ON POINTWISE BINOMIAL APPROXIMATION


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1 Iteratioal Joural of Pure ad Applied Mathematics Volume 71 No , ON POINTWISE BINOMIAL APPROXIMATION BY wfunctions K. Teerapabolar 1, P. Wogkasem 2 Departmet of Mathematics Faculty of Sciece Burapha Uiversity Choburi, 20131, THAILAND Abstract: This paper, we use Stei s method ad wfuctios to give a result i the biomial approximatio to the distributio of a oegative itegervalued radom variable, i terms of the poit metric betwee two such distributios together with its o uiform boud. Furthermore, for applicatios, we use the obtaied result to approximate some distributios such as hypergeometric, egative hypergeometric ad Pólya distributios. AMS Subject Classificatio: 62E17, 60F05 Key Words: biomial approximatio, ouiform boud, poit metric, Stei s method, wfuctios 1. Itroductio May studies of biomial approximatio via Stei s method have yielded useful results i applicatios of probability ad statistics. The first study of biomial approximatio by Stei s method, for approximatig the umber of oes i the biary expasio of a radom iteger ad for problem of coutig Lati rectagles, was preseted by Stei [6]. Ehm [3] gave lower ad upper bouds of the error i the biomial approximatio of a sum of idepedet idicator radom variables, ad he applied the result to samplig with ad without replacemet. Barbour et al. [1] showed that Stei s method could be applied as well i the biomial cotext as i the Poisso. Soo [5] cosidered this approximatio i coectio with a sum of depedet idicator radom variables, Received: May 4, 2011 c 2011 Academic Publicatios, Ltd. Correspodece author
2 58 K. Teerapabolar, P. Wogkasem ad he applied the result to hypergeometric distributio, radom graphs problem ad the classical occupacy problem. Wogkasem et al. [8] used Stei s method ad wfuctios to give a error boud o biomial approximatio to a geeralized biomial distributio, ad Teerapabolar [7] used the same tools as i Wogkasem et al. [8] to give a error boud o biomial approximatio to the beta biomial distributio i the recet paper. However, all bouds as metioed above are the total variatio distace bouds. I this paper, we use Steis method ad wfuctios to give a o uiform boud i the biomial approximatio of a oegative itegervalued radom variable for the poit metric betwee the two distributios. Let X be a oegative itegervalued radom variable with probability fuctio p(x) = P(X = x) > 0 for every x i the support of X, S(x), ad have mea µ ad fiite variace σ 2 (0 < σ 2 < ). It is wellkow that the distributios of some types of X s ca be approximated by a biomial distributio with parameters ad p provided their parameters are satisfied uder certai coditios. For example, a hypergeometric distributio ca be approximated by a biomial distributio provided that the certai coditios o their parameters are satisfied. Let b( ;,p) = ( ) p q deote a biomial probability fuctio with parameters N ad p = 1 q (0,1) at {0,...,}. If we expect p( ) to be closer to b( ;,p), the it is reasoable to estimate p( ) by b( ;,p). For approximatig p( ) by b( ;,p), a boud for the poit metric betwee p( ) ad b( ;,p) is a criterio for measurig the accuracy of the approximatio. I this study, we derive a ouiform boud of the error o the metric betwee p( ) ad b( ;,p). The tools for givig our result cosist of the socalled wfuctios ad Steis method for the biomial distributio, which are i Sectio 2. I Sectio 3, we use Stei s method ad wfuctios to give the result i terms of the poit metric betwee p( ) ad b( ;,p), ad we give some applicatios of the result of this approximatio by usig the result to approximate some distributios such as hypergeometric, egative hypergeometric ad Pólya distributios, which are i the last sectio. 2. Method I order to give the result for this approximatio, we use the same methodology as i Teerapabolar [7], which cosists of Stei s method ad wfuctios. For wfuctios, Cacoullos ad Papathaasiou [2] defied a fuctio w associated
3 ON POINTWISE BINOMIAL APPROXIMATION with oegative itegervalued 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 oegative itegervalued 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 realvalued 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.
4 60 K. Teerapabolar, P. Wogkasem For A = { }, {0,...,}, the solutio g x0 = g {x0 } of (2.5) ca be writte as ( x )p x B 0,p(1 h Cx 1 ) if x x( x)q (x 1) 0 < x, g x0 (x) = (2.7) ( x )p x B 0,p(h Cx 1 ) if x x( x)q (x 1) 0 x 1, where C x = {0,...,x}. To prove the result, the followig lemma is also eed. Lemma 2.1. For {0,1,...,} ad x N, let g x0 (x) = g x0 (x + 1) g x0 (x), the we have the followig. ad g x0 (x) g x0 (x) { 1 q p { } mi 1 p q, 1 p+1 q +1 (+1)pq if > 0 { 1 q p { } mi 1 p q, 1 p+1 q +1 (+1)pq if > 0 (2.8) (2.9) Proof. For = 0, it follows from [1] that g 0 is positive ad decreasig i x {1,...,} ad g 0 (x) = 0 for x = 0 ad x. Therefore, we have g 0 (1) g 0 (x) g 0 (x) for every x {1,...,} ad, by (2.7), g 0 (1) = 1 q which implies g 0 (x) 1 q p ad g 0 (x) 1 q p. For > 0, it follows from [1] that g x0 is positive ad decreasig i x { + 1,...,} ad is egative ad decreasig i x {1,..., } ad g x0 (x) = 0 for x = 0 ad x. Therefore, we have that g x0 (x) g x0 ( ) ad g x0 (x) g x0 ( ) ad 1 g x0 ( ) = ( )p = 1 q 1 q k= +1 1 k= k= +1 ( ) p k q k + 1 k q x 0 1 k=0 ( ) p k q k k ( ) p k 1 q +1 k ( + 1 k) + k 1 k( ) ) x 0 1( ) p k q k + p k q k k ( k k=0 x 0 1 k=0 ( ) k p, p k q k
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 oegative itegervalued radom variable X, i terms of the poit metric ad its ouiform boud, which is obtaied by Stei s method ad wfuctios. Theorem 3.1. Let a oegative itegervalued radom variable X with p(x) > 0 for every x S(x) ad together with correspodig wfuctio 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.
8 64 K. Teerapabolar, P. Wogkasem The X has a hypergeometric distributio with parameters N, ad m ad has probability fuctio as follows: ( m N m ) p(x) = x)( x ( N, x = 0,1,...,mi{,m}. ) Here, its mea ad variace are µ = m N ad σ2 = m(n )(N m), respectively. N 2 (N 1) It is wellkow that the hypergeometric distributio ca be approximated by the biomial distributio. For this applicatio, we give a result of the biomial approximatio to the hypergeometric distributio i terms of the poit metric p( ) b( ;,p), where {0,1,...,mi{,m}}. Followig the relatio (2.2), we have w(x) = ( x)(m x). If mi{,m} =, Nσ 2 we put p = m N i Theorem 3.1, the ( x)p σ2 w(x) = ( x)m N ( x)(m x) N 0 for all 0 x. By applyig Corollary 3.1, a result of this approximatio ca be expressed as the followig. Corollary 4.1. For p = m N, p( ) b( ;,p) { (1 q )q( 1) N 1 { } mi 1 p, 1 p+1 q +1 ( 1)p (+1)p N 1 if 1. Similarly, if mi{,m} = m, we replace ad p i Theorem 3.1 by m ad N, respectively, ad usig Corollary 3.1, we ca obtai a aother result of this approximatio as the followig corollary. Corollary 4.2. If p = N, the we have 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 4.1. It should be oted that each result i Corollaries 4.1 ad 4.2 gives a good biomial approximatio if N is large ad ad m are small, or are small. N ad m N 4.2. Applicatio to Negative Hypergeometric Distributio Let us cosider the process of samplig without replacemet as metioed i previous subsectio. If elemets i a radom sample are draw without replacemet from this populatio util the umber of types II elemets reaches
9 ON POINTWISE BINOMIAL APPROXIMATION a fixed positive iteger r ad let X be the umber of types I elemets i the sample. The X has a egative hypergeometric distributio with parameters N,m ad r, ad its probability fuctio ca be expressed as ) p(x) = ( r+x 1 )( N r x x m x ( N m), x = 0,1,...,m, where r {1,...,N m} ad µ = rm N m+1 ad σ2 = rm(n m r+1)(n+1) are the (N m+1) 2 (N m+2) mea ad variace of X, respectively. It is observed that, if N,r such that r N m+1 teds to a costat θ, the a egative hypergeometric distributio with parameters N, m ad r coverges to a biomial distributio with parameters m ad θ. Therefore, we ca also use the biomial probability fuctio to approximate the egative hypergeometric probability fuctio by usig certai coditios of this covergece. Usig the relatio (2.2), the w(x) = (r+x)(m x). Thus, replacig by m (N m+1)σ 2 r ad p by N m+1 i Theorem 3.1, we have (m x)p σ2 w(x) = r(m x) N m+1 (r+x)(m x) N m+1 0 for all 0 x m. By Corollary 3.1, the followig corollary is obtaied. Corollary 4.3. For r {1,...,N m}, if p = p( ) b( ;m,p) { (1 q m )q(m 1) r N m+1, the N m+2 { } mi 1 p m, 1 pm+1 q m+1 (m 1)mp (m+1)p N m+2 if 1 m Applicatio to Pólya Distributio Suppose that a sigle ur cotai r red ad N r black balls. Draw a ball at radom, ote the color, ad retur it ito the ur together with a additioal ball of the same color. Repeat this way for m draws. Let X be the umber of red balls take out i the m drawigs, the the distributio of X is a Pólya distributio with parameters N, m ad r. The probability fuctio of X is give by ) p ( x) = ( r+x 1 )( N r+m x 1 x m x ( N+m 1 m ), x = 0,1,...,m ad the mea ad variace of X are µ = rm N ad σ2 = rm(n+m)(n r), respectively. N 2 (N+1) Usig the relatio (2.2), we the obtai w(x) = (r+x)(m x). Settig = m Nσ 2 ad p = r N i Theorem 3.1, it follows that (m x)p σ2 w(x) = (m x)x N 0
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 wfuctios, 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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