A Pointwise Approximation of Generalized Binomial by Poisson Distribution

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1 Applied Mathematical Sciences, Vol. 6, 2012, no. 22, A Pointwise Approximation of Generalized Binomial by Poisson Distribution K. Teerapabolarn Department of Mathematics, Faculty of Science, Burapha University, Chonburi 20131, Thailand Centre of Excellence in Mathematics, CHE Sri Ayutthaya Road, Bangkok 10400, Thailand kanint@buu.ac.th Abstract The Stein-Chen method and the w-function associated with generalized binomial random variable to determine a non-uniform bound on pointwise approximation of generalized binomial distribution by Poisson distribution. Some numerical examples have been given to illustrate applications of the obtainable result of this approximation. Mathematics Subject Classification: 62E17, 60F05 Keywords: A generalized binomial distribution; non-uniform bound; Poisson approximation; Stein-Chen method; w-function 1 Introduction A generalized binomial distribution in this study was presented by Dwass [4] in It is a generalized discrete distribution that covers binomial, hypergeometric and Pólya distributions. This distribution depends on four parameters, A, B, n and α, where A and B are positive, n is a positive integer, α is an arbitrary real number satisfying (n 1)α A + B, and A (i) and B (i) are not negative for i =1,..., n, where x (i) = x(x α) (x (i 1)α). The details of background of this distribution can be seen in [4]. Let X be the generalized binomial random variable. Then, following Dwass [4], its probability function is of the form ( ) n A (x) B (n x) p X (x) =,x=0,..., n. (1.1) x (A + B) (n) ( ) n [A(A α) (A (x 1)α)][B(B α) (B (n x 1)α)] =, x (A + B (n 1)α) x =0,..., n. (1.2)

2 1096 K. Teerapabolarn The mean and variance of X are μ = na and A+B σ2 = nab(a+b nα), respectively. (A+B) 2 (A+B α) Dwass [4] pointed out the three special cases of the distribution of X in (1.1) by α as follows: (i). For α = 0, it is a binomial distribution with parameters n, A and B, and its probability function can be expressed as p X (x) = ( )( ) x ( ) n x n A B, x =0,..., n. (1.3) x A + B A + B (ii). For α>0, it is a generalized hypergeometric distribution with parameters A, B, n and α and, when A and B are integers, its probability function α α can be expressed as p X (x) = ( A/α )( B/α x n x ( A/α+B/α n ) ),x=0,..., min{n, A/α}, (1.4) which is the well-known classical hypergeometric distribution. (iii). For α<0, it is a Pólya distribution with parameters A, B, n and α, when A and B are integers, its probability function can be expressed as α α ) p X (x) = ), x =0,..., n. (1.5) ( A/α+x 1 )( B/α+n x 1 x n x ( A/α B/α+n 1 n It is known that both binomial and hypergeometric distributions can be approximated by Poisson distribution under certain conditions on their parameters. Thus, if the conditions on the parameters of generalized binomial and Poisson distributions are satisfied, then the generalized binomial distribution can also be approximated by Poisson distribution. For example, Wongkasem et al. [8] used the Stein-Chen method and w function associated with the generalized binomial random variable X to give a uniform bound on Poisson approximation to generalized binomial distribution as follows: p X (x) λ (x) ( 1 e λ) B(n 1)α + A(A + B α), (1.6) x Ω x Ω where Ω {0,..., n}, λ (x) = e λ λ x, λ = na and A (n 1)( α) when x! A+B α<0. From (1.6), if Ω = { }, where {0,..., n}, then we have p X ( ) λ ( ) ( 1 e λ) B(n 1)α + A(A + B α). (1.7)

3 A pointwise approximation 1097 It is observed that the bound in (1.7) does not depend on, or it is a uniform bound with respect to. So, the uniform bound in (1.7) may not be sufficiently good for measuring the accuracy of the approximation. In this case, a non-uniform bound with respect to is more appropriate. In this paper, we use the Stein-Chen method and w-function associated with the generalized binomial random variable to give a result in the Poisson approximation to generalized binomial distribution, in terms of the point metric p X ( ) λ ( ) together with its non-uniform bound with respect to {0,..., n}. The Stein-Chen method and w-function associated with the generalized binomial random variable are mentioned in Section 2. In Section 3, the main result of this study is derived by the tools in Section 2. In section 4, some numerical examples have been given to show applications of the result, and the conclusion of this work has been presented the last section. 2 Method The tools for giving the result are the Stein-Chen method and w-function associated with the generalized binomial random variable. For Stein-Chen method, Stein [6] introduced a powerful and general method for bounding the error in the normal approximation. This method was first developed and applied in the setting of the Poisson approximation by Chen [3], which is refer to as the Stein-Chen method. Stein s equation for Poisson distribution with mean λ>0 is, for given h, of the form h(x) P λ (h) =λf(x +1) xf(x), (2.1) where P λ (h) =e λ λl l=0 h(l) and f and h are bounded real valued functions l! defined on N {0}. For Ω N {0}, let function h Ω : N {0} R be defined by h Ω (x) = { 1 if x Ω, 0 if x/ Ω. For Ω = { } where N {0}, following Barbour et al. [1] and writing h x0 = h {x0 }, the solution f x0 = f {x0 } of (2.1) is of the form (x 1)!λ x e λ [P λ (h x0 )P λ (h Cx 1 )] if x, f x0 (x) = (x 1)!λ x e λ [P λ (h x0 )P λ (1 h Cx 1 )] if x>. (2.2) 0 if x =0, where C x = {0,..., x}. Let Δf x0 (x) =f x0 (x +1) f x0 (x), then the following lemma gives a nonuniform bound of Δf x0.

4 1098 K. Teerapabolarn Lemma 2.1. Let N {0} and x N, then we have the following. 1. For = 0 (Teerapabolarn and Neammanee [7]), Δf x0 (x) λ 2 (λ + e λ 1). (2.3) 2. For > 0 (Barbour et al. [1]), } Δf x0 (x) min {λ 1 (1 e λ ), 1x0. (2.4) For w-function associated with the generalized binomial random variable, Wongkasem et al. [8] used the recurrence relation in Majsnerowska [5] to give the w-function as the following. Lemma 2.2. Let w(x) be the w-function associated with the generalized binomial random variable X and p X (x) > 0 for every 0 x n. Then w(x) = (n x)(a xα) (A + B)σ 2 0, x=0,..., n, (2.5) where σ 2 = nab(a+b nα) (A+B) 2 (A+B α). The following relation is an important property for proving the result, which was stated by Cacoullos and Papathanasiou [2]. If a non-negative integervalued random variable X have p X (x) > 0 for every x in the support of X and have finite variance 0 <σ 2 <, then E[(X μ)f(x)] = σ 2 E[w(X)Δf(X)], (2.6) for any function f : N {0} R for which E w(x)δf(x) <, where Δf(x) =f(x +1) f(x). For f(x) =x, we have that E[w(X)] = 1. 3 Result The following theorem presents a pointwise approximation of generalized binomial distribution by Poisson distribution, in terms of the point metric between generalized binomial and Poisson distributions together with its non-uniform bound. Theorem 3.1. For {0,..., n}, let λ = na α<0. Then we have the following. A+B > 0 and A (n 1)( α) for p X (0) e λ λ 1 (λ + e λ B(n 1)α + A(A + B α) 1) (3.1)

5 A pointwise approximation 1099 and, for {1,..., n}, { p X ( ) λ ( ) min 1 e λ, λ } B(n 1)α + A(A + B α). (3.2) Proof. Substituting h by h x0, x by X and taking expectation in (2.1), it gives p X ( ) λ ( )=E[λf(X +1) Xf(X)] = λe[f(x + 1)] E[(X μ)f(x)] μe[f(x)] = λe[δf(x)] E[(X μ)f(x)], where f = f x0 is defined as in (2.2). Since E[w(X)] = 1 and E w(x)δf(x) = E[w(X) Δf(X) ] <. Thus, by (2.6), we have For x N, we have and Therefore p X ( ) λ ( ) λe[δf(x)] σ 2 E[w(X)Δf(X)] E{ λ σ 2 w(x) Δf(X) } sup Δf x0 (x) E λ σ 2 w(x). x 1 λ σ 2 w(x) = na (n x)(a xα) A + B A + B [(n x)α + A]x = A + B 0. E λ σ 2 w(x) = λ σ 2 E[w(X)] = λ σ 2 = λ B(n 1)α + A(A + B α). B(n 1)α + A(A + B α) p X ( ) λ ( ) sup Δf x0 (x) λ, x 1 hence, by Lemma 2.1 (1 and 2), the theorem is proved. Corollary 3.1. We have the following.

6 1100 K. Teerapabolarn 1. λ 1 (λ + e λ 1) < 1 e λ. } 2. min {1 e λ, λx0 1 e λ. Remark 3.1. Following Corollary 3.1 (1 and 2), it can be seen that the nonuniform bound in Theorem 3.1 is sharper than the uniform bound in (1.7), that is, the result in Theorem 3.1 is better than the result in (1.7). Immediately from (3.1) and (3.2), additional results of the Poisson approximation are obtained for the three distributions as the following. Corollary 3.2. If α =0, then px (0) e λ λ 1 (λ + e λ A 1) A + B and, for {1,..., n}, { p X ( ) λ ( ) min 1 e λ, λ } A A + B. Corollary 3.3. If α>0, then px (0) e λ λ 1 (λ + e λ B(n 1)α + A(A + B α) 1) and, for {1,..., n}, { p X ( ) λ ( ) min 1 e λ, λ } B(n 1)α + A(A + B α). Corollary 3.4. If α<0 and A (n 1)( α), then px (0) e λ λ 1 (λ + e λ B(n 1)α + A(A + B α) 1) and, for {1,..., n}, { p X ( ) λ ( ) min 1 e λ, λ } B(n 1)α + A(A + B α). A Remark If and/or α are small, then the result of Theorem 3.1 A+B gives a good Poisson approximation. 2. It is noted that, for α>0, B(n 1)( α)+a(a + B + α) (A + B)(A + B + α) < A A + B B(n 1)α + A(A + B α) <, thus, the bound in Corollary 3.4 is always less than the bounds in Corollaries 3.2 and 3.3.

7 A pointwise approximation Numerical examples The following numerical examples are given to illustrate how well Poisson distribution approximates generalized binomial distribution, in the point metric form. Example 4.1. Suppose that n = 10, A = 30, A + B = 1000 and α = 0, then we have λ =0.30, and the numerical results in (1.7) and Theorem 3.1 can be expressed as follows: p X ( ) λ ( ) , =0,..., if =0, p X ( ) λ ( ) if =1, 2, 3, 0.03 if =4,..., 10. Example 4.2. Suppose that n = 30, A = 50, A + B = 1000 and α = 0, then we have λ = , and the numerical results in (1.7) and Theorem 3.1 can be expressed as follows: p X ( ) λ ( ) , =0,..., if =0, p X ( ) λ ( ) if =1, if =2,..., 30. Example 4.3. Suppose that n = 10, A = 30, A + B = 1000 and α = 1, then we have λ =0.30, and the numerical results in (1.7) and Theorem 3.1 can be expressed as follows: p X ( ) λ ( ) , =0,..., 10.

8 1102 K. Teerapabolarn if =0, p X ( ) λ ( ) if =1, 2, 3, if =4,..., 10. Example 4.4. Suppose that n = 30, A = 50, A + B = 1000 and α = 1, then we have λ = , and the numerical results in (1.7) and Theorem 3.1 can be expressed as follows: p X ( ) λ ( ) , =0,..., if =0, p X ( ) λ ( ) if =1, if =2,..., 30. Example 4.5. Suppose that n = 10, A = 30, A + B = 1000 and α = 1, then we have λ =0.30, and the numerical results in (1.7) and Theorem 3.1 can be expressed as follows: p X ( ) λ ( ) , =0,..., if =0, p X ( ) λ ( ) if =1, 2, 3, if =4,..., 10. Example 4.6. Suppose that n = 30, A = 50, A+B = 1000 and α = 1, then we have λ = , and the numerical results in (1.7) and Theorem 3.1 can be expressed as follows: p X ( ) λ ( ) , =0,..., 30.

9 A pointwise approximation if =0, p X ( ) λ ( ) if =1, if =2,..., 30. In view of Examples , it is found that the results (in case of nonuniform bound) are better than the results (in case of uniform bound). 5 Conclusion In this study, the non-uniform bound in Theorem 3.1 is an estimate of the point metric between the generalized binomial and Poisson distributions. This bound is also a criterion for measuring the accuracy of the approximation, that is, if it is small, then a good Poisson approximation to the generalized binomial distribution is obtained. Conversely, if it is large, then the Poisson distribution is not appropriate to approximate the generalized binomial distribution. In view of the result in Theorem 3.1, it is found that the bound on this approximation is small when A A+B and /or α and are small. Acknowledgement. This research is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand. References [1] A. D. Barbour, L. Holst, S. Janson, Poisson approximation, Oxford Studies in Probability 2, Clarendon Press, Oxford, [2] T. Cacoullos, V. Papathanasiou, Characterization of distributions by variance bounds, Statist. Probab. Lett., 7 (1989), [3] L. H. Y. Chen, Poisson approximation for dependent trials, Ann. Probab., 3 (1975), [4] M. Dwass, A generalized binomial distribution, Amer. Statistician, 33 (1979), [5] M. Majsnerowska, A note on Poisson approximation by w-functions, Appl. Math., 25 (1998), [6] C. M. Stein, A bound for the error in normal approximation to the distribution of a sum of dependent random variables, Proc. Sixth Berkeley Sympos. Math. Statist. Probab., 3 (1972),

10 1104 K. Teerapabolarn [7] K. Teerapabolarn, K. Neammanee, A non-uniform bound in somatic cell hybrid model, Math. BioSci., 195 (2005), [8] P. Wongkasem, K. Teerapabolarn, R. Gulasirima, On Approximating a generalized binomial by binomial and Poisson distributions, Internat. J. Statist. Systems, 3 (2008), Received: August, 2011