NEGATIVE BINOMIAL APPROXIMATION TO THE GENERALIZED HYPERGEOMETRIC DISTRIBUTION

This paper uses Stein's method and the w-function associated with the generalized hypergeometric random variable to determine a bound for the total variation distance between the generalized hypergeometric distribu- tion with parameters �, � and N and the negative binomial distribution with parameters r = � + 1 and p = 1 − q = �+�+2 �+�+N+1 . In view of this bound, it is observed that the desired result gives a good negative binomial approximation whenis large.


Introduction
A non-negative integer-valued random variable X is said to have the generalized hypergeometric distribution with parameters α, β and N , GH α,β,N , if its probability mass function is as follows [3]: , respectively.We know that this distribution can be approximated by some appropriate discrete distributions if some conditions of their parameters are satisfied.In this case, Crosu [3] used Stein's method and the w-function associated with the generalized hypergeometric random variable to obtain a bound for the total variation distance between GH α,β,N and a Poisson distribution, P µ , with mean µ = (N −1)(β+1) α+β+2 when β + 2 ≥ N as follows: Later, Teerapabolarn [4] used the same tools to obtain a bound for the total variation distance between GH α,β,N and a binomial distribution, B n,p , with parameters n = N − 1 and p = β+1 α+β+2 as follows: In this paper, we are interested to determine a bound for d T V (GH α,β,N , NB r,p ), where NB r,p is a negative binomial distribution with parameters r and p.

Method
The tools for giving the desired result consist of Stein's method for the negative binomial distribution and the w-function associated with the generalized hypergeometric random variable.Following [1], Stein's equation for negative binomial distribution with parameters r > 0 and p = 1 − q ∈ (0, 1) is, for given h, of the form where N B r,p (h) = ∞ k=0 h(k) Γ(r+k) Γ(r)k! p r q k and f and h are bounded real-valued functions defined on N ∪ {0}.
For A ⊆ N ∪ {0}, let h A : N ∪ {0} → R be defined by Let f A : N ∪ {0} → R satisfy (3), and let x ∈ N and ∆f A (x)=f A (x + 1) − f A (x). Brown and Phillips [1] showed that sup For the w-function associated with the generalized hypergeometric random variable, following [2] and we have E[w(X)] = 1.The following lemma presents this w-function, which obtained from [3].
Lemma 2.1.Let w(X) be the w-function associated with the generalized hypergeometric random variable X.Then, we have the following:

Result
The following theorem presents a bound for the total variation distance between GH α,β,N and NB r,p .
Let δ(G, N) = GH α,β,N {A} − NB r,p {A}, then we obtain Using Lemma 2.1 and (6), we have From which it follows that Therefore, it follows from ( 9) and ( 2), we obtain which completes the proof.If β = 0, then r = 1 and p = α+2 α+N +1 .Thus the approximation in Theorem 3.1 is an approximation of the generalized hypergeometric distribution with parameters α and N by a geometric distribution, G p , with parameter p = α+2 α+N +1 .Corollary 3.1.If β = 0, then we have the following geometric approximation:

Concluding Remarks
In the present study, a bound for the total variation distance between the generalized hypergeometric and negative binomial distributions is obtained by using Stein's method and the w-function associated with the generalized hypergeometric random variable.In view of this bound, it is observed that if β α and N α are small, or α is large, then the result in Theorem 3.1 gives a good negative binomial approximation, that is, the binomial distribution with parameters r = β + 1 and p = α+β+2 α+β+N +1 can be used as an approximation of the generalized hypergeometric distribution with parameters α, β and N when α is sufficiently large.Moreover, it has no any conditions on their parameters in this approximation, which is similar to the binomial approximation in (2) and is different from the Poisson approximation as mentioned in (1).