i=1

c(1)i

wi

1 +k∑

i=1

c∗i wi

−ν+

×

1−

k∑

i=1

(1− c

(1)i

)

wi

1 +k∑

i=1

c∗i wi

νk+1−1

(1 +

k∑

i=1

c∗i wi

)−(k+1)

=Γ

(k∑

i=1

νi

)

k∏i=1

Γ(νi)·

k∏

i=1

wνi−1i

[1 +

k∑

i=1

(c(1)i + c∗i

)wi

]−ν+ {1 +

k∑

i=1

[1−

(c(1)i + c∗i

)]wi

}νk+1−1

=Γ

(k∑

i=1

νi

)

k∏i=1

Γ(νi)·

k∏

i=1

wνi−1i

(1 +

k∑

i=1

c(2)i wi

)−ν+ [1−

k∑

i=1

(1− c

(2)i

)wi

]νk+1−1

(3.2)

defined on the simplex S ={

wi | wi ≥ 0 for i = 1, · · · , k ,k∑

i=1

(1− c

(2)i

)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 isfrom the Dirichlet type 1 to Dirichlet type 2. Similarly, if we let c(1) = 1 and c(2) = 0so that c∗ = −1, then our transformation is from the Dirichlet type 2 to Dirichlet type1.

Example 3.2. If we let c(1) = 0 and c(2) = c so that c∗ = c, then the transformationTc allows a derivation of the distribution GD(·|1,1, c, ν) from the distribution D1, aswas demonstrated in Section 3.1. A similar result follows when we let c(1) = 1 andc(2) = c so that c∗ = c − 1. In this case, the transformation allows a derivation of thedistribution GD(·|1,1, c, ν) from the inverse Dirichlet distribution D2. The distributionD2 is discussed further in Section 5.6.

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 thelatter form Dirichlet type c.

Note that when Dirichlet type 1 is renamed Dirichlet type 0 where 0′ = (0, · · · , 0) andDirichlet type 2 is renamed Dirichlet type 1 where 1′ = (1, · · · , 1), then this new vector-designator is more descriptive in notation than the currently assigned notations of types1 and 2. Likewise, the Dirichlet type 3 notation can now be renamed Dirichlet type 1

2

where 12

′ =(

12 · · · 1

2

). In general, the vector-designator for the general Dirichlet type c

where c′ = (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, orZi ∼ Γ(νi), for i = 1, · · · , k + 1 and that the transformation S is defined as

S :

Xi = Zik+1Pi=1

Zi

for i = 1, · · · , k

Xk+1 =k+1∑i=1

Zi

Define the set Z with random variables Zi ∈ Z. Note that X′ = (X1, · · · , Xk) has theDirichlet 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:

T ◦ S :

Yi =

Zik+1Pj=1

Zj

1−k+1∑i=1

ci

Zi

k+1Pj=1

Zj

=Zi

Zk+1 +k∑

j=1

(1− cj)Zj

for i = 1, · · · , k

Yk+1 =k+1∑

i=1

Zi

The transformations S, T and the composite transformation T ◦S are shown in the figurebelow:

XT

ÂÂ@@@

@@@@

Z

S

??~~~~~~~~

T◦S// Y

Figure 2: Transformations T and S

Special cases of the transformation T ◦ S are demonstrated in the following examples.

Example 3.3. When c′ = (0, · · · , 0), Yi = Zik+1Pj=1

Zj

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

with the Dirichlet type 1 random variable (See Kotz, et al [KBJ00], Chapter 49, Section1).

Example 3.4. When c′ = (1, · · · , 1), Yi = Zi

Zk+1for 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 theinverse transformation S−1 ◦ T−1 which is equivalent to solving for Z1, · · · , Zk in the kequations

c1YjZ1 + · · ·+ (cjYj + 1)Zj + · · ·+ ckYjZk − YjYk+1 = 0 for j = 1, · · · , k.

This requires solving

10

c1Y1 + 1 c2Y1 · · · ckY1

c1Y2 c2Y2 + 1 · · · ckY2

......

. . ....

c1Yk c2Yk · · · ckYk + 1

Z1

Z2

...Zk

= Yk+1

Y1

Y2

...Yk

.

The solution is of the form

S−1 ◦ T−1 :

Zi =YiYk+1

1 +k∑

j=1

cjYj

for i = 1, · · · , k

Zk+1 = Yk+1 −k∑

j=1

Zj =

Yk+1

(1−

k∑j=1

(1− cj)Yj

)

1 +k∑

j=1

cjYj

.

The joint distribution of Z1, · · · , Zk+1 is

g(z1, · · · , zk+1) =1

βν+k+1∏i=1

Γ(νi)

k+1∏

i=1

zνi−1i e−

Pk+1i=1 zi

β (3.3)

where zi ≥ 0 for i = 1, · · · , k + 1. Substituting the solutions found in S−1 ◦ T−1 intoEquation 3.3 we get

g∗(y1, · · · , yk+1) =1

βν+k+1∏i=1

Γ(νi)

k∏

i=1

yiyk+1

(1 +

k∑

i=1

ciyi

)−1

νi−1

yk+1

[1−

k∑i=1

(1− ci)yi

]

1 +k∑

i=1

ciyi

νk+1−1

× e

−

8><>: yk+1

β

1+

kPi=1

ciyi

!" kPi=1

yi+1−kP

i=1(1−ci)yi

#9>=>; × J(3.4)

where the Jacobian J is

11

J =∣∣∣∣∂(x1, · · · , xk)∂(y1, · · · , yk)

∣∣∣∣

=yk

k+1(1 +

k∑i=1

ciyi

)2k

∣∣∣∣∣

(1 +

k∑

i=1

ciyi

)Ik − c · y′

∣∣∣∣∣

=yk

k+1(1 +

k∑i=1

ciyi

)2k

(1 +

k∑

i=1

ciyi

)k−1

=yk

k+1(1 +

k∑i=1

ciyi

)k+1.

The function 3.4 can now be written and simplified to

g∗(y1, · · · , yk+1) =1

βν+k+1∏i=1

Γ(νi)

k∏

j=1

yνj−1i

1 +

k∑

j=1

cjyj

−ν+

1−

k∑

j=1

(1− cj) yj

νk+1−1

× e−yk+1

β yν+−1k+1 .

(3.5)

By integrating g∗ with respect to the random variable Yk+1 we find

g∗∗(y1, · · · , yk) =1

k+1∏j=1

Γ(νj)

k∏

j=1

yνj−1j

1 +

k∑

j=1

cjyj

−ν+

1−

k∑

j=1

(1− cj) yj

νk+1−1

×∞∫

0

e−yk+1

β yν+−1k+1

βν+dyk+1

=Γ (ν+)

k+1∏j=1

Γ (νj)

k∏

j=1

yνj−1j

1 +

k∑

j=1

cjyj

−ν+

1−

k∑

j=1

(1− cj) yj

νk+1−1

.

(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 (Y r11 · · ·Y rk

k ).From this, we then demonstrate several well known special cases of the moment gener-ating function. Following this, we derive the marginal distribution for GD along withseveral 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 hyper-geometric function type D and the Gauss hypergeometric function. See the monographExton [Ext76] for a thorough examination of hypergeometric functions used in this sec-tion.

Using multivariate expected value operations with GD, we get the following:

E (Y r11 · · ·Y rk

k ) =∫· · ·

∫

S

k∏

i=1

yrii GD(y |a,b, c, ν)dy

=∫· · ·

∫

S

Γ(ν+)k∏

i=1

|ai|k∏

i=1

yaiνi+ri−1i

[1−

k∑i=1

(1− ci)(

yi

bi

)ai]νk+1−1

k+1∏i=1

Γ(νi)k∏

i=1

baiνii

[1 +

k∑i=1

ci

(yi

bi

)ai]ν+ dy

(4.1)

Using the transformation

T : yi = bi

(wi

1− ci

) 1ai

(4.2)

with ci < 1 in Equation 4.1 for all i = 1, · · · , k, then E (Y r11 · · ·Y rk

k ) becomes

13

∫· · ·

∫

S

Γ(ν+)k+1∏i=1

Γ(νi)

k∏i=1

|ai|k∏

i=1

[bi

(wi

1−ci

) 1ai

]aiνi+ri−1 (1−

k∑i=1

wi

)νk+1−1

k∏i=1

baiνii

[1−

k∑i=1

(−ci

1−ci

)wi

]ν+

k∏

i=1

bi

|ai|w

1ai−1

i

(1− ci)1

ai

dw

=Γ(ν+)

k+1∏i=1

Γ(νi)

k∏

i=1

brii

k∏

i=1

(1− ci)−�

νi+riai

� ∫· · ·

∫

S

k∏

i=1

wνi+

riai−1

i

(1−

k∑

i=1

wi

)νk+1−1

×[1−

k∑

i=1

( −ci

1− ci

)wi

]−ν+

dw

(4.3)

where S ={

wi |wi ≥ 0, i = 1, · · · , k,k∑

i=1

wi ≤ 1}

.

Using Lauricella function F(k)D , then Equation 4.3 becomes

Γ(ν+)k+1∏i=1

Γ(νi)

k∏

i=1

brii

k∏

i=1

(1− ci)−�

νi+riai

�Γ(νk+1)k∏

i=1

Γ(νi + ri

ai

)

Γ(

ν+ +k∑

i=1

ri

ai

)

× F(k)D

(ν+, ν1 +

r1

a1, · · · , νk +

rk

ak; ν+ +

k∑

i=1

ri

ai;−c1

1− c1, · · · ,

−ck

1− ck

)

=k∏

i=1

brii

Γ(ν+)k∏

i=1

Γ(νi + ri

ai

)

k∏i=1

Γ (νi) Γ(

ν+ +k∑

i=1

ri

ai

)

× F(k)D

(k∑

i=1

ri

ai, ν1 +

r1

a1, · · · , νk +

rk

ak; ν+ +

k∑

i=1

ri

ai; c1, · · · , ck

).

(4.4)

14

Since

F(k)D

(k∑

i=1

ri

ai, ν1 +

r1

a1, · · · , νk +

rk

ak; ν+ +

k∑

i=1

ri

ai; c1, · · · , ck

)

=Γ

(ν+ +

k∑i=1

ri

ai

)

Γ(

k∑i=1

ri

ai

)Γ (ν+)

1∫

0

uPk

i=1riai−1 (1− u)ν+−

Pki=1 ci

�νi+

riai

�−1

du

=Γ

(ν+ +

k∑i=1

ri

ai

)

Γ (ν+)

Γ[ν+ −

k∑i=1

ci

(νi + ri

ai

)]

Γ[

k∑i=1

(1− ci)(νi + ri

ai

)+ νk+1

] .

(4.5)

we have

E (Y r11 , · · · , Y rk

k ) =k∏

i=1

brii

k∏i=1

Γ(νi + ri

ai

)Γ

[ν+ −

k∑i=1

ci

(νi + ri

ai

)]

k∏i=1

Γ (νi) Γ[

k∑i=1

(1− ci)(νi + ri

ai

)+ νk+1

] . (4.6)

Example 4.1. If we let ri = 0 for all i = 1, · · · , k, then from Exton ([Ext76], Equations2.3.5 and 2.3.6), using Equation 4.4, we get

∫· · ·

∫

S

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

=Γ(ν+)

k+1∏i=1

Γ(νi)

∫· · ·

∫

S

k∏

i=1

zνi−1i

(1−

k∑

i=1

zi

)νk+1−1

dz

=Γ(ν+)

k+1∏i=1

Γ(νi)

k+1∏i=1

Γ(νi)

Γ(ν+)

= 1(4.7)

where the right-hand multiple integral has measure

k+1∏i=1

Γ(νi)

Γ(ν+), being a Dirichlet type 1

probability distribution function, defined in Section 5.1.

15

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

E (Y r11 · · ·Y rk

k ) =k∏

i=1

brii

Γ(ν+)k∏

i=1

Γ(νi + ri

ai

)

k∏i=1

Γ(νi)Γ[

k+1∑i=1

(νi + ri

ai

)]

=k∏

i=1

brii

k∏i=1

Γ

(νi + ri

ai

)

Γ(νi)

Γ[

k+1∑i=1

(νi + ri

ai

)]

Γ(ν+)

(4.8)

where rk+1 = 0. This is the moment generating function for the generalized Dirichlettype 1 probability distribution function, defined in Section 5.1. When ai = bi = 1for all i = 1, · · · , k, we have the moment generating function for the Dirichlet Type 1probability 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 (Y r11 · · ·Y rk

k ) =k∏

i=1

brii

Γ(ν+)k∏

i=1

Γ(νi + ri

ai

)

k∏i=1

Γ(νi)Γ[ν+ +

k+1∑i=1

(ri

ai

)]

× F(k)D

(k∑

i=1

ri

ai, ν1 +

r1

a1, · · · , νk +

rk

ak; ν+ +

k∑

i=1

ri

ai; 1, · · · , 1

)

=k∏

i=1

brii

k∏i=1

Γ(νi + ri

ai

)Γ

(νk+1 −

k∑i=1

ri

ai

)

k+1∏i=1

Γ(νi)Γ(νk+1)

=k∏

i=1

brii

k∏i=1

Γ

(νi + ri

ai

)

Γ(νi)

Γ(νk+1)

Γ(

νk+1 −k∑

i=1

ri

ai

)

16

where νk+1 −k∑

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

Example 4.4. If we set ci = 12 for all i = 1, · · · , k, then Equation 4.4 can be written as

E (Y r11 · · ·Y rk

k ) =k∏

i=1

brii

Γ(ν+)k∏

i=1

Γ (νi + ri) Γ(νk+1)

k+1∏i=1

Γ(νi)Γ[

k∑i=1

(νi + ri) + νk+1

]

× F(k)D

(k∑

i=1

ri

ai, ν1 +

r1

a1, · · · , νk +

rk

ak; ν+ +

k∑

i=1

ri

ai;12, · · · ,

12

)(4.9)

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

2 , fori = 1, · · · , k, then

E (Y r11 · · ·Y rk

k ) =k∏

i=1

(12

)riΓ(ν+)

k∏i=1

Γ (νi + ri) Γ(νk+1)

k+1∏i=1

Γ(νi)Γ[

k∑i=1

(νi + ri) + νk+1

]

× F(k)D

(k∑

i=1

ri, ν1 + r1, · · · , νk + rk; ν+ +k∑

i=1

ri;12, · · · ,

12

)

=2−

kPi=1

ri

Γ(ν+)k∏

i=1

Γ(νi + ri)

k∏i=1

Γ(νi)Γ[

k∑i=1

(νi + ri) + νk+1

]

× 2F1

(k∑

i=1

ri,

k∑

i=1

(νi + ri); ν+ +k∑

i=1

ri;12

)

(4.10)

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

17

Equation 4.10 can be rewritten as

E (Y r11 · · ·Y rk

k ) =2−νk+1Γ(ν+)

k∏i=1

Γ(νi + ri)

k∏i=1

Γ(νi)Γ[

k∑i=1

(νi + ri) + νk+1

] 2F1

(νk+1, ν+; ν+ +

k∑

i=1

ri ;12

)

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

4.2 Marginal Distribution of GD

If we apply the transformation T defined in Equation 4.2 to GD, we get

f(w, c, ν) =Γ(ν+)

k+1∏i=1

Γ(νi)

k∏

i=1

(1− ci)− νi

k∏

i=1

wνi−1i

(1−

k∑

i=1

wi

)νk+1−1 [1−

k∑

i=1

( −ci

1− ci

)wi

]−ν+

.

(4.11)

Let c′i = − ci

1− ciand c

(1)i = − c′i

1−m∑

i=1

c′iwi

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

From Equation 4.11, the expression(

1−k∑

i=1

c′iwi

)− ν+

can be written as

18

(1−

m∑

i=1

c′iwi

)− ν+

1−

k∑i=m+1

c′iwi

1−m∑

i=1

c′iwi

− ν+

=

(1−

m∑

i=1

c′iwi

)− ν+(

1−k∑

i=m+1

c(1)i wi

)− ν+

=

(1−

m∑

i=1

c′iwi

)− ν+ ∑

lm+1, ··· , lk

(ν+, lm+1)(ν+ + lm+1, lm+2) · · ·ν+ +

∑

m+1≤ j < k

lj , lk

×[(−c

(1)m+1

)lm+1 · · ·(−c

(1)k

)lk]

wlm+1m+1

lm+1!· · · wlk

k

lk!

=

(1−

m∑

i=1

c′iwi

)− ν+ ∑

lm+1, ··· , lk

(ν+,

k∑

i=m+1

li

) [(−c

(1)m+1

)lm+1 · · ·(−c

(1)k

)lk]

wlm+1m+1

lm+1!· · · wlk

k

lk!

where (a, n) =Γ(a + n)

Γ(a), with a > 0 and n ∈ Z and where

∑lm+1, ··· , lk

is the multiple

sum over lm+1, · · · , lk, where 0 ≤ lj < ∞ for j = m + 1, · · · , k. Then the marginaldistribution of GD, denoted GD(m), for variables (w1, · · · , wm), is

GD(m)(w |1,1, c, ν)

=Γ(ν+)

k+1∏i=1

Γ(νi)

k∏

i=1

(1− ci)−νi

∫· · ·

∫

S

k∏

i=1

wνi−1i

(1−

k∑

i=1

wi

)νk+1−1

×(

1−m∑

i=1

c′iwi

)− ν+ ∑

lm+1, ··· , lk

(ν+,

k∑

i=m+1

li

)[(−c

(1)m+1

)lm+1 · · ·(−c

(1)k

)lk]

wlm+1m+1

lm+1!· · · wlk

k

lk!dw

where dw = dwm+1 · · · dwk.

19

Since

Γ(ν+)k+1∏i=1

Γ(νi)

∫· · ·

∫

S

m∏

i=1

wνi−1i

m∏

i=1

wνi+li−1i

(1−

k∑

i=1

wi

)νk+1−1

dw

=Γ(ν+)

k∏i=m+1

Γ(νi + li)

k+1∏i=1

Γ(νi)Γ(

k+1∑i=m+1

νi +k∑

i=m+1

li

)m∏

i=1

wνi−1i

(1−

m∑

i=1

wi

)Pk+1i=m+1 νi+

Pki=m+1 li−1

=Γ(ν+)

k∏i=m+1

(νi, li)

m∏i=1

Γ(νi)Γ(

k+1∑i=m+1

νi

)Γ

(k+1∑

i=m+1

νi,k∑

i=m+1

li

)m∏

i=1

wνi−1i

(1−

m∑

i=1

wi

)Pk+1i=m+1 νi+

Pki=m+1 li−1

we get

GD(m)(w |1,1, c, ν)

=Γ(ν+)

m∏i=1

Γ(νi)Γ(

k+1∑i=m+1

νi

)k∏

i=1

(1− ci)−νi

(1 +

m∑

i=1

c′iwi

)− ν+ m∏

i=1

wνi−1i

(1−

m∑

i=1

wi

)Pk+1j=m+1 νj−1

×∑

lm+1,··· ,lk

(ν+,

k∑i=m+1

li

)k∏

i=m+1

(νi, li)(

k+1∑i=m+1

νi,k∑

i=m+1

li

)k∏

i=m+1

− c′i

1−

mPi=1

wi

1+mP

i=1c′iwi

li

li!

(4.12)

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

20

GD(m)(y |a,b, c, ν)

=Γ(ν+)

m∏i=1

|ai|m∏

i=1

Γ(νi)Γ(

k+1∑i=m+1

νi

)m∏

i=1

baiνii

×k∏

i=m+1

(1− ci)−νi

[1 +

m∑

i=1

ci

(yi

bi

)ai]− ν+ m∏

i=1

yνi−1i

[1−

m∑

i=1

(1− ci)(

yi

bi

)ai]Pk+1

i=m+1 νi−1

× F(k−m)D

ν+, νm+1, · · · , νk;

k+1∑

i=m+1

νi;− c′k

1−m∑

i=1

(1− ci)(

yi

bi

)ai

1 +m∑

i=1

ci

(yi

bi

)ai

, · · · ,− c′m+1

1−m∑

i=1

(1− ci)(

yi

bi

)ai

1 +m∑

i=1

ci

(yi

bi

)ai

(4.13)

Equation 4.13 is defined on the simplex S ={

yi | yi ≥ 0 for i = 1, · · · ,m ,m∑

i=1

(1− ci)(

yi

bi

)ai ≤ 1}

,

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

Example 4.5. If we let ci = 0 for i = 1, · · · ,m, then

GD(m)(y |a,b,0, ν) =Γ(ν+)

m∏i=1

|ai|m∏

i=1

yaiνi−1i

[1−

m∑i=1

(yi

bi

)ai]Pk+1

j=m+1 νj−1

m∏i=1

Γ(νi)Γ(

k+1∑i=m+1

νi

)m∏

i=1

baiνii

≡ GD(m)1 (y |a,b,0, ν)

(4.14)

Equation 4.14 is defined on the simplex

S =

{yi | yi ≥ 0 for i = 1, · · · ,m ,

m∑

i=1

(yi

bi

)ai

≤ 1

}(4.15)

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

21

Example 4.6. Let ci = 0 for i = 1, · · · , k, and m = 1. Then

GD(1)(y1;a,b, c, ν) =Γ (ν+) |a1|ya1ν1−1

1

[1−

(y1b1

)a1]Pk+1

j=2 νj−1

Γ (ν1) Γ

(k+1∑j=2

νj

)

≡ GB1

y1 : a1, b1, ν1,

k+1∑

j=2

νj

where ba11 > ya1

1 ≥ 0. The function GB1 is the generalized beta type 1 distributionfunction, defined by McDonald and Xu [MX95], Equation 2.1.

Example 4.7. If we let bi = ci = 12 and ai = 1 for i = 1, · · · ,m, then

GD(m)

(y |1,

12,12, ν

)

=Γ(ν+) 2

Pki=m+1 νi

m∏i=1

yνi−1i

(1−

m∑i=1

yi

)Pk+1j=m+1 νj−1

m∏i=1

Γ(νi)Γ(

k+1∑i=m+1

νi

)m∏

i=1

(1 +

m∑i=1

yi

)ν+

× F(k−m)D

ν+, νm+1, · · · , νk;

k+1∑

i=m+1

νi;−

1−m∑

i=1

yi

1 +m∑

i=1

yi

, · · · ,−

1−m∑

i=1

yi

1 +m∑

i=1

yi

=Γ(ν+) 2

Pki=m+1 νi

m∏i=1

yνi−1i

(1−

m∑i=1

yi

)Pk+1j=m+1 νj−1

m∏i=1

Γ(νi)Γ(

k+1∑i=m+1

νi

)m∏

i=1

(1 +

m∑i=1

yi

)ν+

× 2F(k−m)1

ν+,

k∑

i=m+1

νi;k+1∑

i=m+1

νi;−

1−m∑

i=1

yi

1 +m∑

i=1

yi

.

(4.16)

This corresponds to the result of Cardeno, et al., [CNS05] in which the marginal distri-bution 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 (withoutloss of generality) and c ∈ {0, 1}k. Also, define T = {i | ci = 1, i = 1, · · · , k} andT ′ = {i | ci = 0, i = 1, · · · , k}. Let |T | ≥ 1 and |T ′| ≥ 1 with |T |+ |T ′| = k. Then, fromEquation 2.1, we have

GD(y |1,1, c, ν) =Γ (ν+)

∏i∈T ∪T ′

yνi−1i

(1− ∑

i∈T ′yi

)νk+1−1

∏i∈T ∪T ′

Γ (νi) Γ (νk+1)(

1 +∑i∈T

yi

)ν+

defined over the simplex S ={

yi | yi ≥ 0 for i ∈ T ∪ T ′ and∑

i∈T ′yi ≤ 1

}. This proba-

bility distribution function will be called the mixed type 1 - type 2 Dirichlet distributionfunction since it combines properties of both Dirichlet type 1 and Dirichlet type 2. Thisis easily determined to be true through the following two examples:

Example 4.8. The distribution for yT ′ = {yi | i ∈ T ′} is the function f defined by

f(yT ′) =

∞∫

0

· · ·∞∫

0

GD(y |1,1, c, ν)∏

i∈Tdyi

=Γ

(ν+ −

∑i∈T

νi

)

∏i∈T ′

Γ (νi) Γ (νk+1)

∏

i∈T ′yνi−1

i

(1−

∑

i∈T ′yi

)νk+1−1

.

This is the Dirichlet type 1 defined on S1 ={

yi | yi ≥ 0 for i ∈ T ′ and∑

i∈T ′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

g(yT ) =∫· · ·

∫

S1

GD(y |1,1, c, ν)∏

i∈T ′dyi

=Γ (ν+)

∏i∈T

Γ (νi) Γ(

ν+ −∑i∈T

νi

)∏

i∈Tyνi−1

i

(1 +

∑i∈T

yi

)ν+ .

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 ′) = GD(y|1,1, c, ν), where cis suitably chosen from {0, 1}k.

5 Relationships Between GD and Other MultivariatePDFs

The multivariate probability distributions defined in this section are special cases of GDin Equation 2.1 when parameter values for a, b, ν and c from GD are selected andsubstituted 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∏i=1

|ai|k∏

i=1

yaiνi−1i

[1−

k∑i=1

(yi

bi

)ai]νk+1−1

k+1∏i=1

Γ(νi)k∏

i=1

baiνii

(5.1)

where yi > 0 for i = 1, · · · , k, andk∑

i=1

(yi

bi

)ai ≤ 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∏i=1

yνi−1i

(1−

k∑i=1

yi

)νk+1−1

k+1∏i=1

Γ(νi)

(5.2)

where yi ≥ 0 for i = 1, · · · , k, andk∑

i=1

yi ≤ 1.

2. Inverse Dirichlet Type 1

Using Equation 5.1 with ai = −1, bi = 1 for i = 1, · · · , k, the inverse Dirichlettype 1 distribution (denote ID1) is defined as

ID1(y | ν) = GD1(y | -1,1, ν)

=Γ(ν+)

k∏i=1

(1yi

)νi−1(

1−k∑

i=1

1yi

)νk+1−1

k+1∏i=1

Γ(νi)

(5.3)

where yi ≥ 0 for i = 1, · · · , k, andk∑

i=1

y−1i ≤ 1.

3. Independent Generalized Gamma

Let β = (β1, · · · , βk). Substituting bi = ν1

ai

k+1βi for i = 1, · · · , k so that b∗ =(ν

1a1k+1β1, · · · , ν

1ak

k+1βk

)in Equation 5.1, we get

f(y |a, β, ν) = GD(y |a,b∗,0, ν)

=Γ(ν+)

k∏i=1

|ai|k∏

i=1

yaiνi−1i

[1−

k∑i=1

(y

aii

νk+1βaii

)]νk+1−1

k+1∏i=1

Γ(νi)k∏

i=1

ννi

k+1βiaiνi

(5.4)

where yi ≥ 0 for i = 1, · · · , k, andk∑

i=1

(y

aii

νk+1βaii

)≤ 1.

Then the independent generalized Gamma is given by

IGG(y |a, ν, β) = limνk+1→∞

f(y |a, β, ν) =k∏

i=1

|ai| yaiνi−1

i e−�

yiβi

�ai

Γ(νi)βaiνii

where yi ≥ 0 .

25

4. Independent Normal (Special Case)

Let β = σ = (σ1, · · · , σk). By setting ai = 2 and νi = 12 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)

=k∏

i=1

(2e−y2

i /σ2i

σi√

π

)

=2k

πk2 |Σ| 12

e−y′Σ−1y

(5.5)

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

σ21 0

. . .0 σ2

k

. Note that

this is defined only for positive variables.

5.2 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)

=Γ(ν+)

k∏i=1

|ai|k∏

i=1

yaiνi−1i

k+1∏i=1

Γ(νi)k∏

i=1

baiνii

[1 +

k∑i=1

(yi

bi

)ai]ν+

(5.6)

where 0 < yi < ∞ .

1. Dirichlet Type 2

When ai = bi = 1 for i = 1, · · · , k, then the Dirichlet type 2, more commonlyreferred to as the inverse Dirichlet distribution (denoted D2) is defined as

D2(y | ν) = GD2(y |1,1, ν)

=Γ(ν+)

k∏i=1

yνi−1i

k+1∏i=1

Γ(νi)k∏

i=1

(1 +

k∑i=1

yi

)ν+

(5.7)

where 0 < yi < ∞ .

26

2. Generalized Multivariate Cauchy

If we let ai = 2, bi = 2, and νi = 12 for all i = 1, · · · , k and νk+1 = m− k

2 , then wecan say that

Γ(ν+)k+1∏i=1

Γ(νi)=

Γ(

k+12

)k+1∏i=1

Γ(

12

) =Γ

(k+12

)

π( k+12 )

.

Thus, the generalized multivariate Cauchy distribution (denoted GMC) isdefined as

GMC(y |m) = GD2

(y |2,2,

(12, · · · ,

12,

(m− k

2

)))

=Γ(m)

πk2 Γ

(m− k

2

) [1 +

(y12

)2 + · · ·+ (yk

2

)2]m

(5.8)

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

When we take m = k+12 in Equation 5.8, then we can define the multivariate

Cauchy distribution as

MC(y) =Γ(k+1

2 )

πk+12

[1 + (y1

2 )2 + · · ·+ (yk

2 )2] k+1

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 oflength k, and b = 1

θ =(

1θ1

, · · · , 1θk

)where θ = (θ1, · · · , θk), then the generalized

multivariate Lomax (denoted GML) is defined as

GML(y | a, θ, `) = GD

(y |1,

(1θ1

, · · · ,1θk

),1, (`1, · · · , `k, a)

)

=Γ

(k∑

i=1

`i + a

)k∏

i=1

θ`ii

k∏i=1

y`i−1i

Γ(a)k∏

i=1

Γ(`i)(

1 +k∑

i=1

θiyi

) kPi=1

`i+a

.

(5.10)

where 0 < yi < ∞.

When `i = 1 for i = 1, · · · , k, then the multivariate Lomax distribution is

27

defined similarly and we will write

ML(y | a, θ) = GML(y | a, θ,1)

=Γ(k + a)

k∏i=1

θi

Γ(a)(

1 +k∑

i=1

θiyi

)k+a

(5.11)

where 0 < yi < ∞ . Nayak [Nay87] studies the multivariate Lomax distributionwith its generalization and demonstrates its relationship to multivariate f, multi-variate Pareto Type 2, and multivariate Burr.

4. Multivariate f

Using the generalized multivariate Lomax distribution, if we let θi = `i

aifor all

i = 1, · · · , k, then the multivariate f distribution (denoted MF ) is defined as

MF (y | `, a,a) = GML(y | a, θ, `)

=Γ

(k∑

i=1

`i + a

)k∏

i=1

(`i

ai

)`i k∏i=1

y`i−1i

Γ(a)k∏

i=1

Γ(`i)[1 +

k∑i=1

(`i

ai

)yi

] kPi=1

`i+a

(5.12)

where 0 < yi < ∞. For further information on the relationship between multivari-ate 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, thenthe multivariate log-logistic distribution (denote MLL) is defined as

MLL(y | θ) =

(k∏

i=1

θi

)k!

(1 +

k∑i=1

θiyi

)k+1 (5.13)

where 0 < yi < ∞ for i = 1, · · · , k. Note that when k = 1, MLL(y) is the

log-logistic distribution f(y) =θ

(1 + θy)2where 0 < y < ∞.

6. Multivariate Burr

Using GD2, Equation 5.6, we let νi = 1 for i = 1, · · · , k, νk+1 = a and set bi =(1di

) 1ci and ai = ci for i = 1, · · · , k, we get the multivariate Burr distribution

28

(denoted MB), defined as

MB(y | a, c,d) =Γ(k + a)

k∏i=1

ciyici−1

Γ(a)k∏

i=1

[(1di

) 1ci

]ci

1 +

k∑i=1

yi(

1di

) 1ci

ci

a+k

=a(a + 1) · · · (a + k − 1)

k∏i=1

diciyci−1i

[1 +

k∑i=1

diycii

]a+k

(5.14)

where 0 < yi < ∞.

This defines the multivariate Burr distribution discussed by Takahasi [Tak65]. Forfurther information on the relationship between multivariate Lomax and multivari-ate Burr, see Nayak [Nay87], p.172.

7. Multivariate Pareto Type 2

Using the multivariate Lomax distribution Equation 5.11, when we set bi = 1θi

= 1for all i, then the multivariate Pareto Type 2 distribution (denoted MP2) isdefined as

MP2(y | a) = a(a + 1) · · · (a + k − 1)

(1 +

k∑

i=1

yi

)−(a+k)

(5.15)

where 0 < yi < ∞ . The distribution MP2 is derivable from the inverted Dirichletdistribution Equation 5.7 when we set νi = 1 for i = 1, · · · , k and νk+1 = a, aswell as from the multivariate Lomax distribution Equation 5.11 when θi = 1 fori = 1, · · · , k. For further information on the multivariate Pareto distribution seeMardia [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∏i=1

|ai|k∏

i=1

yaiνi−1i

[1−

k∑i=1

12

(yi

bi

)ai]νk+1−1

k+1∏i=1

Γ(νi)k∏

i=1

baiνii

[1 +

k∑i=1

12

(yi

bi

)ai]ν+

(5.16)

29

where yi ≥ 0 for i = 1, · · · , k, andk∑

i=1

12

(yi

bi

)ai ≤ 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)

=Γ(ν+)

k+1∏i=1

Γ(νi)2Pk

i=1 νi

k∏

i=1

yνi−1i

(1−

k∑

i=1

yi

)νk+1−1 (1 +

k∑

i=1

yi

)−ν+

(5.17)

where 0 < yi, νi > 0 for all i,k∑

i=1

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

multivariate generalization of the beta type 3 distribution, denoted B3 (See Car-deno, et al. [CNS05]). In particular, when k = 1, D3(y | ν) = B3(y | ν1, ν2). Foradditional information on B3 and D3, see Cardeno, 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∏i=1

Γ(νi)

2−Pk

i=1 νi

k∏i=1

(1−

k∑i=1

1yi

)νk+1−1

k∏i=1

yνi+1i

(1 +

k∑i=1

1yi

) ν+

(5.18)

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

i=1

1yi≤ 1.

30

6 PDF Taxonomy

A taxonomy is provided to organize the classes of multivariate distributions discussedin Section 5 and illustrate the relationships between the three commonly used Dirich-let distributions and other common multivariate distributions. The general Dirichletdistribution (GD) is defined with the largest number of parameters, 4k + 1, and is de-picted 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) ID3( 5.18)

GD3( 5.16)

OO 77oooooooooooo

GD( 2.1)

ssggggggggggggggggggggggg

²²

OO

GD1( 5.1)

²²xxqqqqqqqqqqqGD2( 5.6)

²² ''OOOOOOOOOOOO

++WWWWWWWWWWWWWWWWWWWWWWW

ID1( 5.3) D1( 5.2)

²²

GML( 5.10)

wwnnnnnnnnnnnn

²²

D2( 5.7)

²²

GMC( 5.8)

²²IG( 5.4)

²²

MF ( 5.12) ML( 5.11)

wwnnnnnnnnnnnn

²² ''OOOOOOOOOOOOMC( 5.9)

IN( 5.5) MLL( 5.13) MB( 5.14) MP2( 5.15)

Figure 3: PDF Taxonomy

Although numerous distributions have been identified and located in this taxonomy, itis uncertain that it is complete.

7 Conclusion

By demonstrating that the Dirichlet distribution encompasses a much broader classof distributions than has been shown to date allows us an opportunity to extend ourknowledge 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-knowndistributions as special cases. We have also found that it is consistent with the otherDirichlet distributions when selected parameters are used.

The down side of this work is that marginal distributions are found to not be in thesame class of distributions as the original distribution. This may limit the possibilityof 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 distribu-tions; a strong possibility exists for this distribution to be extended in several areas ofinvestigation, including:

1. Developing methods for parameter estimation, including maximum likelihood orestimation-maximization. More elaborate methods are certain to be needed as thenumber 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 newdistribution be demonstrated in similar applications.

3. Developing further results that rely on the use of beta functions. This includesextending the results to concepts of neutrality with applications to the generalizedDirichlet distribution defined by Connor and Mosimann.

4. Since this work introduces us to a new type of Dirichlet distribution, we see thepossibility 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 extendingresults in matrix variate beta and gamma distributions and the Lauricellahypergeometric functions;

(c) Extending results in Liouville distribution theory; and

(d) Computing probability integral measures of the generalized Dirichlet distrib-ution.

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 propositionswith a generalization of the Dirichlet distribution. Journal of the AmericanStatistical Association, 64:194–206, 1969.

[CNS05] Liliam Cardeno, Daya K. Nagar, and Luz Estela Sanchez. Beta type 3 distri-bution and its multivariate generalization. Tamsui Oxford Journal of Mathe-matical Sciences, 2005.

[Ext76] Herald Exton. Multiple Hypergeometric Functions and Applications. John Wi-ley and Sons Inc., New York, 1976.

[KBJ00] Samuel Kotz, N. Balakrishnan, and Norman L. Johnson. Continuous Multi-variate Distributions, Volume 1: Models and Applications. John Wiley andSons, Inc., New York, 2nd edition, 2000.

[Mar62] K. V. Mardia. Multivariate Pareto distributions. Annals of Mathematical Sta-tistics, 33:1008–1015, 1962.

[MX95] James B. McDonald and Yexiao J. Xu. A generalization of the beta distributionwith applications. Journal of Econometrics, 66:133–152, 1995.

[Nay87] T. K. Nayak. Multivariate Lomax distribution: Properties and usefulness inreliability theory. Journal of Applied Probability, 24:170–177, 1987.

[Tak65] K. Takahasi. Note on the multivariate Burr’s distribution. The Annals of theInstitute of Statistical Mathematics, 17:257–260, 1965.

[Won98] T. T. Wong. Generalized Dirichlet distributions in Bayesian analysis. AppliedMathematics and Computation, 97:165–181, 1998.