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JOURNAL OF THE
CHUNGCHEONG MATHEMATICAL SOCIETYVolume 22, No. 1, March 2009
A CHARACTERIZATION OF GAMMA DISTRIBUTION
BY INDEPENDENT PROPERTY
Min-Young Lee* and Eun-Hyuk Lim**
Abstract. Let {Xn, n 1} be a sequence of independent
identi-cally distributed(i.i.d.) sequence of positive random
variables with
common absolutely continuous distribution function(cdf) F(x)
andprobability density function(pdf) f(x) and E(X2) < . The
ran-
dom variables Xi Xj(nk=1Xk)
2and nk=1Xk are independent for 1 i 0,0 for x 0,
where >0 and >0. The characteristic function of gamma
distribu-tion is given by
(t; , ) = (1 it).
Here and >0 are two parameters.Let X and Y be two independent
non-degenerate positive random
variables. Then Lukacs(1955) proved that X/Y and X+Y are
inde-pendent if and only ifX and Yare gamma distribution with the
samescale parameter.
Using the moment, Findeisen(1978) characterized the gamma
dis-tribution. Also, Hwang and Hu(1999) proved a characterization
of
Received September 25, 2008; Accepted February 13, 2009.2000
Mathematics Subject Classification: Primary 60E05, 60E10.Key words
and phrases: independent identically distributed, a statistic
scale-
invariant, gamma distribution.Correspondence should be addressed
to Min-Young Lee, [email protected] present research was
conducted by the research fund of Dankkok University
in 2008.
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2 Min-Young Lee and Eun-Hyuk Lim
the gamma distribution by the independence of the sample mean
andthe sample coefficient of variation. Recently, Lee and Lim(2007)
pre-sented characterizations of gamma distribution that the random
vari-
ables nk=1Xk and mk=1Xknk=1Xk
are independent for 1 m < n if and only
ifX1, , Xn have gamma distribution.In this paper, we obtain the
characterization of gamma distribution
by independent property of product and sum of random
variables.
2. Main result
Theorem 2.1. Let {Xn, n 1} be a sequence of i.i.d. sequence
ofpositive random variables with common absolutely cdf F(x) and
pdf
f(x) andE(X2) < . The random variables Xi Xj(nk=1Xk)
2 andnk=1Xk
are independent for 1 i < j n if and only if {Xn, n 1}
havegamma distribution.
Proof. Since Xi Xj
(nk=1Xk)2is a statistic scale-invariant,
Xi Xj
(nk=1Xk)2 and
nk=1Xkare independent for gamma variable [see Lukacs and
Laha(1963)].We have to prove the converse.
We denote the characteristic functions of Xi Xj(nk=1Xk)
2, nk=1Xk and
Xi Xj(nk=1Xk)
2, nk=1Xk
by 1(t),2(s) and(t, s), respectively. The in-
dependence of Xi Xj(nk=1Xk)
2 and nk=1Xk is equivalent to
(1) (t, s) =1(t) 2(s).
The left hand side of (1) becomes
(t, s) =
0
0
exp
is(xi xj)
(nk=1xk)2
+ it(nk=1xk)
dF
where dF = f(x1) f(xn) dx1 dxn. Also the right hand side of(1)
becomes
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A Characterization of gamma distribution by independent property
3
1(t) 2(s) =
0
0
exp
is(xi xj)
(nk=1xk)2
dF
0
0
exp{it(nk=1xk)}dF.
Then (1) gives
(2)
0
0exp
is(xi xj)(nk=1xk)
2+ it(nk=1xk)
dF
=
0
0
exp
is(xi xj)
(nk=1xk)2
dF
0
0
exp{it(nk=1xk)}dF.
The integrals in (2) exist not only for reals t and s but also
for complexvalues t = u+ iv, s = u +iv, where u and u are reals,
for whichv = Im(t) 0, v = Im(s) 0 and they are analytic for all t,
s for
v= I m(t)> 0, v
=I m(s)> 0 [see Lukacs(1955)].Differentiating (2) one time
with respect to s and then two times
respect to t and setting s = 0, we get
(3)
0
0
xi xjexp{it(nk=1xk)}dF
=
0
0
(nk=1xk)2 exp{it(nk=1xk)}dF
where = E Xi Xj
(nk=1Xk)2 for 1 i < j n.
The random variable is bounded. Therefore all its moments
exist.
Note that
= E
X1 X2
(nk=1Xk)2
= E
X1 X3
(nk=1Xk)2
= =E
Xn1 Xn(nk=1Xk)
2
for i.i.d. random variables X1, ,Xn.
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4 Min-Young Lee and Eun-Hyuk Lim
Then we get the following equation by adding of all and
multiplying2
(4)
2 nC2 = E
21i 0, 0 < 21i1.
The solution of this differential equation (5) with the above
initialconditions is
(t) =
1
iE(X)
t
, =
n
1 n2 >0.
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A Characterization of gamma distribution by independent property
5
ThereforeF(x) is a gamma distribution.
References
[1] P. Findeisen, A simple proof of a classical theorem which
characterizes thegamma distribution, Ann. Stat. 6 (1978), no. 5,
1165-1167
[2] T.Y. Hwang and C.Y. Hu,On a characterizations of the gamma
distribution:The independence of the sample mean and the sample
coefficient of variation,
Ann. Inst. Stat. Math. 51 (1999), no. 4, 749-753.[3] M.Y. Lee
and E.H. Lim, Characterization of gamma distribution, J.
Chungcheong Math. soc. 20 (2007), 411-418.[4] E. Lukacs,A
Characterization of the gamma distribution, Ann. Math. Stat. 17
(1955), 319-324.
*Department of MathematicsDankook UniversityCheonan 330-714,
Republic of KoreaE-mail: [email protected]
**Department of Mathematics
Dankook UniversityCheonan 330-714, Republic of KoreaE-mail:
[email protected]
