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Songklanakarin J. Sci. Technol.
41 (1), 96-108, Jan. – Feb. 2019
Original Article
A non-uniform bound on binomial approximation to the beta binomial
cumulative distribution function
Kanint Teerapabolarn1, 2* and Khunakorn Sae-Jeng1
1 Department of Mathematics, Faculty of Science,
Burapha University, Mueang, Chonburi, 20131 Thailand
2 Centre of Excellence in Mathematics, Commission on Higher Education,
Ministry of Education, Ratchathewi, Bangkok, 10400 Thailand
Received: 29 June 2017; Revised: 31 August 2017; Accepted: 2 October 2017
Abstract This paper uses Stein’s method and the characterization of beta binomial random variable to determine a non-uniform
bound for the distance between the beta binomial cumulative distribution function with parameters N, 0n and 0
and the binomial cumulative distribution function with parameters n and
. Some numerical examples are given to illustrate
the obtained result.
Keywords: beta binomial cumulative distribution function, binomial approximation, characterization of the beta binomial
random variable, non-uniform bound, Stein’s method
1. Introduction
Let X be the binomial random variable with parameters
Nn and
(0,1).p Its probability mass function is as
follows:
( ) , {0,..., }x n xnb x p q x n
x
, (1.1)
where 1q p and ( )E X np and ( )Var X npq are its mean and variance, respectively. It is well-known that if the
*Corresponding author
Email address: [email protected]
K. Teerapabolarn & K. Sae-Jeng / Songklanakarin J. Sci. Technol. 41 (1), 96-108, 2019 97
probability of success in the binomial distribution is a random variable and has a beta distribution with parameters 0 and
0, then a new resulting distribution is referred to as the beta binomial distribution with parameters , n and . Let Y
be the beta binomial distribution with parameters , n and . and its probability mass function is of the form
( , )( ) , {0,..., },
( , )
n B y n ybb y y n
y B
(1.2)
where B is the complete beta function and n
and 2
( )2
( 1)( )
n n
are the mean and variance of ,Y respectively.
Some useful applications of this distribution can be found in field such as animal teratology experiments in Gupta and Naradajah
(2004), statistical process control in Sant’Anna and Caten (2012), inferential statistics in Salem and Abu El Azm (2013) and
balancing revenues and repair costs in Ding, Rusmevichientong, and Topaloglu (2014).
For limiting distribution, it follows from (1.2) that if , in such a way
tends to a constant, then the
beta binomial distribution with parameters ,n and converges to a binomial distribution with parameters ,n and
. In
this case, Teerapabolarn (2008) used Stein’s method and the binomial -functionw to give a uniform bound on binomial
approximation to the beta binomial distribution as follows:
1 1 ( 1), (1 )
( 1)( 1)
n nA
n nd Y X p q
n
(1.3)
for every subset A of {0,..., },n where , ( ) ( )Ad Y X P Y A P X A is the distance between the beta binomial
distribution with parameters ,n and and the binomial distribution with parameters n and .
For
0 0{0,..., },xA C x 0 {0,..., },x n the result in (1.3) becomes
0
1 1 ( 1), (1 ) ,
( 1)( 1)x
n nC
n nd Y X p q
n
(1.4)
where 0
0 0, ( ) ( )xCd Y X P Y x P X x is the distance between the beta binomial cumulative distribution function with
parameters ,n and and the binomial cumulative distribution function with parameters n and
at 0.x
We observe that the bound in (1.4) is uniform in 0 {0,..., },x n that is, it does not change along 0 {0,..., }.x n So,
it may be inappropriate for measuring the accuracy of the approximation. In this paper, we are interested to determine a non-
uniform bound with respect to the bounds in (1.4) by using Stein’s method and the characterization of beta binomial random
variable, which are described and determined in Sections 2 and 3, respectively. In Section 4, some numerical examples provided
to illustrate the obtained result, and the conclusion of this study is presented in the last Section.
98 K. Teerapabolarn & K. Sae-Jeng / Songklanakarin J. Sci. Technol. 41 (1), 96-108, 2019
2. Method
Stein’s method and the characterization of beta binomial random variable are both tools which can be used to obtain the
desired result.
2.1 Stein’s method for binomial distribution
Stein (1972) proposed a powerful method for normal approximation, which is called Stein’s method. In 1986, he also
applied this method to binomial approximation (Stein, 1986). Following Barbour, Holst, and Janson (1992), Stein’s equation for
binomial distribution with parameters Nn and 0 1 1p q , for given h, is of the form
,( ) ( ) ( ) ( 1) ( )n ph x h n x pf x qxf x , (2.1)
where ,
0
( ) ( )n
k n kn p
k
nh h k p q
k
and f and h are bounded real valued functions defined on {0,..., }n . For {0,..., }A n ,
let :{0,..., } RAh n be defined by
1 if ,
( )0 if .
A
x Ah x
x A
(2.2)
Following Barbour et al. (1992), let : N {0} RAf satisfy (2.1), where (0)Af (1)Af and ( ) ( )A Af x f n for .x n
Therefore, the solution Af of (2.1) can be expressed as
1 1, , ,
1
( ) ( ) ( )( ) ,x xn p A C n p A n p C
Ax n x
h h hf x
nx p q
x
(2.3)
where 1 {0,..., 1}xC x . Similarly, for
0{ }A x and 0xA C when 0 {0,..., }x n and by setting
0 0{ }x xh h , thus the
solutions 0 0{ }x xf f and
0xCf are as follows:
, ,0 1
1
0, ,0 1
1
( ) ( )
0
( ) (1 )
0
if ,
( )
if
n p x n p Cx
x n x
n p x n p Cx
x n x
h h
nx p q
x
x h h
nx p q
x
x x
f x
x x
(2.4)
and
, ,1 0
1
0 , , 10
1
( ) (1 )
0
( ) (1 )
0
if ,
( )
if .
n p C n p Cx x
x n x
xn p C n p Cx x
x n x
h h
nx p q
x
C h h
nx p q
x
x x
f x
x x
(2.5)
K. Teerapabolarn & K. Sae-Jeng / Songklanakarin J. Sci. Technol. 41 (1), 96-108, 2019 99
Let 0 0 0( ) ( 1) ( )x x xf x f x f x and
0 0 0( ) ( 1) ( )
x x xC C Cf x f x f x . It is seen that 0( ) 0xf x and
0( ) 0
xCf x for 0,x n . Thus, for {1,.., 1}x n and by (2.4), we obtain
0( )xf x
, , ,0 1
, , ,0 1
, , ,0
( ) ( ) ( )
0( )
( ) (1 ) ( )
0( )
( ) (1 ) (1
( )
if ,
if ,
n p x n p C n p Cx x
x n x
n p x n p C n p Cx x
x n x
n p x n p C n p Cx x
x n x
h h h
n x p xqnp q
x
h h h
n x p xqnp q
x
h h h
n x pnp q
x
x x
x x
1)
0 if xq
x x
10 0
0 0 0
10 0
0 1 0
0( )
( )
if ,
if
x xk n k k n kx n x
k k
x n x
n xk n k k n kx n x
k x k
x n x
n nnp q p qp q
k kx
n x p xqnp q
x
n nnp q p qp q
k kx
n x p xqnp q
x
x x
0 0
0 1
0
0( )
,
if .
n nk n k k n kx n x
k x k x
x n x
n nnp q p qp q
k kx
n x p xqnp q
x
x x
x x
(2.6)
By (2.5), we also obtain
0
( )xCf x
, , ,0 1
, , ,0 1
(1 ) ( ) ( )
0( )
( ) (1 ) (1 )
0( )
if ,
if
n p Cx n p C n p Cx x
x n x
n p Cx n p C n p Cx x
x n x
h h h
n x p xqnp q
x
h h h
n x p xqnp q
x
x x
x x
1
10 0 0
0
0 1
0( )
( )
if ,
n x xj n j k n k k n k
j x k k
x n x
x n nj n j k n k k n k
j k x k x
x n x
n n np q p q p q
j k k
n x p xqnp q
x
n n np q p q p q
j k k
n x p xqnp q
x
x x
0 if .x x
(2.7)
The following lemmas present some necessarily properties of 0xf and
0,
xCf which are used to prove the main
result.
100 K. Teerapabolarn & K. Sae-Jeng / Songklanakarin J. Sci. Technol. 41 (1), 96-108, 2019
Lemma 2.1. Let {1,..., 1}x n and 0 {1,..., },x n then the following inequalities hold:
0
0
0
0 ,( )
0 x
if x xf x
if x x
(2.8)
and
0
0
0
0 ,( )
0 .xC
if x xf x
if x x
(2.9)
Proof. Firstly, we have to show that (2.8) holds. For 0 ,x x it follows from (2.6) that
0( ) 0.xf x (2.10)
For 0 ,x x we have to show 0( ) 0xf x when 0x x
and 0x x as follows. From (2.6), for 0 ,x x
0( )xf x
0 0
1
0 0 0
( )
x xx n x k n k k n k
k k
x n x
n n np q p q p q
x k k
n n x p xqp q
x
0 0
10 1 1
1 1 0 0
( ) .
( )
x n x
x xk n k k n k
x n x k k
np q
n nxx p q n x p q
n k kx p q n x
x
Let 1
1 11
0 0
( ) ( ) ,x x
k n k k n k
k k
n nx x p q n x p q
k k
then
1( )x 1
1 1 1 ( 1)
0 0
1 1( 1) ( 1)( )
11 1
x xk n k k n k
k k
n nx n k k n xp q p q
k kn n
1 1
0 0
1 1( 1) ( )
1 1
x xk n k k n k
k k
n nx n k k n xp q p q
k kn n
1
0
1
( 1)1
k n k
x
k
np q
kx n kn
n
0.
Thus for 0 ,x x
0( ) 0.xf x (2.11)
For 0 ,x x
K. Teerapabolarn & K. Sae-Jeng / Songklanakarin J. Sci. Technol. 41 (1), 96-108, 2019 101
0( )xf x
0 0
0 1
( )
n nx n x k n k k n k
k x k x
x n x
n n np q p q p q
x k k
n n x p xqp q
x
0 0
0 1 1
1 1 1
( ) .
( )
x n x
n nk n k k n k
x n x k x k x
np q
n nxx p q n x p q
n k kx p q n x
x
Let 1 12
1
( ) ( ) ,n n
k n k k n k
k x k x
n nx x p q n x p q
k k
then
2( )x 1 1 1 ( 1)
1
1 1( 1) ( 1)( )
11 1
n nk n k k n k
k x k x
n nx n k k n xp q p q
k kn n
1 1
1 1
1 1
1 1( 1) ( )
1 1
n nk n k k n k
k x k x
n nx n k k n xp q p q
k kn n
1
1
1
1
( 1)1
k n k
n
k x
np q
kx n kn
n
0.
Thus for 0 ,x x
0( ) 0.xf x (2.12)
Following (2.11) and (2.12), when 0 ,x x it yields
0( ) 0.xf x (2.13)
Hence, from (2.10) and (2.13), the inequalities in (2.8) hold.
Next, we shall show that (2.9) holds. For 0 ,x x it follows from (2.7) that
0( )
xCf x 0
11 1 1
1 1 0 0
( )
( )
nj n j
x xj x k n k k n k
x n x k k
np q
j n nx p q n x p q
n k kx p q n x
x
0
1
1
1 1 0
1
( 1)1
( )
nj n j k n k
xj x
x n x k
n np q p q
j kx n kn
n nx p q n x
x
0. (2.14)
For 0 ,x x by (2.7), we obtain
102 K. Teerapabolarn & K. Sae-Jeng / Songklanakarin J. Sci. Technol. 41 (1), 96-108, 2019
0( )
xCf x
0
0 1
( )
x n nk n k k n k k n k
k k x k x
x n x
n n np q p q p q
k k k
n n x p xqp q
x
0
0 1 1
1 1 1
( )
( )
xj n j
n nj k n k k n k
x n x k x k x
np q
j n nx p q n x p q
n k kx p q n x
x
01
10
1 1 1
1
( 1)1
( )
xj n j k n k
nj
x n x k x
n np q p q
j kx n kn
n nx p q n x
x
0. (2.15)
Hence, from (2.14) and (2.15), the inequalities in (2.9) hold.
Lemma 2.2. For 0 {1,..., 1},x n
0xCf is an increasing function in 0{1,..., }.x x
Proof. Let 0 0 0
2 ( ) ( 1) ( ).x x xC C Cf x f x f x We shall show that
0
2 ( ) 0xCf x for
0{1,..., 1}.x x It follows from
(2.7) that
0
2 ( )xCf x 0
1
1 0 0
1 1 ( 1) ( 1)
1
n x xj n j k n k k n k
j x k k
x n x
n n np q p q p q
j k k
n n x p x qp q
x
0
1
1 0 0
( )
n x xj n j k n k k n k
j x k k
x n x
n n np q p q p q
j k k
n n x p xqp q
x
0
0
1
11
2 0
1
1
1 1 0
1
( 1)( 1)1
( 1) ( 1)1
1
( 1)1
( )
nj n j k n k
xj x
x n x k
nj n j k n k
xj x
x n x k
n np q p q
j kx n kn
n nx p q n x
x
n np q p q
j kx n kn
n nx p q n x
x
K. Teerapabolarn & K. Sae-Jeng / Songklanakarin J. Sci. Technol. 41 (1), 96-108, 2019 103
0
0
11 1
2 0
1 1 1 ( 1)
2 0
1 ( 1)( 1)
1( )( 1)
1 ( 1)
( )( 1)
nj n j
xj x k n k
x n x k
nj n j
xj x k n k
x n x k
np q
j n x n knp q
n k n xp q n x n
x
np q
j n x n knp q
n k xp q n x n
x
0
11 1
2 0
1 1 ( 1)
0
1 ( 1)( 1)
1( )( 1)
1 1 ( 1)
1 1
nj n j
xj x k n k
x n x k
xk n k
k
np q
j n x n knp q
n k n xp q n x n
x
n k x n knp q
k n k x
0
11 1
2 0
11
0
1 ( 1)( 1)
1( )( 1)
1 ( 1) ( 1)
2
nj n j
xj x k n k
x n x k
xk n k
k
np q
j n x n knp q
n k n xp q n x n
x
n k x n k np q
k n k x
0
11 1
2 0
1 ( 1) 1
1( )( 1)
( 1)
2
nj n j
xj x k n k
x n x k
np q
j n x n kn np q
n k n xp q n x n
x
x n kn n k
n k x
0.
Therefore 0
( )xCf x is an increasing function in 0{1,..., }.x x
Lemma 2.3. Let {1,..., },x n then we have the following inequalities hold:
0
1sup ( )
n
Cx
qf x
np
(Teerapabolarn and Wongkasem, 2011) (2.16)
and
0
1 1
0
1 1sup ( ) min , ,
( 1)x
n n n
Cx
p p qf x
x q n pq
(2.17)
where 0 {1,..., }.x n
Proof. We shall show that the inequality (2.17) holds. Teerapabolarn and Wongkasem (2011) showed that
104 K. Teerapabolarn & K. Sae-Jeng / Songklanakarin J. Sci. Technol. 41 (1), 96-108, 2019
0
1 1
00
1 1( ) min , .
( 1)
n n n
x
p p qf x
x q n pq
(2.18)
From (2.6), we have
( )xf x
1
1 0 .( )
n xk n k k n k
k x k
n np q p q
k k
n x p xq
Substituting 0x by x in the proof of 0 0( )xf x detailed in Teerapabolarn and Wongkasem (2011), we also obtain
1 11 1( ) min , .
( 1)
n n n
x
p p qf x
xq n pq
(2.19)
In order to prove that 1 1
00
1 1
( 1)sup ( ) min , ,
n n n
x
p p qC x q n p
x
f x
it suffices to show
1 1
00
1 1
( 1)( ) min ,
n n n
x
p p qC x q n p
f x
for
every {1,..., 1},x n because
0( ) 0
xCf x as 0, .x n For 01 ,x x we have
0 0
( )xCf x (by (2.9))
00( )
xCf x (by Lemma 2.2)
0
0
0
( )x
k
k
f x
00 0 0( ) ( )xf x f x
0 0( )xf x (by (2.8))
1 1
0
1 1min , ,
( 1)
n n np p q
x q n pq
(by (2.18))
which gives
0
1 1
0
1 1( ) min , .
( 1)x
n n n
C
p p qf x
x q n pq
(2.20)
For 0 1,x x n we obtain
00
( )xCf x (by (2.9))
0
0 00 1 1
( ) ( ) ( )x x x
k k k
k k x k x
f x f x f x
0 1
0
( ) ( ) ( )x
k x x
k
f x f x f x
0 1( ) ( ) ( )xC x xf x f x f x
( )xf x (by (2.8) and (2.9))
K. Teerapabolarn & K. Sae-Jeng / Songklanakarin J. Sci. Technol. 41 (1), 96-108, 2019 105
1 11 1min ,
( 1)
n n np p q
xq n pq
(by (2.19))
1 1
0
1 1min ,
( 1)
n n np p q
x q n pq
,
which gives
0
1 1
0
1 1( ) min , .
( 1)x
n n n
C
p p qf x
x q n pq
(2.21)
Hence, by (2.20) and (2.21), the inequality (2.17) holds.
2.2 The characterization of beta binomial random variable
For the characterization associated with the beta binomial random variable ,Y by applying Lemma 3.1 in Cacoullos
and Papathanasiou (1989), the covariance of Y and 0
( )xCf Y can be expressed as
0 0
0 0
, ( ) ( ) ( ) ( ) ,x x
yn
C C
y k
Cov Y f Y f y k bb k
(2.22)
where .n
Lemma 2.4. Let 0
( ) ( )
( )( ) ,
y
k
k bb k
bb yy
0,..., ,y n be the characterization associated with the beta binomial random
variable ,Y then we have the following.
( )( )
( ) ,n y y
y
0,..., ,y n (2.23)
and
0 0
( )( ), ( ) ( ) .
x x
n Y YC CCov Y f Y E f Y
(2.24)
Proof. We shall show that ( )( )
( )n y y
y
by mathematical induction as follows. It can be seen that (0) n
and
1
0
( ) ( )( 1)( 1)
(1)(1) .k
k bb kn
bb
For 2 ,m n let
0
( ) ( )( )( )
( )( ) ,
m
k
k bb kn m m
bb mm
we have to show that ( 1)m ( ( 1))( ( 1))
.n m m
Since
( 1)m
1
0
( ) ( )
( 1)
m
k
k bb k
bb m
0
( ) ( )( )
( 1)( 1) ( )
m
k
k bb kbb m
mbb m bb m
106 K. Teerapabolarn & K. Sae-Jeng / Songklanakarin J. Sci. Technol. 41 (1), 96-108, 2019
( 1)( 1)( 1)
m n mm
(by 0
( ) ( )( )( )
( )
m
k
k bb kn m m
bb m
)
( ( 1))( ( 1)),
n m m
thus by mathematical induction, (2.23) is obtained.
Substituting (2.23) into (2.22), it becomes
0( , ( ))
xCCov Y f Y 0
( )( )
0
( ) ( )
x
nn y y
C
y
f y bb y
0
( )( )( ) ,
x
n Y YCE f Y
this yields (2.24).
3. Results
The main point of this study is to determine a non-uniform bound for the distance between the binomial and beta
binomial cumulative distribution functions, 0
,xCd Y X as
0 {0,..., }x n . The following theorem presents this desired result.
Theorem 3.1. Let
p
and 0 {0,..., }x n , then we have the following.
0
,xCd Y X
1 1
0
(1 )( 1)01
1 1 ( 1)0( 1) 1
0,
min , 1 .
n
n n n
q n q
p p q n np
x n p
if x
if x n
(2.25)
Proof. Using the same arguments detailed as in the proof of Theorem 2.1 in Teerapabolarn (2008), it follows that
0
,xCd Y X
0 0 0( ) ( ) , ( )
x x xC C CE f Y E pY f Y Cov Y f Y
0 0
( )( )( ) ( ) ( )
x x
n Y YC CE pY f Y E f Y
(by (2.24))
0
( )( )( )
x
n Y YCE pY f Y
0
( )( )( )
x
n Y YCE pY f Y
0
( )( )
x
Y n YCE f Y
0
( )( )
x
n Y YCE f Y
1 1
0
1 ( )0
1 1 ( )0( 1)
if 0,
min , if 1 .
n
n n n
q n Y Y
np
p p q n Y Y
x q n pq
E x
E x n
(by Lemma 2.3)
K. Teerapabolarn & K. Sae-Jeng / Songklanakarin J. Sci. Technol. 41 (1), 96-108, 2019 107
Because 2 2( ) ( 1)
( )( 1),
n Y Y n nnE
we have
0
,xCd Y X
1 1
0
1 ( 1)0( )( 1)
1 1 ( 1)0( 1) ( )( 1)
if 0,
min , if 1 .
n
n n n
q n n
np
p p q n n
x q n pq
x
x n
Hence, the inequality (2.25) is obtained.
Remark. Since 1 11 1
( 1)
n n nq p q
np n pq
and
1 1 1 1
0
1 1 1
( 1) ( 1)min ,
n n n n np p q p q
x q n pq n pq
when
0 {1,..., },x n the bound in Theorem 3.1 is
better than that mentioned in (1.4).
4. Numerical Examples
Teerapabolarn (2008) suggested the result in (1.4) to give a good approximation when and
n are small. So, we
provide two examples to illustrate the result in Theorem 3.1 by setting parameters , n and to satisfy this suggestion.
Example 4.1. Let 30, 10n and 500, then the numerical result in Theorem 3.1 is of the form
0
,xCd Y X
0
0
0
0.0333830
0.024922 if 0,
0.025195 if 1,
if 2,...,30. x
x
x
x
It is better than the numerical result in (1.4),
0
,xCd Y X 00.025195, 0,1,...,30.x
Example 4.2. Let 50, 100n and 1000, then the numerical result in Theorem 3.1 is of the form
0
,xCd Y X
0
0
0
0.2022950
0.040114 if 0,
0.043294 if 1,2,3,4,
if 5,...,50. x
x
x
x
It is better than the numerical result in (1.4),
0
,xCd Y X 00.043294, 0,1,...,50.x
The two examples are indicated that the result in Theorem 3.1 gives a good approximation when n
and
are small,
especially, when
is small. Furthermore, these examples point out that the bound in Theorem 3.1 is better than that shown in
(1.4).
108 K. Teerapabolarn & K. Sae-Jeng / Songklanakarin J. Sci. Technol. 41 (1), 96-108, 2019
5. Conclusions
The bound in this study, non-uniform bound, was determined by using Stein’s method and the characterization of beta
binomial random variable. It is appropriate to approximate the distance between the beta binomial cumulative distribution
function with parameters , 0n and 0 and the binomial cumulative distribution function with parameters n and
,
because it changes along 0 {0,..., }x n . In addition, by theoretical and numerical comparison, the result in this study is better
than that mentioned in (1.4), and it gives a good binomial approximation when n
and
are small.
Acknowledgements
The authors would like to thank the anonymous
referees for their useful comments and suggestions.
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