Use of Moments in Distribution Theory:
N.A. Volodin
Tashkent
U.S.S.R.
Samuel Kotz
University of Maryland
College Park, MD 20742'
U.S.A.
A Mul tivariate Case
N.L. Johnson
University of North Carolina
Chapel Hill, NC 27599-3260
U.S.A.
ABSTRACf
In recent papers, Johnson and Kotz (199Oa,b) have explored the utility of
moment calculations as a simple way of establishing distributional forms.
particular a characterization theorem for beta distributions was proved.
In
In
this paper these methods are extended to multivariate problems, and a result
established for Dirichlet distributions.
Key words:
Characterization, Dirichlet distribution, Limit distribution.
Moments, Random matrices.
INTRODUCfION
In Johnson and Kotz (199Oa), elementary tools - the so-called 'moment
methods' - were employed to obtain distributions and characterizations of
distributions of random mixtures of form
Z = WX 1 +
(l-W)~
(or, more generally
where the X's are mutually independent. and have a common beta distribution.
- 2 -
and the i's are independent of the X's.
In Johnson and Kotz {1990b}, the method was extended to the distributions
of variables of type
Y=
co
:I
j=O
{-l}j
In particular the following result was obtained:
"If XO' Xl' ... are independent and identically distributed {i.i.d.} random
variables, the distribution of Y is beta {a+b,b} if and only if the common
distribution of the X's is beta (a, b}."
Similar, more general, results were recently obtained by Chamayou and
Letac (1991) using different, more advanced methodology.
Example 7 of Chamayou and Letac {page 20}.
See, in particular
Devroye, Letac and Seshadri {1986}
used a moment method for determination of distributions of random intervals which was also the second topic discussed by Johnson and Kotz {199Oa,b}.
In the present paper, we extend this methodology to derive
characterizations of multivariate distributions - in particular, of Dirichlet
distributions.
These distributions have been applied, with increasing
frequency, in statistical modelling, distribution theory and Bayesian
inference.
See, for example, Aitchison {1986} and Fang, Kotz and Ng {1990}.
The essential features of the moment method are:
{i}
{11}
establishing the existence of a limit;
establishing a recurrence relation among variables having the ini tial and
limit distributions;
{iii} using {ii}, obtaining formulae linking moments of the original and limit
distributions, indicating that the moments of either one are determined by the
moments of the other;
- 3 -
(iv)
for a given original distribution. demonstrating that the equations are
satisfied by the moments of a distribution.
Provided the moments determine the distributions (as is certainly the case
when the ranges of variation of the random variables are finite. which is so
for Dirichlet distributions). it follows that this distribution is the limit
distribution if and only if the original distribution is the one whose moments
have been used in the calculations.
2.
THEOREM:
Let
RESULTS
y~n} = (y~~} •...• y~~}) (i=l •...• k; n=1.2 •... ) be i.i.d.
(k-vector) random variables and
yen}
-1
-yen} =
The the limit distribution of each row
(n)
~1
(n)
•...• ~
of
x(n} = yen} y(n-1} ... y(l}
""
""
'"
'"
is Dirichlet with parameters kat kat .... ka (D(ka.ka •.... ka}) if and only if
the distribution of each
REMARK (1):
y~n} is D(a.a •... a}.
(n)
(n)
If the joint distribution of Yi1 •...• yik is D(a.a •... a}. then
the joint probability density function is
f
y~n}
(l) = f(ka)
~ y~-l
{f(a}}k i=l
k
(0 ~ Yi;
I Yi = 1)
i=l
The range of variation of each yen} is finite (in fact [0.1]).
ij
(1)
- 4 k
()
k
] y n
j=l ij
REMARK (2):
(n)
( )
( )
= 1 = ] X • so that X n and Y n
j=l ij
--
are stochastic
matrices.
REMARK (3):
When k = 2. D(a.a) is a beta (a.a) distribution and D(2a.2a) is a
beta (2a.2a) distribution.
PROOF:
The proof will be given in four stages. corresponding to (i) -(iv) of
the Introduction
(i)
Existence of the limit
Denoting the (i.j)-th element of ~(n) by X~~).
X(1) _ y(1)
ij
-
ij
and
k
X(n+1) = ] yen) X(n)
ij
u=l iu uj
(2)
(n)
so that X(n+1) is a weighted mean of X(n) •...• Xkj
• if
ij
ij
k
] Yi(n) = 1 and yi(n ) ~ 0
U
u=l
u
as in (1).
(u=l ..... k)
Hence
and further
~ max(X(l»
u
As n
nj
(u.i.j = 1•...• k)
(3)
~~. min(X~~» and max(X~~» must each tend to a limit. We now show that
u
u
these limits coincide with probability 1. and hence this must also be
lim X(n) = X • say
ij
ij
~
(for all i)
(4)
- 5 -
provided
pr[max(Y~~» < 1] ) O.
u
If min(X(n»
uj
u
= max(X(n»
then all {Xu(n )} (u=l •...• k) are equal (to this
j
uj
•
u
common value). and this is also their common limit value X1j =
~j
= ... =
~j.
Otherwise. defining
we have. from (4)
pr[D~n+1) ~ (l~}D~n}I{X~~)}. j = 1.... k] ) ~ ) 0
for some fixed
~ ) 0 and ~ ) O. for all {X~~}}.
Hence
(5)
(since D~l) ~ 1) where Z has a binomial distribution with parameters (n.~).
So. for any 6 ) 0 and any
*
~
) 0
pr[D~n) < 6 for all n ) nO (6.~.~*)] ) 1 - ~*
(Note that
r ) n.
(6)
D~n) cannot decrease with n. so if D~n} < 6. then D~r} < 6 for all
Choose m in (5)
greater than {log
6}/{log{1~}}
and nO large enough to
make
Pr[Z
Note that each
as n
~~
~
m] ) 1 -
~
* .}
x~~} (i=l •...• k) tends to the same limiting value (see (4})
(though not. in general. the same for all j).
distribution of ~~n) is the same for all i.
A fortiori. the limit
Further. there is no need to
investigate the joint limit distribution of ~~n) •...• ~n) since given any set
of
~i
values. the values of each of the other rows
same as those of
~i.
{~j}
(j
~
i) will be the
- 6 -
{ii}
Recurrence relation among variables with original and limit
distribution
If X{n} has a limit distribution. this will also be the limit distribution
-i
{as n ~ m} of the i-th row of y{n} y{n-1} ... y{2}.
Hence we have
- -
X* = xy
{7}
where X* and X each have the limit distribution. Y has the y{1} distribution.
"'"
""wI
,...,
,...,
-
and X and Yare mutually independent.
{iii}
Formulae linking moments of the original and limit distributions
From {7}
{S}
Since each!u has the same distribution we see that each row
*
~i
= {X
i1
..... X }
ik
has the same distribution.
We denote
{for all i}
and
E[j=1~ Ytujj ]
by u
t1···~
{for all u}.
From {S}
J.L
s1···~
=E
rr
[klk
j=1
~ X
=1
iu
krr {X
}S]
uj
Y
j
u=1
iu
Yuj }h ju}]
{9}
where
[hjl~~·hjk] = (Sj!)~l(hjul)}-l
and
~
hj
denotes summation over all nonnegative integers
hj
= {h
j1
•...• h
jk
}
- 7 subject to
k
1
u=l
h ju = Sj
(j = 1•...• k).
Equation (9) can be rearranged as
J.L
where h
·u
k
h.u] k
=
1 ... 1 E rr Xi
rr (h j1
s
sl··· k h
~
=1
u
j=l
1
L
=
k
1
j=l
~~ .hju] E L~ ~ Yh~u]
=1 j=l
(10)
UJ
hj .
u
Noting that
EL~=l j=l
~ yhujju]
=
~
u=l
E[
~
j=l
~
ju
v
yh ] =
uj
u=l h 1u •···
·hm
k
= rr
j=l
v
h 1j • .. "~j
we have {for all (sl •...• ~»
J.L
sl· .... ~
=1
h
""1
... 1
h..
"'lc
~
°. 1 ..... h ·k j=l
u.
.
(h j 1•
~~ .. h jk]
vh1 J.' ...• h..
--k j
( 11 )
From the equations (II). the J.L·s can be obtained from the v·s. and
conversely.
(There are. of course. relationships among the J.L·s. and among the
v·s. arising from Remark (2) following the statement of the theorem.)
(iv)
Checking that the moment equations are satisfied
If !~n) has the symmetrical Dirichlet distribution D{a.a •... a) for all i
and all n. then
k
= r(ka)
{r(a)}k
Equations (11) now become
rr r{a + hUj )
,;;.u=;;;,;l=-_
rea + h. j )
(12)
- 8 -
We now show that equations (11) are sa ti s fi ed if
~i
has the
D(ka.ka •...• ka) distribution. with
k
~
n (ka + s )
_ r(k a) j=1
j
2
s1·····~ - {r(ka)}k r(k a + s)
2
(14.1)
k
where s = }; Sj' and
j=1
k
~.l·····h.k
n r(ka + h. j )
_
.l&,j=....;l:.-.."...._
2
- {r(ka)}k r(k a + s)
2
r(k a)
k
k k
k
(since }; h j = }; }; h j = }; s = s).
j=l·
u=l j=l u
u=l u
The right-hand side of (13) can be written as
k
(because
k
n n
j=1 u=1
k
rea + hUj ) =
k
n n
j=1 u=1
rea + h ju »
where (y1 •...• yk) has a Dirichlet D(a.a •... a) distribution (See (1).)
Thus the right-hand side of (13) is equal to
(14.2)
- 9 -
k
~
(because
u=l
Y
u
= 1).
The statement of the theorem now follows from the last paragraph of the
Introduction.
REMARK (4):
The distributions D(a,a, ... ,a) and D(ka,ka, ... ,ka) have the same
marginal expected values _k- 1 - and the same pairwise correlation coefficients
- -(k-1)
-1
.
However, the marginal variances of D(a,a, ... ,a) are
k-2 (ka + 1)-1(k-1), while those of D(ka,ka, ... ,ka) are k- 2 (k2a + 1)-1(k-1).
3.
CX>Na.USING OOMMENTS
The result obtained in the theorem is remarkable in that the relationship
between original and limit distributions is so simple in the case of
symmetrical Dirichlet distributions.
The methodology can be extended to the case of general exponential
families, prOViding simplified and more 'transparent' proofs of the results of
Chamayou and Letac (1991), mentioned earlier.
- 10 REFERENCES
Aitchison. J. (1986)
The Statistical Analysis of Compositional Data. London:
Chapman and Hall.
Chamayou. J.F. and Letac. G. (1991)
Explicit stationary distributions for
compositions of random functions and products of random matrices.
]. !heor. Prob .• ,.,
4. 3-36.
Devroye. L.• Letac. G. and Seshadri. V. (1989)
The limit behavior of an
interval splitting scheme. Stattst. Prob. Letters. ,.,
4. 183-186.
Fang. K.-T .• Kotz. S. and Ng. K.W. (1990)
Synunetrical Multivariate and Related
Distributions. New York: Chapman and Hall.
Johnson. N.L. and Kotz. S. (199Oa)
Randomly weighted averages:
--
Some results
and extensions. Amer. Stattsttctan. 44. 245-249.
Johnson. N.L. and Kotz. S. (l990b)
Use of moments in deriving distributions
--
and some characterizations. Math. Sctenttst. 15. 42-52.