AN IMPROVEMENT OF POINTWISE NEGATIVE BINOMIAL APPROXIMATION BY w-FUNCTIONS

and has mean E(Y ) = rq p and variance Var(Y ) = rq p . Let X be a non-negative integer-valued random variable with probability function pX(x) > 0 for every x in the support of X, S(x). Observing that if parameters of the distribution of X and the negative binomial distribution are given under appropriate conditions, then the negative binomial distribution can be used as an approximation of the distribution of X. Furthermore, if we expect the distribution of X to be


Introduction
The negative binomial random variable Y with parameters r > 0 and p ∈ (0, 1) has the probability function as p Y (y) = Γ(r + y) Γ(r)y! q y p r , y = 0, 1, ..., and has mean E(Y ) = rq p and variance Var(Y ) = rq p 2 .Let X be a non-negative integer-valued random variable with probability function p X (x) > 0 for every x in the support of X, S(x).Observing that if parameters of the distribution of X and the negative binomial distribution are given under appropriate conditions, then the negative binomial distribution can be used as an approximation of the distribution of X.Furthermore, if we expect the distribution of X to be closer to the negative binomial distribution than other distributions, then it is reasonable to approximate the distribution of X by the negative binomial distribution.In the past few years, there has been some research on topics related to the negative binomial approximation for non-negative integer-valued random variable, which can be found in [1], [6] and [7].Recently, [5] gave a result in pointwise negative binomial approximation to the distribution of X, when r ∈ [1, ∞), as follows: where µ and σ 2 ∈ (0, ∞) are mean and variance of X and w(X) is the wfunction associated with X.However, this result could not be applied to the case of r ∈ (0, 1).In this study, we focus on determining a non-uniform bound on this approximation for r ∈ (0, 1).

Method
We use the same methodology as in [5], which consists of Stein's method and w-functions.For w-functions, [2] first defined a function w associated with nonnegative integer-valued random variable X and [3] expressed this function in the simple relation as follows: where w(0) = µ σ 2 and p X (x) > 0 for every x ∈ S(x).For Stein's method, [4] introduced a method for bounding the error in the normal approximation.The method was applied to the negative binomial approximation by [1].Stein's equation for negative binomial distribution with parameters r > 0 and p = (1 − q) ∈ (0, 1), 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}.

Result
The following theorem presents the desired result for r ∈ (0, 1).

Conclusion
In this study, a result in pointwise negative binomial approximation to the distribution of a non-negative integer valued random variable for r ∈ (0, 1) was obtained.It could be applied to approximate the Pólya and beta-negative binomial distributions when their parameters satisfy the parameter r ∈ (0, 1), which could not be applied for r ∈ [1, ∞).Therefore, the result makes this approximation to be more complete.

1 .
Approximation of the Pólya distribution The Pólya random variable X with positive integer parameters N, m,c and s has the probability function as p X (x) = 0, 1, ..., m, and has mean µ = sm N and variance σ 2 = sm(N +cm)(N −s) N 2 (N +c)

Corollary 4 . 1 .
If r = s c < 1 and p = N N +cm , then we have the following: