eu NEGATIVE BINOMIAL APPROXIMATION TO THE BETA BINOMIAL DISTRIBUTION

This paper determines a bound on the approximation of the beta binomial distribution with parameters n, � andby a negative binomial distri- bution with parametersand �+� �+�+n . With this bound, it is indicated that the beta binomial distribution can be well approximated by the negative binomial distribution whenis large.


Introduction
The beta binomial random variable X with parameters n ∈ N, α > 0, and β > 0, with distribution written BB n,α,β , has probabilities and has mean µ = nα α+β and variance σ 2 = nαβ(n+α+β) (α+β) 2 (1+α+β) , where B is the complete beta function.It is well-known that the beta binomial can be approximated by some discrete distributions, such as Poisson and binomial distributions, provided that their parameters are given under appropriate conditions.
Received: August 18, 2014 c 2015 Academic Publications, Ltd. url: www.acadpubl.euFor example, Teerapabolarn [3] gave a bound for the total variation distance between the binomial and beta binomial distributions as follows: for A ⊆ {0, ..., n}, where B n,p is the binomial distribution with parameters n and p = 1−q = α α+β .Later, he also gave a bound for the total variation distance between the Poisson and beta binomial distributions [4] as follows: , where P λ is the Poisson distribution with mean λ = nα α+β and α ≥ (n − 1).Let us consider (1.1), we have It is observed that α + β → ∞ while n α+β remains constant, then the beta binomial distribution with parameters n, α and β converges to a negative binomial distribution with parameters α and α+β α+β+n .Therefore, it is reasonable to approximate the beta binomial distribution by the negative binomial with these parameters when α + β is sufficiently large.In this paper, we use Stein's method and the w-function associated with the random variable X to give a bound for the total variation distance between the beta binomial and negative binomial distributions.

Method
The following lemma is the w-function associated with the beta binomial random variable X, which is directly obtained from [3].
Lemma 2.1.We have where The next relation is an important property to obtain the desired result, which was stated by [2].If a non-negative integer-valued random variable X has p X (x) > 0 for every x in support of X and 0 for any function g : N ∪ {0} → R for which E|w(X)∆g(X)| < ∞, where ∆g(x) = g(x + 1) − g(x) and E[w(X)] = 1.For Stein's method in the negative binomial approximation, following [1], it is applied for real number r > 0 and 0 < p = 1 − q < 1, for every subset A of N ∪ {0} and the bounded real valued function g = g A : N ∪ {0} → R defined in [1], yields (2.3) For any subset A of N ∪ {0}, [1] showed that sup (2.4)

Concluding Remarks
In the present study, a bound for the total variation distance between the beta binomial and negative binomial distributions is derived by using Stein's method and the w-function associated with the beta binomial random variable.With this bound, it is observed that the result in Theorem 3.1 gives a good approximation when β is large, that is, the negative binomial distribution with parameters α and α+β α+β+n can be used as an approximation of the beta binomial distribution with parameters n, α and β when β is sufficiently large.Additionally, it has no any conditions on their parameters of the approximation, which is similar to that mentioned in (1.2) and is different from the Poisson approximation as mentioned in (3.1).