MULTIVARIATE LIOUVILLE DISTRIBUTIONS
by
1
Rameshwar D. Gupta
Division of Mathematics.
Computer Science and Engineering
University of New Brunswick
St. John. New Brunswick. Canada
Donald St. P. Richards 2 ,3
Department of Statistics
University of North Carolina
Chapel Hill. N.C. 27514
1partially supported by the NSERC research grant A-4850.
2partially supported by the National Science Foundation on grant
MCS-8403381.
3Portions of this research were completed during a recent leave of
absence at the University of Wyoming.
AMS 1980 Subject Classification:
28A01.
Primary 62H10, Secondary 62E15,
Key Words and Phrases: Dirichlet distribution, Dirichlet integral,
regression, totally positive, reverse rule, stochastic representation,
characterization, multivariate Dirichlet distribution, fractional integral.
ABSTRACT
A random vector (Xl' •.. , X ), with positive components, is said
n
to have a Liouville distribution if its joint probability density
aC I
an-l
function is of the form f (xl + ... + x ) xl
•.. x
with the a
n
n
i
all positive.
Examples of these are the Dirichlet and inverted Dirichlet
distribution.
In this paper, a comprehensive treatment of the Liouville
distributions is provided.
The results developed pertain to stochastic
representations, transformation properties, complete neutrality,
marginal and conditional distributions, regression functions, total
positivity and reverse rule properties.
Further, these topics are
utilized in various characterizations of the Dirichlet and inverted
Dirichlet distributions.
Matrix analogs of the Liouville distributions
are also defined, and many of the results obtained in the vector
setting are extended appropriately.
1.
INTRODUCTION
During recent years, increasing attention has been paid to families
of probability distributions which are defined through various functional
form assumptions.
Johnson and Kotz [12; p. 294ff]
review a number of
articles which classify distributions through functional form requirements
on density functions.
Even more contemporary results begin with
functional form restrictions on characteristic functions, rather than
on the densities.
We refer the reader to Cambanis et al [1], [2] and
Richards [25] for examples of the latter approach.
There are two important consequences of these studies.
First, they
develop unified analytical treatments of all the distributions in the
particular class.
In addition, features common to the family are dis-
covered and highlighted; for example, the review paper [3] (see also
[16]) fully illustrates that certain important sampling properties of
the multivariate normal distributions are also valid for the larger
family of elliptically contoured distributions.
In the paper at hand we study another family, the Liouville distributions, which is defined through functional form requirements on
density functions.
An (absolutely continuous) random vector (Xl' ... , X ) is said to
n
have a Liouville distribution if the corresponding joint probability
density function is proportional to
n
f (Xl
a.-1
+ ... + x n ) II x.~ ~
i=l
(1.1)
-2-
where the variables range over the generalized octant R~ = {(xl' •.• , x ) :
n
x.>O,
~
i=l, ••.• ,n}; a , ... ,a are positive numbers and f(·) is a suitably
1
n
chosen nonnegative continuous function.
f(o) are listed in Section 2.)
(The precise assumptions on
If f(.) has noncompact support, we say
that (Xl' .•• , X ) has a Liouville distribution of the first kind.
n
If the
support of f(·) is compact then we may assume, after a suitable scaling,
that f(·) is supported on (0,1); then, the variables in (1.1) range
n
l
xi < l}, and
i=l
we shall say that (Xl' ••• , X ) has a Liouville distribution of the
n
over the simplex Sn = {(xl' ... , x ) : xi> 0,
n
i=l, ... , n ;
second kind.
Brief treatments of the Liouville distributions appear in Marshall
and 01kin [19; p. 308] (under the name Liouville-Dirichlet distributions)
and also in Sivaz1ian [28].
In [19] it is shown that if (Xl' ••• , X ) has
n
a Liouville distribution and
~(.)
n
on R+,
then E 1J.!(X , ••• , X )
n
1
is a Schur-convex function ([19, p. 54])
is a Schur-convex function of the parameters
(aI' •.• , an) whenever the expectations exist.
[28] presents some results
on marginal distributions and transformation properties.
1.1
Examples.
a 1 bt ,
f(t) = t - e-
(Gamma margina1s with correlation [19; p. 308])
t> 0 , a> 0 , b > 0 ,
distribution of the first kind.
If
then (Xl' ••• , X ) has a Liouville
n
The joint density funciton of (Xl' ..• , X )
n
is
a.-1 -bx.
n
(l
i=l
x. )
~
a-I
n
Xi
~
e
(1. 2)
II
i=l
~
r(a. )
~
If a = 1 then Xl' •.. , X are independent gamma variables with common scale
n
parameter b
0
-3-1
a
1 2.
0< t < 1,
Example.
(The Dirichlet distribution)
a + > 0,
n 1
then the joint density function of (Xl' ••• , X ) is
n
n+1
L
r(
n
(1-
r (an+ 1 )
f (t) = (1-t) n+1
,
a.-1
a )
i
i=l
If
I
xi)
a n+ 1-1
i=l
n
II
i=l
x.
1.
1.
(1. 3)
rea. )
1.
This is a Liouville distribution of the second kind.
1.3.
Example. (The inverted Dirichlet distribution) If
-(a +"·+a
)
f(t) = (1 + t)
1
n+1 , t> 0, a + > 0, then we obtain a Liouville
n 1
distribution of the first kind.
The joint density function is
n+1
r(
L
a. )
1.
i=l
a.-1
n
II
i=l
X.
1.
1.
(1.4)
rCa. )
1.
In this paper we provide a comprehensive treatment of the Liouville
distributions, extending and completing the range of topics discussed in
[19], [28].
The results obtained pertain to stochastic representations,
transformation
propertie~ complete
neutrality, marginal and conditional
distributions, regression functions, total positivity and reverse rule
properties, and characterizations based on the above topics.
The Examples 1.1, 1.2 and 1.3 playa significant role in these
developments; throughout they consistently play the role of landmarks
in an unexplored habitat, pointing towards new and fertile results.
For example, results of Karlin and Rinott [15] on the multivariate
reverse rule properties of the Dirichlet distributions are extended to
a large subset of the Liouville distributions of the second kind.
In
return, we shall present characterizations of the three examples using
-4independence. regression and neutrality properties.
We shall also consider matrix analogs of the Liouville distributions.
The positive definite (symmetric) m x m matrices AI' •.• , An are said
to have a Liouville distribution of the first kind if their joint
density function exists and is proportional to (IAI :: determinant of A)
f (A
1
+ ... + A
n
a.-p
n
)
II IA.I
J.
i=1
(1. 5)
J.
mxm
where f (.) is supported on R+
'
matrices; p = (m+ 1)/2,
the cone of positive definite m x m
and a. > p - 1 ,
J.
i=I, •••• n.
By analogy with
the vector situation the matrix Liouville distributions are those for
n
I
which I -
Ai is also positive definite. where I denotes the m x m
i=1
identity matrix.
1.4.
Example.
Let SO' ••• , Sn be independent m x m Wishart matrices
having common covariance matrix L,
respectively, where a. > (m - 1) /2,
J.
and degrees of freedom a ' ••. , an
O
= 0,
i
... , n •
Let
S~
denote the
n
unique positive definite square root of S =
L
i=O
S.
J.
Then Olkin and
Rubin [20] show that the matrices
A.
J.
= S-~ S. S-~,
J.
.
J.=
1,
•••
,n,
have the joint density function which is proportional to
~a
n
II - L
i=1
Ai
I
-p
0
~a
n
II
i=1
IA.I
J.
-p
i
(1. 6)
-5n
I A. all positive definite. This is a Liouville
i=l 1
distribution of the second kind, generalizing Example 1.2.
for AI' ••• , An '
1.5.
•
I -
Example.
Let SO' ••• , Sn be defined as before.
Olkin and
Rubin [20] show that the random matrices
have the joint density function proportional to
-~(a1 +
n
11+
I
i=l
A.
1
I
.•. + a )
n
n
II
i=l
IA.I
1
~a.-p
(1. 7)
1
This is a Liouville distribution of the first kind, generalizing Example
1. 3 •
•
e
We shall extend some results obtained for Liouville distributed
vectors to their matrix counterparts; however, the execution of this
program requires rather sophisticated techniques.
For examples, in
characterizations of (1.6) through complete neutrality we first need
to classify all scalar-valued characters of the general linear group of
mx m matrices; in characterizations based on regression functions,
we shall apply Deny's theorem [4] on locally compact Abelian groups.
Central to our exposition is the unconvential fractional calculus
[27].
The fractional calculus, and especially Weyl's fractional
integrals, provides a natural setting in which we develop some of
our results; especially those results related to marginal and conditional distributions and characterizations based on regression
functions.
In the matrix space setting we shall apply the fractional
calculus, defined by Richards [23], to obtain results on marginals
and regression functions.
-6Finally, we describe the layout of our results.
In Section 2,
we list some notation and other preliminary material.
presentation is divided in two parts.
After this, the
Part I, consisting of Sections
n
3-6, deals with the Liouville distributions on R+.
Sections 7-9,
•
containing the results on the matrix analogs, comprises Part II.
Stochastic representations, transformation properties and their
consequences are presented in Section 3.
As one application we develop
a new proof of a result, due to Rao [21], concerning the distributions
of certain products of independent beta variables.
In Section 4,
we obtain results on marginal and conditional distributions and
regression functions.
Uing results of Karlin [13] and Karlin and
Rinott [14], [15], Section 5 develops necessary and sufficient criteria
for a Liouville distribution to be multivariate totally positive or
reverse rule of order two.
As corollaries, we obtain results on
correlation and probabilistic inequalities.
Section 6 contains several
characterizations of (1.2) - (1.4) based on the results of Sections
3 and 4.
Section 7 contains stochastic representations and trans-
formations for the matrix distributions, extending some of the results
in Section 3.
In a similar vein, Sections 8 and 9 are appropriate
extensions of Sections 4 and 6.
•
-72.
P'PELIMINARIES
Throughout, all random vectors and matrices are absolutely continuous
with continuous density functions.
In many instances, the precise values of normalizing constants
(such as the constant in (2.2) below) will not be necessary for our
treatment of the Liouville distributions.
Therefore, we shall usually
denote these constants by the symbols c , Co ' c ' c , etc.
1
j
We invariably write (Xl' ••• , Xn ) - Ln [f (t) ; aI' ••• , a n ] whenever
(Xl' ••• , X ) follows a Liouville distribution.
n
Whenever there is a
necessity to distinguish between the two kinds of Liouville distributions, we write (Xl' ••• , Xn ) -
L~k) [f(t); aI'
••• , a ] , k
n
according as the distribution is of the first or second kind.
= 1 or 2 ,
In the case
of the Dirichlet distributions (1.3) and (1.4), we write (Xl' ••• , X )
n
••• ,a
n
i f n = 1,
;
an + 1) or ID (aI' ••. , an ;
respectively.
However
we use the notation B(a ;a ) and IB(a ;a ) for the beta and
1 2
l
2
inverted beta distributions.
Precisely the same notation will be used
for the matrix Liouville distributions, since the context will always
eliminate any possible confusion.
The Liouville distributions are adopted from Liouville's extension
of Dirichlet's integral [6; p. 160]: if aI' ••• , an are positive,
a = a 1 + ... + an'
and
fa t a - 1 f(t)dt<oo,
(2.1)
-8-
then
n
n
J...
J f( L
n
R
+
n
xi)
IT
i=l
xi
a i -1
i=l
f(a.)
i=l
~
dX = - - - i
f(a)
IT
r..
a 1
t - f(t)dt.
(2.2)
0
In the setting of mx m positive definite matrix space R: x m •
i f p:: (m + 1) /2; a. > p - 1
Liouville's integral extends as follows:
for i = 1. 2. . ..• n;
a = a + ••• + an;
1
~
and
J mxm ITla-Pf(T)dT<oo
(2.3)
R+
then
n
J···Jf( . L1 A.)
~
(Rm x m)n
~=
n
IT
. 1
~=
a
IA I
i
i
- p
dA.
~
m
II
._
f
f
(a)
=_~-_1
+
m
m
(a.)
~
_
(2.4)
xJ
In (2.3) and (2.4).
IAI
mxm
R+
denotes the determinant of
IT/a-Pf(T)dt.
A;
dA is the Lebesgue
mxm
measure on R+
; and f m(.) is the multidimensional gamma function [10].
If f : R+ + R.
then the Weyl fractional integral of order a. a> 0 •
of f(.) is
a
1
W f(t)= f(a)
Jt
00
(s-t)
whenever the integral exists.
a-I
f(s)ds.
t>O.
(2.5)
Background results on the properties of the
a
operator W are provided by Rooney [27] and earlier authors.
The only
properties utilized here are (i) if a continuous function f(t) satisfies
(2.1) then there is a one-one correspondence between f(t) and Waf(t)
-9ll
(this -elationship is usually referred to as "fractional differentiation
(ii) wo. satisfies the semigroup proper.ty
I n t h e sett i ng
0
f Rm+ x m the ana1 og
statistical problems, by
Ri~hards
0
[23].
~ + P = wo. wP , 0. > 0,
(3 >
);
0•
f wo. was app Ii e,
d to certa i n
If a continuous function
f: R:xm-+R satisfies (2.3), then
(2.6)
is the Weyl fractional integral of order
means that 8 - T is positive definite.
correspondence between
0.
of f (T), where "8> T"
As in (2.5), there is a one-one
wO' f(T) and f(T); indeed, in both situations
the bijective relationship between Wo.f(T) and f(T) may be proven
through the formal use of Laplace transforms.
0.>0,
B
Also, wo.w :c.:wo.+
13 ,
13>0.
In stating Deny's theorem [4], we require the following notation.
Let G be a locally compact Abelian group with a countable basis, and
V a Radon measure on the Borel subsets of G such that the smallest
closed subgroup of G generated by the support of v is G itself.
let
Further
G be the set of all continuous ¢:G-+R+ for which ep(s+t)=¢(s)¢(t)
for all s, t
in G, and
G(I-I) = {¢
€
G:
fG ¢(t)dv(t) = l}
With the topology of uniform convergence on compact subsets of G ,
G is
a locally compact space.
Further, G(v) is a Borel subset of
G.
With these conventions, we may now state Deny's theorem.
Theorem.
f(s) = f
Let f : G-+ R+ be a continuous function which satifies
G
f(s+ t)dv(t),
S€
G.
(2.7)
-10Then there exists a unique positive measure ~ on G(v) such that
f(s) =
(2.8)
fG(v) ¢(s)d]..l(¢) •
Applications of Deny's theorem to the characterization of probability
distributions appear in Rao [22] and Richards [24].
We shall be con-
cerned with applications of Deny's theorem in which G is the group of
mxm
mx m synnnetric matrices, and f : G -+- R+ is supported on R+
; as we
shall see, the solutions (2.8) of (2.7) are Laplace transforms of
positive measures on certain hypersurfaces in R: x m •
-111.
3.
LIOUVILLE DISTRIBUTIONS ON R: •
TRANSFORMATIONS AND STOCHASTIC REPRESENTATIONS
We begin with a result which establishes a one-one correspondence
between the two kinds of Liouville distributions.
3.1.
(2)
Proposition •
Suppose that (Y 1 , •.• , Y ) - L n
n
[g(t); aI'
Define
•••, a ] •
n
n
X.=Y./(l1.
1.
L
j=l
Y.),
J
i=1,2, •• .,n.
f ( t) = (l + t) - (a 1 + ••• + an + 1) g ( t / (l + t) ),
(3.1)
t >0 •
(3.2)
In particular, the correspondence between f(t) and g(t) is one-one.
Proof.
Solving the equation
(3.1), we obtain
n
Y. =X./(l+
1.
1.
L
j=l
X ),
j
i= 1,2, •• ., n,
(3.3)
and it is straightforward to show that the Jacobian of the transformations (3.3) is (l +
L y.) -n- 1 •
n
Then, the conclusion follows by
i=l 1.
applying (3.3) to the density function of (Y , ••• , Y ) and carrying out
n
1
routine simplifications.
It goes without saying that the converse
result is also valid.
3.2.
Theorem.
Let (Xl' ... , X ) - L [f(t); aI' ..• , a ].
n
n
n
the following stochastic representations are valid:
Then,
-12n-1
(i) (Xl"'"
X )
n
k (Y l'
L
.•• , Y _ l' 1 Y.) Y
n
i=l 1
n
where
(Y , .•. ,Y
1) and Yare mutually independent, and (Y , .•• ,Y
1)
nn
n1
1
-D(a , · · · , an- l ; an) ;
1
L
(ii) (Xl' ... ,X)
n
n-1
n-1
(IT Y · , (l-Y1) IT Y., ... , 1-Y
l)Y
i=l 1
i=2 1
nn
i
where Y1 , ""Y are mutually independent, and Y - B(.L1aj;ai+1)
n
i
J=
i=1,2, ••• ,n-1;
n-1
(iii) (Xl' ... , Xn)
n-1
IT "<i+y.)-l
1 i=2
1
~ (IT (l+y )-l, Y
i=l
i
,
... ,
Y
l(l+Y
l)-1.)y where Y , ••• , Y are mutually independent, and
nnn
1
n
i
Y. - IB (a i
+ 1;
1
I
j=l
a .)
J
(iv) (Xl'"'' Xn)
i=l, 2, ... ,n-1
k
n-2
(Y l' Y2 (l - Y1)' ... , Yn _ 1 IT (l-Y.) ,
1
i=l
n-1
IT (l- Y.»Y where Y , .•• , Yare mutually independent, and
1
i=1
1
n
n
n
Y. - B(a.;
a ), i=1, 2, ... ,n-1.
1
1
j=i+1 j
I
n
In all four cases, Y - L [f(t)
n
1
L a.]
i=l
1
Before proving the theorem, we review some of its implications.
From each of the four stochastic representations, it follows that
n
LX..
Y ~
Further, for known values of aI' .•• , an' the function
n. i=l 1
f(t) and hence the distribution of (Xl' ••. , X ) is uniquely determined
n
by the distribution of Y .
n·
If, for example, Y
n
has a gamma distribution
-13then (Xl' ••• , X ) has a distribution of the type listed in Example 1.1.
n
For an application of the stochastic representations, suppose that
a
-1
(Xl' ••. , X ) - D(a , •.• , an; an + 1); equivalently, f(t) = (1- t) n+1
n
1
0< t < 1.
,
Then in particular, Proposition 4.1 below shows that
Xl .. B(a ; a 2 + ••• +a + 1)'
1
n
Further, the random variables
in Theorem 3.2(ii) are independent beta variables,
Y
i
Y
1
, ••. ,
Y
n
- B(a + ••• +a ;a +
1
i
i
1
),
i = 1, 2, ••• , n. From the stochastic representation we also see that
n
n
Xl ~ II Y. , and therefore II Yi .. B(a 1 ; a + ••• +a + 1)' This argument
2
i=l J.
i=1
n
provides a new proof of a result, due to Rao [21; p. 168],on the distribution
of certain products of independent beta variables.
3.3.
Proof of Theorem 3.2.
Each stochastic representation defines a
transformation from (Xl' ••• , X ) to (Y l' ••• , Y ).
For (i), we substitute
n
n
n-1
This
x. = y .y (i = 1, 2, ••• , n -1) and x = (1 - L y.)y into (1. 1).
J.
J. n
n
i=1 J. n
n-1
transformation has the Jacobian y
After carrying out some routine
n
simplifications, we observe-that the joint density of (Y , ••. , Y ) is
1
n
proportional to
which is the desired result.
The remainder of the theorem may be established similarly.
In
particular we remark that the Jacobians of the transformations in
n-1
n
. 1
J., y~ - 1 II (1 + y i) -i - 1
(ii), (iii) and (iv) are, respectively, II y.
. 2 J.
J.=
i=1
and y~ - 1
n-l
II (l _ yi) n - i - I .
i=1
-14In ending this section we relate the Liouville distributions with
the spherically symmetric distributions [1], [16], and derive generalizations
of results arising from the relationship.
n
A random vector (Y , ••• , Y ) in R is spherically
1
n
n
2
symmetric i f its joint density function is of the form f (
y.) for
i=l ~
some func tion f ( .) •
3.4. Definition.
I
The following result is due to Guttman and Tiao [8].
3.5. Proposition. If (Y , ••• , Y ) is spherical with density function
n
1
n 2
222
f(i~1 Yi) , then (Y 1 , Y2 , ••• , Yn ) - Ln[f(t); ~, ... ,~].
Using this relationship, we can obtain a number of interesting
results.
One of these is
~,
r
••• ,
~],
then
n
(i~1 Xi)/(i~l Xi) - B(r/2; (n- r)/2) , r< n.
By Proposition 3.5, (Xl'"'' X ) ==L (Y 2 , ... , Y2 ) where
n
1
n
I!
2
(Y 1 , ••• , Yn ) is spherically symmetric with density f( L Y.)
i=1 ~
r
n
r
n
Therefore (
x.)/(
x.) ~
Y~)/(
Y:) - B(r/2) ; (n - r) /2) ,
i=1 ~
i=1 ~
i=1 ~
i==1 ~
Proof.
I
L
(L
L
by a result of Kelker [16].
As we now show, Corollary 3.6 extends to arbitrary Liouville distributions even though no relationships exist with the spherical distributions for general values of aI' .•. , a
n
.
~
-15?
7. Proposition.
••• , a ]
n
Proof.
3.2(i).
We use the stochastic representation given in Theorem
n-1
There, it was shown (X ,···,X ) ~ (Y 1 ,···,Y
l'
Y.)Y
1
n
ni=l ~ n
I
••• , a
a ) independently of
n- 1; n
r
n
- L [f(t);
1
n
Y
I
i=l
Therefore,
a-i].
~
••• , a
r
then
L
r
Y. - B(
i=l ~
L
i=l
r
n
ai ;
I
a )·
i=r+1 i
;
(I
n
x.) / (
i=l ~
I
i=l
Xi) ~
r
L Y .•
i=l ~
(by Proposition 4.1 below)
-16-
4.
MARGINAL AND CONDITIONAL DISTRIBUTIONS
(Xl' •.• ,X r ) - Lr[fr(t); aI' ••• ,a r ], r<n, where a=ar+l+ ••• +a
n
and f r (t) = Waf(t) is the fractional integral of order a of f(t) •
Proof.
By definition, the marginal density function of (Xl' .•• ,X )
r
is proportional to
r
(IT
a - 1
i
Xi
)
i=l
J•.. J
n- r
R+
r
f(
n
LX. + LX. )
i=l ~
n
IT
a - 1
x .i
dx .
i=r+l ~ i=r+l ~
~
(4.1)
r
Applying Liouville's integral (2.2) to the function f(t+
I
x.) , we
i=l ~
see that (4.1) is proportional to
(
r
a.-l
~
roo
IT Xi
) )"0 f(t+
i=l
r
\'
a-I
LX.) t
dt.
i=l ~
(4.2)
r
Replacing t by t -
LX.
i=l ~
in (4.2) shows that the density of (Xl' ••• , X )
r
is proportional to
4.2. Remark. (i)
For the Liouville distributions of the second
kind, the marginal distribution of (Xl' ... , X ) was previously derived
r
in [28].
(ii)
The unicity of the fractional integral operators shows that,
for given values of aI' ••• , an' there is a one-one correspondence
between f (t) and f (t).
r
In particular, the extreme case r = 1 shows
that the distribution of (Xl' ••• , X ) is uniquely determined by the
n
distribution of Xl •
-17-
(iii) A curious property of the class
L~2),
earlier noted by
Johnson and Kotz [12; p. 305] in the case of the Dirichlet distributions,
is that at most one of the univariate margina1s can be uniformly
distributed on (0,1) •
(2)
To this end,suppose that (X ,X )- L2
1 2
n = 2.
where
In proving this result, it is enough to assume
a1~a2
[f(t)
a
with no loss of generality.
2
Then, Xl - Li )[W 2f (t)
_ (2) a 1
X2
L
[W f(t); a ].
If Xl is uniformly distributed on (0, 1) then
2
1
a -1 a
a
-a + 1
1
2
1
2
t
W f(t)=c, equivalently, W f(t)=ct
,0<t<1. Hence,
the density of X is proportional to
2
a -1 a
a -1 a -a
a
t 2
W _1 f ( t) .. t 2
W 1
2 (W 2 f (t) )
= c t
a2 - 1 a - a
-a + 1
2
1
1
W
(t
) ,
which is not constant on (0, 1).
That is, X is not uniformly
2
distributed on (0, 1) •
Following on Proposition 4.1, we now derive the conditional
distributions of the Liouville vectors.
4.3. Corollary.
If (Xl' ••• , X ) - Ln[f(t);
n
conditional distribution of (X r + l '
r<n,
aI' ... , an] then the
... , X ),
given {Xl = xl' .•• , X = x } ,
r
r
r
r
is L
[g (t); a +1' ••• , a ] where g (t) =f(t+ L x.)/f ( I x.).
n-r r
r
n
r
. 1 ~
r .~= 1 ~
~=
n
The proof follows straightforward from Proposition 4.1.
that the conditional distribution depends on xl' .•• , x
r
LX..
r
Note also
only through
r
Therefore, we lose no generality by conditioning on { L x. =y},
i=l ~
i=l ~
r
where y =
Below, we shall use this result repeatead1y in developing
i=l ~
the multiple regression properties of the class L •
n
LX..
-18a , .•• , an]'
1
12. r < n ,
and the expectations exist, then
(4.3)
n
n
L
where a =
a. ,
i=r+1
1.
L
j=
n
ji
j
II
X
is replaced by X.,
i-r+1 i
1.
Proof.
In particular. (4.3) remains valid if
ji'
i=r+1
r + 1<i <n •
By Corollary 4.3,
n
j
i
E(II
X.
i=r+1 1.
r
I
X
i=1
i
= t) f
r
(t) = c
n
j.+a.-1
(II
x 1.
1.
)
i
n - r i=r+1
J••• f
1
R+
n
X f (t +
I
i=r+1
x .) dx + 1 ••• dx •
1.
r
n
Applying Liouville's integral (2.2) shows that the last integral above
equals
roo j+a-1 f (y + t)dy = c Wj+a f (t) •
c )0 Y
2
Since f (t) = Waf(t),
r
result for
E(XI)
a-I
f(t) = (1- t) n + 1
n
The corresponding
is proven analogously.
4.5. Example.
E( II
i=r+1
then we have obtained (4.3).
If (X1~-""Xn) - D(a , •••• a
1
0 < t < 1.
n
;
a + ).
n 1
then Proposition 4.4 shows that
r
j .
x.1.1
1.
Xl =x . . . . . X =x)
1
r
r
so that
is proportional to (1-
I
i=l
.
x.)J.
1.
-19A particular instance of this result was noted by Johnson and Kotz
[ 12; p. 304].
4.6. Proposition.
Under the same hypotheses assumed in Proposition
4.4,
n
E(h(
L
i=1:+1
r
Xi)
I I
i=l
r:
Xi = t) f (t) = c
(y - t)a - 1) h(y - t) f(y)dy,
r
t
(4.4)
where h(t) is a real-valued function for which the relevant expectations
exist.
The proof is almost identical to the proof of Proposition 4.5.
particular (4.4), with h(t) = t j ,
r
In
shows that for l.::r< k'::n,
l X. = t) = c j
i=l 1
.
r
I L X. = t)
--k i=l 1
E(x..J
.
(4.5)
-20-
5.
TOTAL POSITIVITY AND DEPENDENCE PROPERTIES
In this section we use the results of Karlin [13] and Karlin and
Rinott [14], [15] to relate the Liouville distributions with the multivariate totally positive and reverse rule distributions.
Then, we make
applications to probability and correlation inequalities and positive
dependence properties.
n
On R , we introduce the lattice operations It and V:
i f x = (xl' ••• , x ),
n
n
y = (Yl' ••• , y ) are in R ,
-
n
then
xlty=(min(xl,y ), ••• ,min(x,y».
l
n
n
5.1.
n
Definition.
A function g : R
-+
[0,00)
is multivariate totally
n
positive of order 2 (MTP 2) i f for all x, y in R ,
g(x) g(y) < g(x It y) g(x V y) •
""u
""u
-
.....,
_
.....
.....,
(5.1 )
A random vector (Xl' ••• , X ) is MTP 2 i f its density function is MTP 2 •
n
In order to verify (5.1), it is sufficient [26] to check that g(x) > 0
is MTP 2 in every pair of variables where the remaining variables are held
fixed.
With regard to the Liouville distributions, we have the following
result.
5.2. Proposition.
Let (Xl' ••• ,X) - L [f(t);a , •.• , a ] .
l
n
n
n
the following are equivalent:
Then
-21(i)
(ii)
n
(II
i=l
2
f (x + y) is TP 2 on R+
;
(iii)
f(t) is logarithmically convex on R+ •
Proof.
If a function g (xl' ••• , x ) is MTP 2'
n
then so is the function
h (x ))g(x , ••• ,x ), where each hi (xi) is a nonnegative function.
1
n
i i
n
a.-1
Therefore, the density which is proportional to ( II x . ~
i=l ~
n
is
) f ( LX.)
i=l
~
n
MTP
2
if and only if f(
MTP 2'
L xi)
is MTP •
2
Since MTP 2 is equivalent to pairwise
i=l
2
then we have f(x 1 +x 2 ) TP 2 on Ri"'
are equivalent.
This proves that (i) and (ii)
Finally, the equivalence of (ii) and (iii) was proven
by Karlin [13; p. 160]; indeed the condition (5.1), when n = 2, is
~
•
precisely the definition of logarithmic convexity.
5.3. Example.
Let f(t) = t a - 1e -bt,
t> 0, a> 0, b> O.
The
corresponding Liouville distribution was encountered in Example 1.1.
function f(t) is logarithmically convex i f and only if 0 < a < 1.
The
Therefore
the density (1.1) is MTP2 for all 0 < a'::' 1 and b> 0 •
5.4. Example.
Let f(t)=t a - 1 0+t)-b,
t>O,
a>O, b>O.
The
function f(t) is logarithmically convex, and hence the corresponding
Liouville dis tribu tion is MTP 2' i f 0 < a < 1 •
Having determined necessary and sufficient criteria for a Liouville
distribution to be MTP 2' we may now apply the results of Karlin and
Rinott [14; Section 4] to obtain far-reaching information on probability
and expectation inequalities.
n
order on R as follows:
In order to do this, we introduce a partial
i f x = (xl' ••• , x ) and y = (Y1' ••• , y ) then x < y
n
n
---
-22if x. < y.,
1--
1
n
i = 1, 2, .•• , n. A function ¢ : R -+ R is increasing (decreasing)
i f x~ y implies ¢(x)
< ¢(y) (¢(x)
> ¢(y».
...., - . . . . ,
...., .....
Then, we obtain the following
results from [14; pp. 484-487].
5.5.
MTP
2
Theorem.
Let (Xl' ••• , X ) - Ln[f(t); aI' ••• , an] and have a
n
density function.
(i)
(ii)
Then,
the marginal distribution of (Xl' ••• , X ) is MTP 2'
r
for any increasing function ¢ : R~ -+ R,
r = 1, 2, •.• , n - 1
r = 1, 2, •.• , n, the multiple
regression
is increasing in (x
(iii)
r
+ I'
••• , x ) ;
n
if ¢ and 1JJ are both increasing or both decreasing on R~,
1JJ(X ' .... X » > o.
1
n
-
Cov(¢(X , ••• , X ),
1
n
then
In particular if ¢ l' ••• , ep n
n
are all increasing or all decreasing on R+'
then
(5.2)
(iv)
if a
i
c (a)
r
= a,
i
= 1,
= P (Xl -< a,
2, ... , n,
Co (a) = 1,
••• , X < a) , r
r-
= 1,
and
2, ••• , n ,
then
c
r-
1 (a) c
r
+ 1 (a) ->
2
c (a),
r
r = 1, 2, .•• , n - 1 •
It is customary to apply (5.2) to the development of inequalities
for the distribution function of (Xl' ••• , X ).
n
positive numbers x. and define
1
To this end, choose
-Z3-
1, X. < x. ,
(Xl' ••• , X )
1
n
<P •
i = 1, Z, ••• , n.
={
1-
1
0, otherwise,
Then (5.Z) becomes
n
(5.3)
P(X1-.S.x1, ••• , X < x ) > II P(X 1 < x.) •
n- n - i=l
- 1
Replacing <Pi by 1 -
<P i
leads to
n
P(Xl ~ xl' ••• , X > x ) > II P (X. > x.) •
n- n - i=l
1- 1
(5.4)
For the inverted Dirichlet distribution, the inequalities (5.3) and
(5.4) were obtained by Kimball [17] (see also Johnson and Kotz [lZ; p. Z40]).
5.6. Definition.
A
function g:Rn+(O,oo) is multivariate reverse rule
of order Z (MRR Z) if g(.) satisfies the reverse of (5.1); that is,
g(x)
g(y)
> g(x A y) g(x,.., Vy)
,
.....
...., -.....
.....
(5.5)
n
for all ~, y in R • A random vector (Xl' ••• , X ) is MRR Z i f its density
n
function is MRR •
Z
Suppose that a nonnegative function g(.) has the property that
g(x) g(~) 1:0 implies g(~) I: 0 for all ~ such that ~-.S.~2.~.
to the MTP
Z
case,
g(~)
is MRR Z if and only if
g(~)
Then similar
is RR Z in every pair
of variables where the remaining variables are held fixed [Z6].
-245.7. Proposition.
Let (Xl' •• qX) - L [f(t);a , ... , a ]
l
n
n
n
is montone increasing or decreasing.
where f(t)
Then (Xl' ••• , X ) is MRR 2 if and
n
only if f(t) is logarithmically concave.
a. - 1
n
Proof.
(Xl' ••• , X ) is MRR if and only if the function ( IT
2
n
i=l
n
f(
I
xi) is MRR
i=l
2
on R~;
J.
J.
).
n
I
by (5.5), this is equivalent to f(
xi) being
i=l
n
Since f(t) is monotone, then f(
MRR 2 •
X.
'i.
n
x.) f(
i=l
I
Yi):/: 0 implies
i=l
J.
n
f(
2
i=l
z.):/:O for all x.<z'<Yi'
J.
J.-
i=l,2, .. qn.
J.-
Hence, (Xl' ... ,X ) is
n
n
MRR
RR
if and only if f(
2
2
L
xi) is pairwise RR
i=l
or in f ( t)
2
;
that is, f(x
l
+ x 2)
is
is concave.
As noted in [15], the MRR
2
property does not suffice to imply expectation
and probabilistic inequalities analogous to (5.2) - (5.4).
To this end,
e
we are forced to strengthen the reverse rule requirements.
5.8 Definition.
(i) A function <p : R + R is a Polya frequency function
of order 2 (PF ) if <p(s-t) is TP
2
2
in the variables s,t, _oo<s, t<oo.
(ii) A random vector (Xl' ••• , X ) or its density function g(x) is stronglyn
MRR
2
(S - MRR ) i f for any set {<Pi} of PF 2 functions the marginal
2
h (x. , •• q x. ) =
J. "
J.
1
r
is MRR
J... J g (x)
Rr
n-r
IT <P. (x. ) dx.
i=l J i J i
Ji
(5.6)
in the variables (x. , .•• , x. ), where {iI' ••• , i } and {jl' ... , j
}
r
n- r
J.
J.
1
r
are complementary sets of indices drawn from {l, 2, ••• , n} •
2
-25A- example of a S - MRR
D(a , .. .,a ;
1
n
a + ),
n 1
distribution is the Dirichlet distribution
2
wherea .::1 for all i=1,2, ... ,n+1.
i
result was established by Karlin and Rinott [15; p. 508].
This
Indeed, a
close inspection of their analysis yields the following general principle.
5.9. Theorem.
a. > 1,
Let (Xl' ... , X ) - L(2)[f(t); aI' •.• , a ] where
n
n
n
i = 1, 2, ••• , n, and f(t) be monotone increasing or decreasing.
1-
Then (Xl' ••• , X ) is S - MRR i f and only i f f (t) is logarithmically
2
n
concave.
Proof.
Since S - MRR
2
entails MRR ,
2
is validated by Proposition 5.7.
then the necessity of the assertion
In proving the sufficiency we shall
modify the argument of Karlin and Rinott [15; Proposition 2.4], presenting
explicit details for the convenience of the reader.
If n
= 2,
then (Xl' X )
2
is RR 2 i f and only i f f (xl + x ) is RR Z in (xl' x ) •
2
Z
This holds by virtue of Proposition 5.7.
a
a
Let t+ = t for t> 0 and 0 for t < 0 •
for any PF
2
When n = 3 we have to show that
function </>(x ) ,
1
(5.7)
Since </> is PF
.
is TP
Z
in (t,x )
2
in (t, x )
Z
a - 1
1
Also, (t-x )+
Z
a- 1
Therefore </>(t-x ) (t-x )+1
is TP
Z
Z
2
then </>(t-x ) is TP in (t, x ).
Z
2
2
2
since a .::1.
1
-26The logarithmic concavity of f(t) shows that f(t+x ) is RR in
2
3
(t, x ).
3
Consequently, the basic composition formula [13; p. 98] shows
that the integral (5.7) is RR 2 in (x 2 ' x ) .
3
Assume by induction that the function
1
1
J ••• J
a
is RR
Z
a
n
a-I
n
r-1
xii
) f( L x.) II <p. (x.)dx.
i=1
i=l ]. i=l ]. ].
].
(5.8)
(II
in every pair of variables (xi' x.) , with r < i < j < n,
J
set of PF
Z
functions <PI' •• ., <p - •
r 1
--
for any
The integral in (5.8) is clearly
of the form
r
a - 1
n
i
x.]. )
xi
) h(
i=1
i=r
I
(II
for some function h.
Multiplying (5.8) by a PF
Z function <P r (x r ) and
proceeding as in the case n = 3 we deduce tha t
is RR
Z
in (x., x ) , r + 1 < i < j < n.
].
j
--
This establishes the inductive step,
hence completing the proof.
5.10. Corollary.
Under the hypotheses of Theorem 5.9, if <PI' ••• , <P
are all increasing (or decreasing) PF
n
r
<Pi (Xi» 2 E( II <Pi (Xi»
i=1
i=l
E( II
Z
functions then
n
• E( II
<Pi (Xi»
i=r+1
r = 1, Z, ••• , n , provided the expectations exist.
n
n
E(II <P . (X.» < II E(<p i (Xi»
i=l ]. ]. - i=l
•
n
,
(5.9)
In particular,
(5.10)
-27Pr~of.
This is a direct application of [15; Theorem 2.1J.
The indicator functions of finite or infinite intervals are PF •
2
Therefore with the hypotheses of Theorem 5.9, the inequality (5.10)
entails
n
P(a1~X1~b1,
for 0 < a. < b < 00
1.- i -
..
•
.... a <x <b)< II P(a.<X.<b.)
n- n- n - i=1
1.- 1.- 1.
-286.
CHARACTERIZATIONS
In the preceding sections, we have treated several aspects of the
Liouville distributions.
Here, we show how these distributions may be
characterized using the topics considered earlier.
Throughout, we shall assume that lim f(t) = 1.
t-+O+
6.1. Proposition.
Then
. •• , a ] •
n
Xl' ••• , X are mutually independent if and only if each Xi has a gamma
n
distribution with a common scale parameter.
Proof.
Mutual independence holds if and only if for all x. > 0 ,
1.
n
(II
a - 1
i
xi
) f
i=l
n
n
a. - 1
x .) = II x i .1.
hi (xi) ,
i=l 1.
i=l
(I
where hI' ••• , h n are continuous.
n
f
Equivalently,
n
(I
xi) = II hi (xi)'
i=l
i=l
By symmetry, hi =h
1
x. > 0,
for all i= 1, ••• , n.
n
we obtain hI (0) = 1; hence, hI (0) = 1.
f(x 1) = hi (xl)' x1~ 0 •
(6.1)
i = 1, 2, •.• , n •
1.
Further, when xl' ..• , x -+0+
n
When x ' ••• , x -+ 0 + we obtain
2
n
Thus, (6.1) entails
(6.2)
which is the multiplicative analog of Cauchy's functional equation.
Every bounded, continuous solution of (6.2) is of the form f(t)=e
t> 0, b> 0;
-bt
,
then, Xl' ••• , X are independent gamma variables with common
n
scale parameter b > 0 •
-29We note that mutual independence is never possible for the class
L~2).
Our next result shows that in
L~l),
the independence of a pair
entails mutual independence.
6.2. Corollary.
n> 3.
Let (Xl' ••• , X ) - L(l)[f(t); aI' ••• , a ] with
n
n
n
If any pair Xi and X are independent then Xl' •.. , X are mutually
j
n
independent.
Proof.
We may suppose, without losing generality, that Xl and X
2
By Proposition 4.1, the marginal distribution of (X ,X )
l
2
(1)
a
is L
[f (t); aI' a ] where f (t) = W f(t), a = a + ..• + an'
Since Xl and
2
2
3
2
n
a
-bt
X2 are independent then by Proposition 6.1, W f(t) = c e
for some
a -bt
-bt
b> 0 •
I t is easy to check that W (e
) =c e
; hence the unicity of
l
a
-bt
the operator W entails f(t) = e
, t > O.
Then, Xl' •• ., X are mutually
n
are independent.
independent.
6.3. Theorem.
Let l2.k2.r<n,
The the regression
f(t) = e
-bt
Proof.
, t > 0,
j+a
n
(1)
n
[f(t);a , • •• , a ] •
l
n
I L
X. = t) is constant (a. s.) if and only if
i=rtl ~
b> 0 •
That is, Xl' ••• , X are mutually independent.
n
If f(t)=e-
constant a.s.
4.4, clW
E(~
j>O and (Xl' ... ,Xn)-L
bt
,
then it is clear that the regression is
Conversely if the regression is constant, then by Proposition
a
f(t) = W f(t) where a
=a r +
1 + .•• + an'
fractionally we obtain f(t)=cWjf(t), t>O.
Differentiating
Writing this criterion in
the form
f(t) =c
ro
. 1
f (s + t) sJ - ds, t> 0 ,
we may apply Deny's theorem to deduce that f(t) = e
(6.3)
-bt
, t > 0,
b> 0 •
- 30-
6.4. Remark.
(i) From Propositions 4.4 and 4.6 it follows that Theorem
xt
6.3 remains valid if
r
is replaced by
(L
.
X.)J or h(
r
L
X.) for any
i=l 1
i=l 1
continuous, nonnegative function h(·) for which the expectations exist
and the support of h(·) is not a lattice.
(ii) If j is a positive integer, then Theorem 6.3 can be established
Writing (6.3) in the form
without recourse to Deny's theorem.
roo
1
f(t)=c j f(j) J t (y-t)
where c. > 0,
j-l
f(y)dy,
(6.4)
t>O,
then repeated differentiation of (6.4) entails
J
(6.5)
From (6.5) we deduce that f(t) is infinitely differentiable. and even
that f(t) is completely monotone; that is.
(-1)
i
f
(i)
(t)~O,
.
1=0,1.2 ....
By the well-known Hausdorff-Bernstein theorem. there exists a unique,
nonnegative. finite, Borel measure
f(t) =
j
o
The uniqueness of
therefore. f(t) = e
e-
~
ty
~
such that
d~(y) •
and the relationship (6.5) prove that
-bt
• t> O. b> 0 •
(iii) For the Dirichlet distributions D(a • • • • , a ;
1
n
Preposition 4.4 shows that
~
is singular;
- 31-
E(xt I
where j>O,
n
1:
i=r+l
x.1. = t)
l~k~r<n.
= c(l- t)j,
0<t <1 ,
(6.6)
We conjecture that (6.6) characterizes the
Dirichlet distributions among the class L(2).
n
available for arbitrary j > 0,
Although no results are
we can validate the conjecture for certain
integral values of j •
The problem of characterizing the Dirichlet distributions through
a
(6.6) is, by Proposition 4.4 and the definition of the W operators,
equivalent to solving the integral equation
1
f(a+j)
Jl
(y_t)a+ j - l
t
where a = a + 1 + ••• + an'
r
condition (2.1).
h(z)
= (l +
s > 0;
0 < t < 1,
Joos
(l + z)
-j
(l-t)j
r(a)
Jl
(y-t)
a-I
f(y)dy,
= zl (z +
1),
t
= sl (s +
1) and
transform (6.7) into
h(z) (z - s)
a
+j
- 1
1
dz = cjf(a)
Joos
h(z) (z - s)
a - 1
that is,
(6.8)
Differentiating fractionally in (6.8) we get
Wj
~f
« 1 + s) - j
h (s»
=
(6.7)
t
subject to f(t)..::. 0 and the integrability
The substitutions y
z) -(a + 1) f(z/z + 1»
1
r{a+ j)
f(y)dy=c j
c h (s) •
j is an integer, then repeated differentiation of (6.9) produces
the ordinary differential equation
(6.9)
dz ,
-3Z-
(6.10)
If j = 1,
(6.10) yields h(s) = (1 + s)-b, b> 0,
(1- t)8 - 1, 0 < t < 1,
8> 0 •
For j
= Z,
in which case f(t) =
(6.10) is sometimes referred
to as Cauchy's equation [18]; then, the general solution is of the
form
where Al and A are the two distinct solutions of the indicial equation
Z
Z
A - A-
Co = 0 •
A
Since cO> 0 we may assume that Al > 0> A •
Z
This entails h(s) = c (l + s)
1
ruled out by (Z.l).
0< t < 1,
8> 0 •
The term c (l + 5) 1 is
A
1
Z
, or f (t)
= (1 -
t) 8 - 1 ,
These observations prove the conjecture for j = 1, Z •
Higher values of j seem to require more complex arguments.
In this
regard, Hartman and Wintner [9] have shown that (6.10) always possesses
a completely monotone solution.
Of course, this is not new since we
know that f(t) = c(l + t)-d solves (6.8) for some c, d.
However it seems
difficult to even determine whether every completely monotone solution
is of the type above.
Next, we characterize the Dirichlet distributions among the class
L(Z) using the concept of complete neutrality (Doksum [5]).
n
6.5. Definition.
in the simplex Sn.
Let (Xl' ••• , X ) be a random vector taking values
n
(Xl' ••• ,X ) is completely neutral if there exist
n
mutually independent nonnegative random variables Y , ••• , Y such that
1
n
- 33-
(Xl' •• .,Xn )
1 (Y 1 ,Y 2 (l-Y 1 ),
n-1
•• "Yni~l (1-Y i
»·
(6.11)
The notion of complete neutrality can be related with the tailfree
distributions of Freedman [7].
Recent characterizations of the Dirichlet
distributions, using complete neutrality and related concepts, were
presented by James and Mosimann [11].
6.6. Proposition.
. .• , a ] , then
n
(Xl' ••. , X ) is completely neutral if and only i f (Xl' ••• ,X ) n
n
D(a 1 , •• ., an; a n + 1) •
Proof.
We apply the transformation defined by (6.11),
i-I
IT (1-YJ.)'
j=l
x. =y.
1.
1.
i=1,2, ••• ,n,
to the density function of (Xl' ••• , X ).
The corresponding Jacobian
n
n
n-1
equals IT (1- Yi)
; hence, the joint density function of Y , ••• , Y
1
i=l
n
is proportional to
n
( IT
i=l
a.-1
Yi
1.
a.
(1 _ y.)
O<y.<l, i=l, ••• ,n.
1.
a-I
is, f(t)=(1-t) n+1
1.
1.
+···+a
n
+1
n) f (1 - IT (1
i=l
-
Yi
))
'
If (Xl' ••• ,X ) - D(a 1 , .. .,a ; a + );
n
n
n 1
(6.12)
that
then (6.12) shows that Yl' ••• , Y are mutually
n
independent beta variables.
Conversely if Y , ••• , Y are independent then again by (6.12), there
1
n
exist continuous functions hI' ••• , h
n
such that
n
n
f(l- IT (1-Y.»= IT h (1-y.),
1
i=l i
1
i=l
O<Yi<l,
i=1,2, ... ,n.
Using the assumption lim f(t) = 1, we can conclude (as in the proof of
t-+O+
Proposition 6.1) that f(t) = h. (1- t),
1
Therefore
n
n
f(l-
i = 1, 2, ••• , n, 0 < t < 1.
IT (1-y.»= IT f (y .),
1
1
i=l
i=l
0 < y. < 1,
1
i = 1, 2, •.. , n •
The substitution f (t) = f (1- t) entails
n
f(
IT (1-y.»=
1
i=l
which is the multiplicative version of Cauchy's functional equation.
~
an + C 1
Therefore, f (1 - t) = (1 - t)
:: f (t) for some an + 1.
Necessarily,
an + 1 is positive since (6.12) is a density function.
In particular,
(Xl' ••• , Xn ) ~ D(al' ••• , an; an + 1) •
A similar result may be established for the class L(l) and the
n
inverted Dirichlet distributions.
In view of the preceding result,
we may state the following result.
6.7. Proposition.
(1)
Suppose that (Xl' ••• , X ) '" L
[f (t) ; a ' • •• , a ] •
n
1
n
n
Then there exist mutually independent positive random variables Y , ••• , Y
n
1
such that
n-1
(Xl' ••• ,X ) L (Y ,Y (1+Y ), .... Y II (1+Y.»
1
2
1
n
n i=l
1
-35-
LIOUVILLE DISTRIBUTIONS ON R~ x n •
II.
7.
TRANSFORMATIONS AND STOCHASTIC REPRESENTATIONS
Throughout, the unique positive definite (symmetric) square root
of a positive definite matrix T will be denoted by
..
T~ .
Suppose that (B , ••. , B ) - L~2) [g(T); a , ••. , a J •
1
n
1
n
7.1. Proposition.
Define
A. = (I J.
n
L
B )
j=1 j
f(T) = II+TI
_~
~_~
Bi (I -
L.
j=1
Bj )
, i=1,2, ••• ,n.
(7.1)
-(a 1 + ••• + a + p)
1
n
g(T(I+T)-), T> O.
(7.2)
In particular, there is a one-one correspondence between f(T) and g(T).
Proof.
The statement and proof of this result are natural extensions
n
of those given in Proposition 3.1.
n
Let A = LA., B = LB..
O i=1 J.
O i=1 J.
It
follows from (7.1) that
the last equality holding since B commutes with any rational function
O
of B '
O
Simple manipulation of these identities shows that B = (I + A )
O
O
and 1- A = (I + B )
O
O
-1
.
-1
A
O
Therefore, we may invert the transformation (7.1),
obtaining
(7.3)
The Jacobian of (7.3) may be shown to equal
the rest of the proof is standard.
II + AO1- (m + 1) p.
Af ter this,
-36-
When g(T) = I1- TI a - P,
7.2. Remark.
b
f(T) = II+TI- , T> 0,
0 < T < I, or equivalently,
the transformation (7.3) was utilized by Olkin
and Rubin [20; Theorem 3.4] to transform the inverted Dirichlet distribution into a Dirichlet distribution; their method was based on a set
of successive transformations.
Consequently Proposition 7.1 not only
extends [20; Theorem 3.4] but also provides a more direct proof •
7.3. Proposition.
•••, a ] .
n
Then
B.)B~ ,
1
n
where (B 1 , •.. , B _ 1) - D(a , ... , an _ 1; an) independently of
1
n
Proof.
The Jacobian of the transformation
B~ BiB~
Ai=n
n'
A =
n
i
1 2 , ... , n 1,
=,
n-l
B~(I -
L
n
1=1
may be shown to equal
B)
1
B~
n
IBn I (n-l)p.
Then, the result may be proven using
standard arguments.
7.4. Proposition.
Let (AI' ... , A ) - L [f (T) ; aI' ••• , a ] •
n
n
n
Define
_~
n
B.=(
1
L
j
A )
j
~i
n
B=l:
n
j=1
A
j
•
_~
n
A.(
1
l:
.
J=1
A )
j
,
i=I,2, ... ,n,
(7.4)
-37n
Then B , ••• , B are mutually independent, with B - B(a ;
i
i
n
1
n
L
j=i+l
a )
j
and B - L [f(T); La.].
1
n
i=1 ~
Proof.
•
We use the method of successive transformations given in
[20; Theorem 3.4] .
Let C =A +A + 1 + ••• +A ,
i
n
i
i
B = C-~ Ai C-~ '
i
i
i
i = 1, 2, ••. , n.
i = 1, 2, ••• , n - 1.
Now we make the successive transformations:
(AI' ••• , An- 2' Cn- l' Bn- 1)
B _ 1)
n
(C , B ' B , ••• , B _ 1)'
1
2
1
n
-+ ••• -+
Therefore B = C and
n
1
-+
(AI' •.• , An- 3' Cn- 2' Bn- 2'
The Jacobian of the combined
transformation may be shown to equal
n-l
(n-i-i)p
IB I(n-l)p II II-Bil
n
i=1
The conclusion now follows from these observations by way of standard
arguments.
Similar results may be obtained using the transformation ([20; Theorem 7.3])
i=I,2, ••• ,n-l
(7.5)
n
B =
n
L Ai'
i=1
7.5. Proposition.
Let (AI' ••• , An) - Ln[f(T); aI' ... , an] , and
define B , ••• , B through (7.5).
1
n
Then, B , .•• , B are mutually independent
1
n
i
wi th B. - B(
~
•
La ;
j=1 j
n
a.
~
+ 1)
,
i = 1, 2, ••• , n - 1,
and B
n
- L 1 [f (T) ;
La.].
. 1
~=
~
-35-
7 6. Proposition.
(Generalization of Rao's theorem)
i
independent B. - B(
1.
B~ B B~
2 1 2
•••
B~
B~
n - 1 n
I a.
.-1 J
; a i + 1) , i==I,Z, •••
Jn+l
- B( a ·
a.).
l'
I
i=2
,n,
then
I f mutually
B~B~
n n-l
1.
"
Proof.
prove.
We proceed by induction.
When n == 2,
For n == I,
there is nothing to
define Al and A in terms of B and B using the
2
Z
I
transformation (7.4).
By reversing the argument which establishes
Proposition 7.4, we obtain (AI' A ) - D(a
2
B~ B B~ == Al - B(a ; a + a )·
2
l
2 1 2
3
l
; a
2
+ a 3)
•
In particular,
The inductive step is also established
through similar arguments.
When n == I,
this result becomes Rao' s theorem (see the remarks below
Theorem 3.2).
Finally, we generalize Proposition 3.7.
7.7. Proposition.
Proof.
I f (AI' ••• , An) - L [f (T)
n
aI' ••• , an] and r < n ,
Mimic the proof of Proposition 3.7 using the stochastic
representation of Proposition 7.3.
-39-
8.
MARGINAL AND CONDITIONAL DISTRIBUTIONS
8.1. Proposition.
If (AI' ••• , A ) - L [f (T);
n
n
a , ••• , a ] then
n
1
and f (T) = Waf (T) is the fractional integral of order a of. f (T). ,In
r
r
particular, there is a one-one correspondence between f(T) and f (T) .
r
Proof.
Mimic the proof of Proposition 4.1, using the generalized
Liouville-Dirichlet integral (2.4) and the definition (2.6) of the
a
fractional integral operators W •
8.2. Corollary.
Under the hypotheses above, the conditional dis-
r
r
where g (T) = f (T + l Ai) / f ( l Ai).
r
i=l
r i=l .
Thus, as in the vector case, it is sufficient to condition throughout
r
on {
l
i=l
A. = T} •
1
8.3. Corollary.
If the expectations exist, then
(8.1)
n
where j =
l
i=r+1
n
j., a =
1
L
a •
i=r+1 i
Further i f h(T) is real-valued and the
relevant expectations exist, then
r
l A =T)· Waf(T) =
i=l i .
cJ
S>T
IS-Tla-Ph(S-T) f(s)dS.
-40-
9.
CHARACTERIZATIONS
In this section we extend the results of Section 6, characterizing the
Liouville matrices through the properties of independence, constant
regression and complete neutrality.
Since the zero matrix is a limit
mxm
point of the convex cone R+
,we may assume throughout that lim f(T) = 1 ,
T-+O+
where the notation "T -+ 0 +" means that T -+ 0 through R: x m •
9.1. Proposition.
•••, a ] •
n
Then
AI' ••• , An are mutually independent i f and only i f f (T) = exp (- tr L: T) ,
T> 0, for some L: > 0 •
Proof.
It is clear that independence holds if and only if f(T) satisfies
(9.1)
If f(T) is of the hypothesized form then (9.1) is certainly valid.
Conversely, suppose that (9.1) holds.
equation in the
~(m+
Regarding (9.1) as a functional
1) distinct entries t
ij
of the matrix T, then we
have
m
f(T) = exp(-
m
L L
i=l 1=1
0ij t ij ) :: exp(- tr L: T)
where L: = (0ij).
If there exists TO> 0 such that tr (L: TO) < 0 then the
sequence fen TO)
n = 1, 2, .•• , is unbounded, contradicting the integrability
criterion (2.3).
Consequently treE T)
~
0 for all T> 0 and therefore
[10; p. 478] E is positive semidefinite.
of E is guaranteed by (2.3).
Further the positive definiteness
_41-
and A are independent. then AI' •••• An are mutually independent.
2
This result may be established by arguing along the lines of proof
of Corollary 6.2.
9.3. Theorem.
Let l<k<r<n.
- - -
The the regression E( I~ Ij
j<O. and (AI' •••• A) - L [f(T);a •.••• a].
n
n
l
n
n
I I
i=r+l
A.
= T)
is constant (a. s.) i f and only i f
1.
m
there exists a probability measure l.l on R x m such that
+
f(T) =
J
exp(- tr z:: T) dl.l(Z::).
T> O.
(9.2)
mxm
R
+
Further. the representing measure l.l is concentrated on a hypersurface
of the form {Z::: Iz::l = c} •
Proof.
By proceeding as in Theorem 6.3. it may be seen that the a.s.
constancy of the regression function is equivalent to f(T) satisfying the
integral equation
f (T) = c
f (8 + T) 18 Ij - P ds.
J
T>0 •
mxm
R
+
which is Deny's equation.
In order that we may apply Deny's theorem. we
need to find all continuous bounded solutions ¢ of the functional
equation
-42-
In the course of proving Proposition 9.1, it was shown that every such
cP
is of the form
cP (T)
= exp (- tr L: T),
T> 0 ,
L: >
o.
Hence the represen t~
ation (9.2) follows from Deny's theorem, along with the uniqueness of
and the statement regarding the support of
~
•
Similar to the vector distributions, results paralleling Theorem 9.3
may be derived (from Deny's theorem and Corollary 8.3) if 1~lj is
r
replaced by h( LA.) ; here, h(·) will be an unbounded, nonnegative,
i=l 1.
continuous function for which the relevant conditional expectations
exist and for which the support is not a lattice.
9.4. Definition:
Let AI' ••• , An be random matrices such that
n
'\l.
mxm
A. ta k e va 1ues 1.n
. R+
Al' ••• , A , I . Then (AI' ••• , An) is
n
i=l 1.
completely neutral if there exists mutually independent B , ••• , B '
1
n
0< B < I (i = 1, 2, ••• , n) , such that
i
(AI' ... , An) ~ (B 1 , (I - B]) ~ B2 (I - B1 ) ~ ,
n-1
••• ,( II
i=l
(9.3)
n-l
(I-B.)~)B (II (I-B .)~))
1.
n i=l
n - 1.
9.5. Proposition.
Let (AI' ••• , An) ..
L~2) [f(T)
aI' .•. , an]'
Then
(AI' ••• , An) is completely neutral if and only i f (AI' ••. , An) - D(a , .•• , an;
1
Proof.
The Jacobian of the transformation
i-I
A. = (II
1.
j=l
i-I
(I-Bj)~) Bi ( II (I-B i _ .)~)
j=l
J
i=l, •.• ,n,
- 43n
II -
equa1 ::,
IT
i=l
say) that
Bi
In -
i •
Also, it can be established (using induction,
n
~
n
~
A. = I - (IT (I - Bi ) ) ( IT (I - Bn +1-1? ) •
i=l 1
i=l
i=l
n
I
Therefore, since (A , ••• , An) l
L~2) [f (T);
a , ••• , an]'
1
the joint
density function of (B , ••• , B ) is proportional to
1
n
(9.4)
n
n
X f (I - ( IT
i=l
0< Bi < I,
(I - B . )~) (IT (I - B +1· )~) ) ,
1
i=l
n-1
i = 1, ••• , n •
a
-p
If f (T) = 1- Tin + 1
,
I
0 < T < I,
an + 1 > 0 ,
then (9.4) shows that B , ••• , B are independent multivariate beta
l
n
matrices; hence, (A , ••• , An) is completely neutral.
l
Conversely, the complete neutrality of (A , ..• , An) entails
1
n
f(I- (IT
i=l
n
( II (I-lh+l-t~»
i=l
n
= IT h.(I-B.)
i=l 1
1
for continuous, nonnegative functions hi (.),
i = 1, 2, ••• , n.
(I-Bi)~)
what is now a standard argument, we find that hi (I - B) = f (B),
i = 1, ... , n.
Thus, (9.5) entails (with B
i
=0
for i2:. 3)
equivalently, the function geT) = f(I - T) satisfies
for 0 < B ,
1
B <I •
2
(9.5)
Using
0 < B< I ,
-44-
Tr2 validity of the propostion will then follow from the following
result.
9.6. Lemma.
xm
Let g(.) be a nontrivial, continuous function on R:
such that
g(B~AB~) =g(A)
Then g(A)=IAl
Proof.
k
g(B),
A> 0,
(9.6)
B> O.
for some k in R.
Since g(.) is nontrivial, then (9.6) entails g(I)=l.
Every nonsingular m x m matrix X possesses the "polar coordinates"
decomposition [10; p. 482] X = V B~ , where B> 0 and V belongs to O(m),
~
the group of mxm orthogonal matrices. Then B=X'X and V'X=B = (B~)
,
=X'V.
Substituting these relations into the functional equation (9.6), we
obtain
g(V'XAX'V) =
with X = I,
g(B~AB~) = g(A)
g(B) = g(A) g(V'XX'V) •
(9.7) shows that g(V'AV) = g(A),
g(A) is orthogonally invariant.
V in O(m), A> 0;
(9.7)
that is,
Consequently (9.7) reduces to
g(XAX') = g(A) g(XX') •
(9.8)
The substitution A=YY' in (9.8) shows that the function p(X) = g(XX')
satisfies
p(XY) = p(X) p(Y)
for all X, Y in GL (m),
the group of mx m nonsingular matrices.
(9.9)
The
functions p(X) in (9.9) are the scalar-values homomorphisms of GL(m).
-45-
Under the assumption that p(X) is a polynomial, Weyl [29; pp. 23-26]
shows that p(X)
= Ixl k
for some integer k; this settles a special case
of the general claim of our lemma.
In classifying the general solutions of (9.9), we proceed as follows.
First, pel) = 1.
Further since p(X) is a nonnegative, continuous
homomorphism; and the group Oem) is compact in the usual topology,
then p(O(m»
is a compact multiplicative group of nonnegative numbers.
That is, p(O(m»
= {l}.
Next, every X in GL(m) is of the form X = UDV
where U, V belong to Oem) and D = diag(d , ••. , d ).
1
m
p(X)
= p(UDV) =
p(U) p(D) p(V)
= p(D)
Then,
•
Denoting p(D) by p(d , ••• , d ), we obtain from (9.9) the functional
1
m
equation
k
k
Therefore, p(d , ••• , d ) = d 1 ••• d m ' with k , ••• , k in R •
m
m
m
1
1
1
Since
p(D) = p(DV) for all permutation matrices V, we even have the k. invariant
1.
under permutations; hence k. = k for all i = 1, ... , m.
1.
P (X)
= 1X Ik
,
or, g (A)
= IAIk/2 •
Therefore
This concludes the proof of the lemma,
hence, of the proposition.
Paralleling the vector case, we can also obtain an analog of Proposition
9.5 for the Liouville matrices of the first kind.
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e
1.
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Chmielewski, M.A. (1981). Elliptically symmetric distributions: a
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Herz, C. (1955).
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