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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}, 8934 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. 8936 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