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A Simple Representation of the Weighted
Non-Central Chi-Square Distribution
Dr. Elsayed A. Elsherpieny
1, Dr. Sahar A. Ibrahim
2, Dr. Yassmen, Y. Abdelall
3Professor, Dept. of Mathematical Statistics, Institute of Statistical Studies and Research, Cairo University, Cairo, Egypt1
Lecture, Dept. of Mathematical Statistics, Institute of Statistical Studies and Research,Cairo University, Cairo, Egypt2, 3
Abstract: In this paper we consider the probability density function (pdf) of the weighted non-central chi-square distribution with
degrees of freedom. This pdf presented in the literature as an infinite sum. Herein, we present an alternative simple expression to this pdf which may be useful in the future studies. The results of Kettani (2006) and Andras and Baricz (2008) can be obtained as a special case from ours.Keywords: Weighted non-central chi-square distribution, Modified Bessel function, Modified Struve function, Hoppe’s formula.
I. INTRODUCTION
Grau (2009) introduced the weighted non-central chi-square distribution with
degrees of freedom,
2
,
1,
2
, which has application in a wide spectrum of quality control areas. The probability density function (pdf) of such distribution is expressed as an infinite sum of central chi-square distributions as follows:
)
1
.
1
(
.
0
,
0
,
1
1
2
1
2
2
02 2 2
2 2
1 2
1 2
2
2
2
f
z
f
z
z
k
k
e
z
f
k
k k
k k
Where
1and
2 are two positive numbers,
is the non-centrality parameter,
.
is the gamma function, 2
.
k
f
is the central chi-square distribution with
k
degrees of freedom. Now in (1.1) substitutek
2
j
1
for oddk
and denote it byf
O
z
and
k
2
j
for evenk
and denote it byf
E
z
,j
0
,
1
,
2
,
...
, we get
)
2
.
1
(
1
2
2
1
2
1
2
2
1
2
2
02 1
2 2
2 1
2 2
1 2
1
2
2 2
2 1 2
j
i
i i
j
i
i i
i j
z
f
j
j
z
f
j
j
e
z
f
j j
z
f
z
.
f
O E
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1
,
(
1
.
3
)
2
2
1
2
2
02 1
2 2
1 2
1 2
/
2 1 2
j i
i i
i j
O
z
f
j
j
e
z
f
j
and
2
2
/ 2 2
2 2
0 1
1
2
2
.
(1.4)
2
1
2
jj
E
i i
j i
j
e
f
z
f
z
j
When
1
2
1
, (1.1) reduced to the pdf of the non-central chi-square distribution with
degrees of freedomand non-centrality parameter
,
2
. When
1
2
1
and
0
,
(1.1) reduced to the pdf of the centralchi-square distribution with
degrees of freedom,
2, given by
,
0
.
(
1
.
5
)
2
2
/22 1 2
2
z
e
z
z
f
z
Kettani (2006) expressed the pdf of the non-central chi-square distribution for odd degrees of freedom in terms of the mth-derivative of the hyperbolic cosine function. Andras and Baricz (2008) got the same results of Kettani (2006) but through an alternative proof. In addition, they deduced the corresponding formulas for even degrees of freedom. Their proofs are based on simple properties of modified Bessel functions of the first kind and on Hoppe’s formula for the mth -derivative of a composite function.
As the non-central chi-square distribution arises frequently in various applications including finance, estimation theory, decision theory and time series analysis, the weighted non-central chi-square distribution arises in quality control and statistical analysis. Elsherpieny et. al. (2012) proved that the pdf of the weighted non-central chi-square distribution with υ degrees of freedom has an expression that does not explicitly involve an infinite sum. This expression is given by
,
1
2
if
4
;
1
,
2
3
;
1
1
2
2
2
if
4
;
2
3
,
2
3
;
1
1
2
2 1
2 2
1 2 2
1 2
/
2 1
2 2
1 2 2
1 2
/
2 2 2 2
3 2
m
z
m
F
z
f
e
m
z
m
F
z
f
e
z
f
i
i i
i i i
i i
i i
O
m m
and
)
7
.
1
(
.
1
2
if
cosh
2
2
2
)
6
.
1
(
2
2
if
2
4
2
1
2 2
1 2
/
2
1
2 0
2 2 2
/
2 2
m
z
d
z
d
e
z
z
e
m
z
d
z
I
d
e
z
e
z
f
i
m i m
z i m i
m i m
z i m E
i i
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Where
m
is a non negative integer, 1F
2
.
is the generalized hypergeometric function,I
0
.
is the modified Bessel function of the first kind of zero order andcosh
.
is the hyperbolic cosine function.In this paper, for numerical evaluation purpose, we interested to simplify the infinite sum in the pdf of the weighted non-central chi-square distribution with υ degrees of freedom. In Section 2, we present some required special functions and notations to facilitate understanding this paper. In Section 3, we prove that the pdf of the
2
,
1,
2
distribution with υ degrees of freedom can be represented as the sum of odd and even terms each has a simple form expression. Our proof is based on the modified Struve and Bessel functions. For the even terms we follow the same argument as that given in Andras and Baricz (2008). That is, our proof is based on some simple properties of modified Bessel functions of the first kind and on Hoppe’s formula for the mth-derivative of a composite function.II. SOME SPECIAL FUNCTIONS AND NOTATIONS
The following are some special functions that will be used in our derivation. 1-
2
n
is the gamma duplication function and is defined as
,
1
,
2
,
3
,...
.
(
2
.
1
)
2
1
2
2
1 2
n
n
n
n
n
2-
J
p(
x
)
is the Bessel function of the first kind of orderp
and is defined as
,
and
0
.
(
2
.
2
)
1
!
2
/
1
0
2
p
R
x
m
p
m
x
x
J
m
p m m
p
3-
I
p(
x
)
is the modified Bessel function of the first kind of orderp
and is defined as
,
and
0
.
(
2
.
3
)
1
!
2
/
0
2
p
R
x
m
p
m
x
x
I
m
p m
p
In particular
,
.
!
2
/
0
2 2
0
x
R
m
x
x
I
m
m
4-
H
p
x
is the Struve function of the first kind of orderp
, which is valid for allp
,and is defined as
,
.
(
2
.
4
)
2
3
2
3
2
/
1
0
1 2
R
x
m
p
m
x
x
H
m
p m m
p
5-
L
p
x
is the modified Struve function of the first kind of orderp
, which is valid for allp
, and is defined as
,
.
(
2
.
5
)
2
3
2
3
2
/
0
1 2
R
x
m
p
m
x
x
L
m
p m
p
6- The derivative formula of the modified Struve function is defined as
x
L
x
x
L
1
x
,
x
R
.
(
2
.
6
)
dx
d
p p p
p
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7-The relation between the modified Struve function
L
n
x
and the modified Bessel functionI
n(
x
)
for non-negativeinteger n is given by
,
,
0
,
1
,
2
,
...
.
(
2
.
7
)
2
!
!
2
!
2
1
2
20
2 2
1 2
1
x
R
n
x
t
n
t
t
x
x
I
x
L
t n n
t
t t
n
n
8-
n
!
!
is the double factorial function and is defined as
(
2
.
8
)
.
0
,
1
if
1
even
is
0
if
2
.
4
...
2
odd
is
0
if
1
.
3
...
2
!
!
n
n
n
n
n
n
n
n
9-
cosh
j
x
is the alternate hyperbolic cosine/sine function and is defined as
.
2
1
)
9
.
2
(
odd
is
if
sinh
even
is
if
cosh
cosh
x j x
j
e
e
j
x
j
x
x
For more details about the above special functions see Andrews (1992).
III.ASIMPLE FORM OF THE PDF OF
2
,
1,
2
.In this section, we prove that the pdf of the weighted non-central chi-square distribution with υ degrees of freedom has a simple form expression. This expression will be given in the following theorem.
The Main Theorem
The probability density function of the weighted non-central chi-square distribution with
degrees of freedom,
1 2
2
,
,
, is given by:
f
z
f
O
z
f
E
z
where
,
1
if
sinh
1
2
2
)
2
.
3
(
2
2
if
1
2
)
1
.
3
(
1
2
if
)
(
1
2
2 1
2 2
1 1 2
/ 2
1
2 1
1 2 2
1 1
2 /
2 1
2 1 2 2
1 2
1 2 /
2 2
2
i
i z
i i i
i m
m
i z
i m
i
i m
i z
i
m
O
z
e
z
e
m
z
L
z
dz
d
e
e
m
z
A
z
dz
d
e
e
z
f
i i
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and
.
1
if
cosh
2
2
)
4
.
3
(
2
2
if
4
)
3
.
3
(
1
2
if
cosh
2
2
2 1
2 2
1 2
/ 2
1
1 0
2 0
) ( 2
2 / 2 2 2 /
2 1
1 0
2 2
2 / 2 1 2
/
2
2 2
i
i z
i i
m
l
l m
i l
m
l z m
i i i
m
l
i l m l
z m
i i
E
z
e
z
e
m
z
d
z
I
d
z
s
e
z
e
m
z
z
s
e
z
z
e
z
f
i
i i
Where
,
1
,
2
and
0
,
1
,
2
,
...,
2
!
!
!
2
1
cosh
21 4 2 2
0 0
2
2
z
i
m
t
m
t
t
z
z
a
z
A
t m i m
t
m t m
l
i l
m l
i
21 2 2
!
!
1
2
1
l
i l m
l m l
l
z
l
C
z
a
,
2
!
!
1
!
!
1
1
2 /2
l
m
l
l
m
z
z
s
i ll
l
,and
m
is a non negative integer.Proof:
It is worth mentioning here that when putting
1
2
1
, thenf
( )
z
f
E( )
z
, andin (3.3) we get the pdf of the non-central chi-square distribution which is the same as that given in Kettani (2006) but with completely different proof and as that of Andras and Baricz (2008). Also when
1
2
1
, then( )
E( )
f
z
f
z
, and in (3.4) we get the pdf of the non-central chi-square distribution as that given in Andras and Baricz (2008).First, for
m
1
, considerf
O
z
given in (1.3). Using the gamma duplication function in (2.1) withn
j
1
and using (1.5), we get
0
2
1 2
1 2
2 1 2
1 2 2 2 1 2
1 2
/
2
1
2
2
1
2
3
2
2
2
j i
j
z j
i i i j
O
j
e
z
j
e
z
f
i
.
(
3
.
5
)
2
1
2
2
3
4
/
1
2
21 0
2 2 1 2 1
2 2 2 1 1
2 2 /
2
i j
z j
i i
i i
j
j
e
z
z
e
i
Using (2.5) of the modified Struve function of the first kind of order
1
2
p
and with2
,
1
,
2
i
z
x
i , then (3.5) becomes:
2
2
14 2 2
/ 2z
z
e
i
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1
.
(
3
.
6
)
4
21
2 1
2 2 2 2
2 2 1 2
/ 2
/ 2
i
i i
z
i i i
z
L
z
e
z
e
i
Using the derivative formula of the modified Struve function (2.6) with
2
p
andx
i2z
,
i
1
,
2
, we get
,
1
,
2
.
2
2 1
2 2 2 2
2 2
2 2
2
i
z
L
z
z
z
L
z
dz
d
i i
i i i
i
Hence, Equation (3.6) takes the form
1
.
(
3
.
7
)
2
21
2 2
2 2 2
1 2
/ 2
/ 2
i
i i
z i
O
z
L
z
dz
d
e
e
z
f
i
Let
be odd, i.e.,
2
m
1
, m=0,1,2,... in (3.7) we get
1
.
(
3
.
8
)
2
2 1
2 2
1 2 1 2 2
1 2
1 2 / 1
2
2
i
i m
m
i z
i
m O
m
z
L
z
dz
d
e
e
z
f
i
Using (2.7) with
n
m
andx
i2z
,
i
1
,
2
,we get
)
9
.
3
(
.
...
,
2
,
1
,
0
,
2
!
!
2
!
2
1
/
2
2 2 0
2 2
/ 1 2 2
2 1 2
2 1
m
z
t
m
t
t
z
z
I
z
L
t m
i m
t
t t
i i
m i
m
Consider the following formula of the modified Bessel function of the first kind of order
2
1
1
km
, given in Kettani (2006),
1
2
1
!
!
cosh
.
(
3
.
10
)
2
1 0
2 1
1
x
x
l
C
x
x
I
m l kl l m
l m m
l l
m
k
Taking k =1 and
x
i2z
,
i
1
,
2
in (3.10), we get
2
1
2
1
!
!
cosh
2
2
.
2/ 2 0
2 / 1 2 2
2
1
l
C
z
z
z
z
I
m l il
i l m
l m m
l l
i i
m
Substituting in (3.9) and then in (3.8) we get the result in (3.1).
Let
be even, i.e.,
2
m
2
, m=0,1,2,... in (3.7) we get the result in (3.2).
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1
.
4
21
2 2
1 2 1 2 2
2 2 1 2
/ 1
2
i
i i
z
i i i O
z
L
z
e
z
e
z
f
i
It is known that
2
sinh
.
21
x
x
x
L
Taking
x
i2z
,
i
1
,
2
,we get the result.Second, for
m
1
, considerf
E
z
given in (1.4). Let
be odd, i.e.,
2
m
1
,m
0
,
1
,
2
,...
and recall the new form in (1.7). Also consider the following Hoppe's formula for the mth-derivative of a composite function given in Andras and Baricz (2008),
1
,
1
,
2
.
(
3
.
11
)
!
0 0
) (
i
z
g
z
d
d
z
g
C
k
z
g
f
z
g
f
z
d
d
mk
k
l
l i m m l k i l k l k i
k
i m m
Let
f
z
cosh
z
andg
i
z
i2z
,
i
1
,
2
, in (3.11), we get
)
12
.
3
(
.
2
,
1
,
1
!
cosh
cosh
0 0
2 2
2 2
i
z
z
d
d
z
C
k
z
z
z
d
d
mk
k
l
l
i m m l k
i l k l k i
k i
m m
It is clear that
f
k
i2z
cosh
k
i2z
.Also
2
2
2
/2
2
...
2
2
2,
1
,
2
.
(
3
.
13
)
i
z
m
l
l
l
z
z
d
d
l l mi l
i m
m
m
Consequently (3.12) becomes
)
14
.
3
(
.
2
,
1
,
!
!
2
2
...
2
1
2
cosh
cosh
2 20 0
2 2
i
z
l
k
l
m
l
l
l
z
z
z
z
d
d
ki m
k
k
l
l k
m i k i
m m
Consider the following assertion given in Andras and Baricz (2008), i.e.
.
(
3
.
15
)
!
1
!
!
2
2
!
1
2
1
!
!
2
2
...
2
1
0
m
m
k
k
k
l
l
k
m
l
l
l
m kk
l
l k
Where
k
0
,
1
,
2
,...,
m
andm
is a natural number. Then (3.14) becomes:
,
1
,
2
.
2
cosh
!
1
!
!
2
2
!
1
2
1
cosh
2 20
2
2
i
z
z
z
k
k
m
k
m
z
z
d
d
ki m
k
m i k k
m i
m m
Substituting in (1.7) we get
cosh
.
!
1
!
!
2
2
!
1
2
1
2
2
2 2
2 2
2 1 2
/ 1
2
2
z
z
k
k
m
k
m
e
z
e
z
f
k ik
i m
k m z
i m
E m
i
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Changing
k
in (3.16) with(
m
l
)
, we obtain the following formula
cosh
.
2
2
2 2
1 0
2 2 / 2 1 2
/ 1
2
2
z
z
s
e
z
z
e
z
f
m l ii
m
l l z m
i i E
m
i
Where
.
!
1
!
!
2
!
1
1
2 /2
l
m
l
l
m
z
z
s
i ll
l
Now rewriting
s
l
z
as follows:
.
!
!
!
2
!
1
2 /2l
m
l
m
l
l
m
l
m
z
z
s
l l i l
At
l
m
,s
l
z
0
and consequently the last term in the summation overl
is zero. Hence we get the result (3.3). Let
be even in (1.4), i.e.,
2
m
2
,m
0
,
1
,
2
,...
and recall the corresponding new form in (1.6). Now recall Hoppe’s formula in (3.11) withf
z
I
0
z
andg
i
z
i2z
,
i
1
,
2
, i.e.,
)
17
.
3
(
.
2
,
1
,
1
!
0 0
2 2
2 0
2
0
i
z
z
d
d
z
C
k
z
I
z
I
z
d
d
mk
k
l
l
i m m l k
i l k l k i
k
i m
m
It is clear that
I
z
z
d
d
z
I
z
f
k ii k
i k i
k 2
0 2 2
0
2
.Using (3.13) and (3.15) then (3.17) becomes:
,
1
,
2
.
2
!
1
!
!
2
2
!
1
2
1
2 20
2 )
( 0 2
0
i
z
z
z
I
k
k
m
k
m
z
I
z
d
d
ki m
k
m i k k
m i
m m
Substituting in (1.6) we get
)
18
.
3
(
.
!
1
!
!
2
2
!
1
2
1
4
2 0
2 2 2
1 0
2 2 2 / 2
2
2
z
I
z
k
k
m
k
m
e
e
z
f
k ik
i i
m
k
k m z
i m E
m
i
Changing
k
in (3.18) with(
m
l
)
, we obtain the following formula
.
4
2 0
2
1 0
2 2 / 2 2 2
/ 2
2
2
z
I
z
s
e
z
e
z
f
l m l ii
m
l z m
i i E
m
i
Where
.
!
1
!
!
2
!
1
1
2 /2
l
m
l
l
m
z
z
s
i l ll
ISSN: 2319-8753
I
nternational
J
ournal of
I
nnovative
R
esearch in
S
cience,
E
ngineering and
T
echnology
(An ISO 3297: 2007 Certified Organization)
Vol. 2, Issue 9, September 2013
REFRENCES
[1] Andras, S. and Baricz, A. "Properties of the Probability Density Function of the Non-Central Chi-Squared Distribution", Journal of Mathematical Analysis and Applications, 346, 395-402, 2008.
[2] Andrews, L. C. "Special Functions of Mathematics for Engineers", Second Edition, McGraw-Hill, Inc. New York 1992.
[3] Elsherpieny, E. A., Ibrahim, S. A. and Abdel-all, Y. Y."Contributions to the Weighted Non-Central Chi-Square Distribution", The Egyptian Statistical Journal, June, Institute of Statistical Studies and Research, Cairo University, Egypt. 2012.
[4] Grau, D. " Moments of the Unbalanced Non-Central Chi-Square Distribution", Statistics and Probability Letters, 79, 361-367, 2009.
[5] Kettani, H. "Contributions to the Theory of the Non-Central Chi-Square Distribution", International Conference on Scientific Computing, Las Vegas, Nevada, 2006.
APPENDIX
The proof of (1.6) and (1.7) given in Elsherpieny et. al. (2012) is as follows. Using the following formula given in Andras and Baricz (2008)
)
1
(
,
,...
2
,
1
,
0
,
!
2
!
2
2
1
2
n
n
n
n
n
and using also (1.5) in (1.4), we get
/
4
.
(
2
)
2
2
!
1
2
2 1
2 2
1 2 2 2 0
1 2
2
/ 2
i
z j i i
i j
E
i
e
z
z
j
j
e
z
f
Using (2.3) with
1
2
p
andx
i2z
,
i
1
,
2
, we get
/
.
(
3
)
4
21
2 1
2 2 2 1 4 2 2 2
/ 2
i
i z
i i E
z
I
e
z
e
z
f
i
It is clear that when
1
2
1
, (3) reduced to the formula of the pdf of the non-central chi-square distributiongiven in Kettani (2006) and in Andras & Baricz (2008).
To simplify (2), consider the normalized modified Bessel function of the first kind of order
p
,
0
,
1
,
:
p
, given in Andras and Baricz (2008), i.e.,
x
2
p
p
1
x
p/2I
p
x
.
(
4
)
p
Taking
1
2
p
andx
i2z
,
i
1
,
2
, then (3) can be written as follows:
.
(
5
)
2
2
2
2 1 2 2
1 2
2 1 2 2 2 2
/
2
z
e
z
e
z
f
ii
z
i i E
i
Therefore,
.
2
2 1
2 1 2 2 2
2 /
2
i
i i
i E
z
z
f
e
z
f
Again, putting
