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Applied Mathematical Sciences, Vol. 8, 2014, no. 179, 8933 - 8936
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ams.2014.410838
Binomial Approximation for a Sum of
Independent Binomial Random Variables
K. Teerapabolarn
Department of Mathematics, Faculty of Science
Burapha University, Chonburi 20131, Thailand
c 2014 K. Teerapabolarn. This is an open access article distributed under
Copyright the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
Abstract
This paper uses Stein’s method and the binomial w-functions to determine a bound for approximating the distribution of a sum of n independent binomial random variables, each withP
parameters ni and pP
i , by a bin
1
nomial distribution with parameters m = ni=1 ni and p = m
i=1 ni pi .
When all pi are small or all pi are close to p, the result of the study
gives an accurate approximation.
Mathematics Subject Classification: 60F05, 60G50
Keywords: Binomial approximation, Binomial w-function, Stein’s method
1
Introduction
There has been much methodological research on topics related to the binomial
approximation, which have yielded useful results in applications of probability
and statistics, and the most valuable findings have concerned the binomial
approximation for sums of independent and dependent Bernoulli random variables. In the past few years, some authors have sought to propose a good error
bound for measuring the accuracy of this approximation. Many accurate results are derived from the well-known Stein’s method, can be found in [1] and
[3]−[5]. In this context, we extend the approximation to a sum of independent
binomial random variables.
Let X1 , ..., Xn be independently distributed
binomial random variables,
each with probability function pXi (k) = nki pki qini −k for k ∈ {0, ..., ni ; ni ∈ N},
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K. Teerapabolarn
2
and with
Pn mean µi = ni pi and variance σi = ni pi qi , where qi = 1 − pi . Let
Sn = Pi=1 Xi and Bm,p denote
P the binomial
P random variable with parameters
m = ni=1 ni and p = m1 ni=1 µi = m1 ni=1 ni pi . In this paper, we focus on
determining a bound for the total variation distance between the distribution
of Sn and the distribution of Bm,p is denoted and defined as follows:
dT V (Sn , Bm,p ) = sup |P (Sn ∈ A) − P (Bm,p ∈ A)|,
(1.1)
A
where A is a subset of {0, ..., m}.
The tools for giving the desired result consist of Stein’s method and the
binomial w-functions, which are in Section 2. In Section 3, we determine the
desired result, a bound for dT V (Sn , Bm,p ), and the conclusion of this study is
presented in the last section.
2
Method
The following lemma gives the binomial w-functions, which are directly obtained from [6].
Lemma 2.1. For 1 ≤ i ≤ n, let wi be the w-function associated with the
binomial random variable Xi , then we have the following:
wi (k) =
(ni − k)pi
, k ∈ {0, ..., ni }.
σi2
(2.1)
The following relation is an important property for proving the result, which
was stated by [2].
!!
n
X
X
Cov(Sn , f (Sn )) =
Cov Xi , f Xi +
Xj
=
i=1
n
X
j6=i
σi2 E[wi (Xi )∆f (Sn )],
(2.2)
i=1
for any function f : N ∪ {0} → R for which E|wi (Xi )∆f (Sn )| < ∞, where
∆f (x) = f (x + 1) − f (x).
For Stein’s method in the binomial approximation, it can be applied for
every m ∈ N and 0 < p = 1 − q < 1, for every A ⊆ {0, ..., m} and bounded
real-valued function f = fA : N ∪ {0} → R defined as in [1], where f (0) = f (1)
and f (x) = f (m) for x ≥ m. So, Stein’s equation for these conditions is as
follows:
P (Sn ∈ A) − P (Bm,p ∈ A) = E[(m − Sn )pf (Sn + 1) − qSn f (Sn )].
(2.3)
Binomial approximation for a sum of binomial ...
8935
For A ⊆ {0, ..., m} and x ∈ N ∪ {0}, Ehm [3] showed that
sup |∆f (x)| ≤
A,x
3
1 − pm+1 + q m+1
.
(m + 1)pq
(2.4)
Result
The following theorem presents a bound on the error of binomial approximation to the distribution of Sn .
Theorem 3.1. With the above definition, we have the following:
n
1 − pm+1 + q m+1 X
dT V (Sn , Bm,p ) ≤
|pi − p| ni pi .
(m + 1)pq
i=1
(3.1)
Proof. From (2.3), it follows that
dT V (Sn , Bm,p ) = |E[(m − Sn )pf (Sn + 1) − qSn f (Sn )]|
= |E[mpf (Sn + 1) − pSn ∆f (Sn ) − Sn f (Sn )]|
= |E[mp∆f (Sn )] − pE[Sn ∆f (Sn )] − Cov(Sn , f (Sn ))|
n
X
=
{E[µi ∆f (Sn )] − pE[Xi ∆f (Sn )] − Cov(Xi , f (Sn ))} .
i=1
Using (2.2) and Lemma 2.1, we have
n
X
2
dT V (Sn , Bm,p ) = E [(µi − pXi )∆f (Sn )] − σi E[wi (Xi )∆f (Sn )] i=1
≤
n
X
E{|ni pi − pXi − σi2 wi (Xi )||∆f (Sn )|}
i=1
≤ sup |∆f (x)|
A,x
≤ sup |∆f (x)|
A,x
n
X
i=1
n
X
E |ni pi − pXi − (ni − Xi )pi |
|pi − p|ni pi .
i=1
Hence, by (2.4), (3.2) is obtained.
It is noted that if ni = 1 for every i ∈ {1, ..., n}, then Sn has the Poisson
binomial distribution with parameter p = (p1 , ..., pn ). Thus, an immediately
consequence of Theorem 3.1, a binomial approximation to the Poisson binomial distribution is also obtained.
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K. Teerapabolarn
Corollary 3.1. If n1 = · · · = nn = 1, then the following inequality holds:
n
1 − pm+1 + q m+1 X
dT V (Sn , Bm,p ) ≤
|pi − p| pi .
(m + 1)pq
i=1
4
(3.2)
Conclusion
In this study, a bound on the error of binomial approximation to the distribution of a sum of n independent binomial random variables was derived by
Stein’s method and the binomial w-functions. It is a good measurement of the
approximation when all pi are small or all pi are close to p.
References
[1] A.D. Barbour, L. Holst, S. Janson, Poisson approximation, Oxford Studies
in Probability 2, Clarendon Press, Oxford, 1992.
[2] T. Cacoullos, V. Papathanasiou, Characterization of distributions
by variance bounds, Statist. Probab. Lett., 7 (1989), 351–356.
http://dx.doi.org/10.1016/0167-7152(89)90050-3
[3] W. Ehm, Binomial approximation to the Poisson binomial distribution,
Statist. Probab. Lett., 11 (1991), 7–16. http://dx.doi.org/10.1016/01677152(91)90170-v
[4] Y.T. Soon Spario, Binomial approximation for dependent indicators,
Statist. Sinica, 6 (1996), 703–714.
[5] C.M. Stein, Approximate Computation of Expectations, IMS, Hayward
California, 1986.
[6] R. Kun, K. Teerapabolarn, A piontwise Poisson approximation by wfunctions, Appl. Math. Sci., 6(2012), 5029-5037.
Received: October 21, 2014; Published: December 12, 2014