# ON POINTWISE BINOMIAL APPROXIMATION

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3 ON POINTWISE BINOMIAL APPROXIMATION with o-egative iteger-valued radom variable X i the relatio w(x)p(x) = 1 σ 2 x (µ i)p(i), x S(x) (2.1) i=0 ad, afterwards, Majserowska [4] expressed the relatio (2.1) as the form w(0) = µ σ 2, w(x) = 1 { } σ 2 µ + σ2 w(x 1)p(x 1) x, x S(x) \ {0} (2.2) p(x) ad w(x) 0, x S(x), (2.3) where p(x) > 0 for every x S(x). The followig relatio is a importat property for provig the result, which was stated by Cacoullos ad Papathaasiou [2]. If a o-egative iteger-valued radom variable X is defied as i Sectio 1, the E[(X µ)g(x)] = σ 2 E[w(X) g(x)], (2.4) for ay fuctio g : N {0} R for which E w(x) g(x) <, where g(x) = g(x + 1) g(x). For g(x) = x, we have that E[w(X)] = 1. For Stei s method, we start it by usig Stei s equatio i Barbour et al. [1]. Stei s equatio for the biomial distributio with parameters N ad p (0,1) is, for give h, of the form ( x)pg(x + 1) qxg(x) = h(x) B,p (h), (2.5) where B,p (h) = k=0 h(k)( k) p k q k ad g ad h are bouded real-valued fuctios defied o {0,1,...,}. For A {0,1,...,}, let h A : {0,1,...,} R be defied by h A (x) = { 1 if x A, 0 if x / A. (2.6) By followig Barbour et al. [1] o pp. 189, let g A : N {0} R satisfy (2.5), where g A (0) = g A (1) ad g A (x) = g A () for x.

5 ON POINTWISE BINOMIAL APPROXIMATION ad g x0 ( ) = gives ad = 1 p q = k= +1 k= +1 1! k!( k)!( ) pk q +1 k pq + (+1)!(+1 k) k!(+1 k)!( ) pk q +1 k ( + 1 k= +1 = 1 p+1 q +1, k x0 1 k=0 + x0 1 k=0 ) p k q +1 k + 1 q! k!( k)! pk+1 q k pq (+1)!(k+1) (k+1)!( k)! p k+1 q k k=1 g x0 (x) mi{ 1 p q, 1 p+1 q +1 } g x0 (x) mi{ 1 p q, 1 p+1 q +1 }. So, from both cases, (2.8) ad (2.9) are obtaied. ( ) + 1 p k q +1 k k 3. Result The followig theorem shows a result i the biomial approximatio to the distributio of a o-egative iteger-valued radom variable X, i terms of the poit metric ad its o-uiform boud, which is obtaied by Stei s method ad w-fuctios. Theorem 3.1. Let a o-egative iteger-valued radom variable X with p(x) > 0 for every x S(x) ad together with correspodig w-fuctio w(x) be defied as above. The the followig iequalities hold: 1. For = 0, p(0) b(0;,p) 1 q p { E ( X)p σ 2 w(x) + p µ } (3.1)

6 62 K. Teerapabolar, P. Wogkasem ad, if p = µ, the p(0) b(0;,p) 1 q p E ( X)p σ 2 w(x). (3.2) 2. For {1,...,}, { 1 p p( ) b( ;,p) mi q, 1 p+1 q +1 } { E ( X)p σ 2 w(x) } + p µ (3.3) ad, if p = µ, the { 1 p p( ) b( ;,p) mi q, 1 p+1 q +1 } E ( X)p σ 2 w(x). (3.4) Proof. Substitutig h by h {x0 }, x by X ad takig expectatio i (2.5), we obtai p( ) b( ;,p) = E[( X)pg(X + 1) qxg(x)], (3.5) where g = g x0 is defied i (2.7) ad E[( X)pg(X + 1) qxg(x)] = E[pg(X + 1) px g(x) Xg(X)] = E[pg(X + 1)] pe[x g(x)] E[Xg(X)] = pe[g(x + 1)] pe[x g(x)] E[(X µ)g(x)] µe[g(x)] = pe[ g(x)] pe[x g(x)] E[(X µ)g(x)] + (p µ)e[g(x)] Sice E[w(X)] = 1 ad g(x) is bouded, the E w(x) g(x) <. Thus, by (2.4), it follows that E[( X)pg(X + 1) qxg(x)] = pe[ g(x)] pe[x g(x)] which, by (3.5), yields σ 2 E[w(X) g(x)] + (p µ)e[g(x)] = E{[( X)p σ 2 w(x)] g(x)} + (p µ)e[g(x)], p( ) b( ;,p) = E { [( X)p σ 2 w(x)] g(x) } + (p µ)e[g(x)]

7 ON POINTWISE BINOMIAL APPROXIMATION E{ ( X)p σ 2 w(x) g(x) } + p µ E g(x). Hece, by usig Lemma 2.1, the theorem is proved. The followig corollary is a cosequece of Theorem 3.1. Corollary 3.1. If ( x)p σ 2 w(x) / < 0 for every x S(x), the 1. For = 0, ad, if p = µ, the p(0) b(0;,p) 1 q p p(0) b(0;,p) 1 q p { ( µ)p σ 2 + p µ } (3.6) µq σ 2. (3.7) 2. For {1,...,}, { 1 p p( ) b( ;,p) mi q, 1 p+1 q +1 } { ( µ)p σ 2 } + p µ (3.8) ad, if p = µ, the { 1 p p( ) b( ;,p) mi q, 1 p+1 q +1 } µq σ 2. (3.9) 4. Applicatios This sectio, we apply the result i Theorem 3.1 to approximate some distributios such as hypergeometric, egative hypergeometric ad Pólya distributios Applicatio to Hypergeometric Distributio Suppose a radom sample of size is draw without replacemet from a fiite populatio cotaiig N elemets of two types of which m are of type I ad N m are of type II. Let X be the umber of type I elemets i the sample.

10 66 K. Teerapabolar, P. Wogkasem for all 0 x m. The followig corollary is directly obtaied from Corollary 3.1. Corollary 4.4. If p = r N, the we have the followig. p( ) b( ;m,p) { (1 q m )q(m 1) { N+1 } mi 1 p m, 1 pm+1 q m+1 (m 1)mp (m+1)p N+1 if 1 m. Remark 4.2. Each result i Corollaries 4.3 ad 4.4 yields a good approximatio as N is large ad m is small, or m N is small. Ackowledgmets The authors would like to thak Faculty of Sciece, Burapha Uiversity, for fiacial support to do this research. Refereces [1] A.D. Barbour, L. Holst, S. Jaso, Poisso Approximatio, Oxford Studies i Probability 2, Claredo Press, Oxford (1992). [2] T. Cacoullos, V. Papathaasio, Characterizatio of distributios by variace bouds, Statist. Probab. Lett., 7 (1989), [3] W. Ehm, Biomial approximatio to the Poisso biomial distributio, Statist. Probab. Lett., 11 (1991), [4] M. Majserowska, A ote o Poisso approximatio by w-fuctios, Appl. Math., 25 (1998), [5] Y.T. Soo Spario, Biomial approximatio for depedet idicators, Statist. Siica, 6 (1996), [6] C.M. Stei, Approximate Computatio of Expectatios, IMS, Hayward Califoria (1986). [7] K. Teerapabolar, A boud o the biomial approximatio to the beta biomial distributio, It. Math. Forum, 3 (2008), [8] P. Wogkasem, K. Teerapabolar, R. Gulasirima, O approximatig a geeralized biomial by biomial ad Poisso distributios, Iterat. J. Statist. Systems, 3 (2008),

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