Non-uniform Bound on the Point Metric of the Beta Binomial

Applied Mathematical Sciences, Vol. 8, 2014, no. 173, 8647 - 8650
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ams.2014.410818
Non-uniform Bound on the Point Metric of
the Beta Binomial and Binomial Distributions
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 specific w-function to determine
a non-uniform bound on the point metric of the beta binomial distribution with parameters n, α and β and a binomial distribution with
α
. Numerical examples are provided to illustrate
parameters n and α+β
the result obtained.
Mathematics Subject Classification: 62E17, 60F05
Keywords: Beta binomial distribution, Binomial approximation, Stein’s
method, w-function
1
Introduction
The beta binomial random variable X with parameters n ∈ N, α > 0, and
β > 0 has probabilities
n B(x + α, n − x + β)
, x = 0, 1, ..., n,
(1.1)
bb(x) =
x
B(α, β)
nαβ(n+α+β)
nα
and variance σ 2 = (α+β)
and has mean µ = α+β
2 (1+α+β) , where B is the complete beta function. Because the beta binomial distribution obtained from the
binomial distribution with parameters n and p, where p is a random variable
that has a beta distribution with shape parameters α and β, it is natural to
8648
K. Teerapabolarn
speculate that the beta binomial distribution can be approximated by the binomial distribution. In this case, [2] gave a uniform bound on the total variation
distance between two such distributions
n(n − 1)
(1.2)
dT V (BB n,α,β , Bn,p ) ≤ 1 − pn+1 − q n+1
(n + 1)(1 + α + β)
for A ⊆ {0, ..., n}, where dT V (BB n,α,β , Bn,p ) = sup |BB n,α,β {A} − Bn,p {A}|,
A
BB n,α,β is the beta binomial distribution and Bn,p is the binomial distribution
α
with parameters n and p = α+β
. Correspondingly, when A = {x0 } for every
x0 ∈ {0, ..., n}, (1.2) becomes the point metric of two such distributions and
its uniform bound as follows:
n(n − 1)
,
(1.3)
|bb(x0 ) − b(x0 )| ≤ 1 − pn+1 − q n+1
(n + 1)(1 + α + β)
where b(x0 ) is the binomial probability function at x0 . Note that the uniform
bound in (1.3) does not depend on x0 , which may not be sufficiently good for
measuring the accuracy of this approximation. In this paper, we are interested
to determine a non-uniform bound on the point metric |bb(x0 ) − b(x0 )| at
x0 ∈ {0, ..., n}.
2
Method
The tools for determining the desired result are Stein’s method and the wfunction associated with the beta binomial random variable. The following
lemma gives that w-function, which is directly obtained from [2].
Lemma 2.1. We have
w(x) =
(n − x)(α + x)
, x = 0, 1, ..., n,
(α + β)σ 2
(2.1)
nαβ(n+α+β)
where σ 2 = (α+β)
2 (1+α+β) .
For Stein’s method in the binomial approximation, following [1], it can be
applied for n ∈ N and 0 < p = 1 − q < 1, for every x0 ∈ {0, ..., n} and
bounded real-valued function g = g{x0 } : N ∪ {0} → R, where g(0) = g(1) and
g(x) = g(n) for x0 ≥ n, So, Stein’s equation for these conditions is as follows:
bb(x0 ) − b(x0 ) = E[(n − X)pg(X + 1) − qXg(X)].
For x, x0 ∈ {0, ..., n}, let ∆g(x) = g(x + 1) − g(x), [3] showed that
( 1−qn
if x0 = 0,
np n
o
sup |∆g(x)| ≤
1−pn 1−pn+1 q n+1
min x0 q , (n+1)pq
if x0 > 0.
x≥0
(2.2)
(2.3)
Non-uniform bound on the point metric
3
8649
Result
The following theorem presents a non-uniform bound for the point metric of
the the beta binomial and the binomial distributions.
α
, then we have the following:
Theorem 3.1. For x0 ∈ {0, ..., n}, if p = α+β
( (1−qn )(n−1)q
if x0 = 0,
n(1+α+β)
o
n
|bb(x0 ) − b(x0 )| ≤
(3.1)
(n−1)n
(1−pn )p 1−pn+1 −q n+1
,
if
x
>
0.
min
0
x0
n+1
1+α+β
Proof. By (2.2) and using the proof in Theorem 3.1 of [3], we have
|bb(x0 ) − b(x0 )| = |E[(n − X)pg(X + 1) − qXg(X)]|
≤ E{|(n − X)p − σ 2 w(X)||∆g(X)|} + |np − µ|E|g(X)|
α(n − X)
2
=E − σ w(X) |∆g(X)|
α+β
α(n − X) (n − X)(α + X) −
=E |∆g(X)|
α+β
α+β
(n − X)X =E |∆g(X)|
α+β nµ − σ 2 − µ2
≤ sup |∆g(x)|
α+β
x≥0
(n − 1)nαβ
≤ sup |∆g(x)|
.
(α + β)2 (1 + α + β)
x≥0
Hence, by (2.3), (4.2) is easily obtained.
4
Numerical example
The following examples are given to illustrate how well an improved binomial
distribution approximates a beta binomial distribution.
10
Example 4.1 Let n = 10, α = 10 and β = 90, then p = 100
and the numerical
result is as follows:
(
0.00522347
o if x0 = 0,
n
|bb(x0 ) − b(x0 )| ≤
(4.1)
0.08910891
min
,
0.05558690
if
x
>
0.
0
x0
30
Example 4.2 Let n = 30, α = 30 and β = 470, then p = 500
and the numerical
result is as follows:
(
0.00153030
n
o if x0 = 0,
|bb(x0 ) − b(x0 )| ≤
(4.2)
0.10419162
min
, 0.04778921
if x0 > 0.
x0
8650
K. Teerapabolarn
From the Examples 4.1 and 4.2, it is seen that the result in Theorem 3.1
gives a good binomial approximation when αβ or βn is small.
5
Conclusion
A non-uniform bound for the point metric of the beta binomial distribution
with parameters n, α and β and a binomial distribution with parameters n and
α
p = α+β
was derived by using Stein’s method and the w-function associated
with the beta binomial random variable. With this bound, it is pointed out
that the result obtained in the present study gives a good binomial approximation when αβ or βn is small, that is, β α or β n.
References
[1] A.D. Barbour, L. Holst, S. Janson, Poisson approximation, Oxford Studies
in Probability 2, Clarendon Press, Oxford, 1992.
[2] K. Teerapabolarn, A bound on the binomial approximation to the beta
binomial distribution, Int. Math. Forum, 3(2008), 1355–1358.
[3] K. Teerapabolarn, P. Wongkasem, On pointwise binomial approximaion by
w- functions, Int. J. Pure Appl. Math., 71(2011), 57–66.
Received: October 15, 2014, Published: December 3, 2014

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