Songklanakarin Journal of Science and Technology SJST R1 Teerapabolarn

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1 Soglaaari Joural of Sciece ad Techology SJST--.R Teeraabolar A No-uiform Boud o Biomial Aroimatio to the Beta Biomial Cumulative Distributio Fuctio Joural: Soglaaari Joural of Sciece ad Techology For Review Oly Mauscrit ID SJST--.R Mauscrit Tye: Origial Article Date Submitted by the Author: -Aug- Comlete List of Authors: Teeraabolar, Kait; Buraha Uiversity, Mathematics Keyword: Beta biomial cumulative distributio fuctio, biomial aroimatio, characterizatio of the beta biomial radom variable, o-uiform boud, Stei s method For Proof Read oly

2 Soglaaari Joural of Sciece ad Techology SJST--.R Teeraabolar Page of Origial Article A No-uiform Boud o Biomial Aroimatio to the Beta Biomial Cumulative Distributio Fuctio Kait Teeraabolar,* ad Khuaor Sae-Jeg Deartmet of Mathematics, Faculty of Sciece, Buraha Uiversity, Choburi,, Thailad For Review Oly Cetre of Ecellece i Mathematics, CHE, Sri Ayutthaya Road, Bago, Abstract Thailad * Corresodig author, address: ait@buu.ac.th This aer uses Stei s method ad the characterizatio of beta biomial radom variable to determie a o-uiform boud for the distace betwee the beta biomial cumulative distributio fuctio with arameters N, α > ad > ad the biomial cumulative distributio fuctio with arameters ad eamles are give to illustrate the obtaied result. α α +. Some umerical Keywords: Beta biomial cumulative distributio fuctio, biomial aroimatio, characterizatio of the beta biomial radom variable, o-uiform boud, Stei s method.. Itroductio Let X be the biomial radom variable with arameters Its robability mass fuctio is as follows: For Proof Read oly N ad (,).

3 Page of Soglaaari Joural of Sciece ad Techology SJST--.R Teeraabolar where q ad E( X ) b( ), {,..., } ad Var( X ) For Review Oly (.) q are its mea ad variace, resectively. It is well-ow that if the robability of success i the biomial distributio is a radom variable ad has a beta distributio with arameters α > ad >, the a ew resultig distributio is referred to as the beta biomial distributio with arameters, α ad. Let Y be the beta biomial distributio with arameters, α ad, ad its robability mass fuctio is of the form B( y, + y) bb( y), y {,..., }, y B( α, ) where B is the comlete beta fuctio ad α µ α ( + ) (.) ad σ are the ( + )( ) mea ad variace of Y, resectively. Some useful alicatios of this distributio ca be foud i field such as aimal teratology eerimets i Guta ad Naradajah (), statistical rocess cotrol i Sat Aa ad Cate (), iferetial statistics i Salem ad Abu El Azm () ad balacig reveues ad reair costs i Dig, Rusmevichietog, ad Toaloglu (). For limitig distributio, it follows from (.) that if α, i such a way α teds to a costat, the the beta biomial distributio with arameters, α ad coverges to a biomial distributio with arameters ad. I this case, Teeraabolar () used Stei s method ad the biomial α w-fuctio to give a uiform boud o biomial aroimatio to the beta biomial distributio as follows: ( ) d A( Y, X) ( + q + ) ( + )( + ) For Proof Read oly (.)

4 Soglaaari Joural of Sciece ad Techology SJST--.R Teeraabolar Page of for every subset A of {,..., }, where ( ) d Y, X P( Y A) P( X A) is the distace For Review Oly A betwee the beta biomial distributio with arameters, α ad ad the biomial α distributio with arameters ad. For result i (.) becomes where ( ) C A C {,..., }, {,..., }, the ( ) dc ( Y, X) ( + q + ), ( + )( + ) (.) d Y, X P( Y ) P( X ) is the distace betwee the beta biomial cumulative distributio fuctio with arameters, α ad ad the biomial cumulative distributio fuctio with arameters ad α α + at. We observe that the boud i (.) is uiformly i {,..., }, that is, it does ot chage alog {,..., }. So, it may be iaroriate for measurig the accuracy of the aroimatio. I this aer, we are iterested to determie a o-uiform boud with resect to the bouds i (.) by usig Stei s method ad the characterizatio of beta biomial radom variable, which are described ad determied i Sectios ad, resectively. I Sectio, some umerical eamles rovided to illustrate the obtaied result, ad the coclusio of this study is reseted i the last Sectio.. Method Stei s method ad the characterizatio of beta biomial radom variable are both tools, which ca be used to obtai the desired result.. Stei s method for biomial distributio Stei () roosed a owerful method for ormal aroimatio, which is called Stei s method. I, he also alied this method to biomial aroimatio For Proof Read oly

5 Page of Soglaaari Joural of Sciece ad Techology SJST--.R Teeraabolar (Stei, ). Followig Barbour, Holst, ad Jaso (), Stei s equatio for biomial distributio with arameters form, N ad < q<, for give h, is of the h( ) B ( h) ( ) f ( + ) qf ( ), (.) where B, ( h) h( ) ad f ad h are bouded real valued fuctios For Review Oly defied o {,..., }. For A {,..., }, let h :{,..., } R be defied by if A, ha( ) if A. A (.) Followig Barbour et al. (), let f A : N {} R satisfy (.), where f A() f A () ad f A( ) f A( ) for. Therefore, the solutio f A of (.) ca be eressed as f A B, ( ha C ) B, ( ), ( ) ha B h C ( ), + where C {,..., }. Similarly, for A { } ad by settig h ad h, thus the solutios { } f B, ( h ) B, ( hc ) + ( ) B, ( h ) B, ( hc ) + f f ad { } if, if > For Proof Read oly (.) A C whe {,..., } ad f C are as follows: (.)

6 Soglaaari Joural of Sciece ad Techology SJST--.R Teeraabolar Page of that obtai Let B, ( hc ) B, ( h ) C + C ( ) B, ( hc ) B, ( hc ) + f For Review Oly if, if >. f ( ) f ( + ) f ( ) ad f ( ) ad C C C C (.) f ( ) f ( + ) f ( ). It is see f ( ) for,. Thus, for {,.., } ad by (.), we B, ( h ) B, ( hc ) B, ( h ) C + if < ( ), f ( ) B, ( h ), ( C ), ( C ) B h B h + if ( ), B, ( h ) B, ( hc ) B, ( h C ) if > ( ) q if ( ), q + < + + if ( ) q, + if ( ). q > By (.), we also obtai f C ( ) B, ( hc ) B, ( h C ) B, ( h ) C if ( ), B, ( hc ) B, ( h C ) B, ( h C ) if > ( ) q For Proof Read oly (.)

7 Page of Soglaaari Joural of Sciece ad Techology SJST--.R Teeraabolar j j j j + ( ) q j j j j + ( ) q For Review Oly if, if >. The followig lemmas reset some ecessarily roerties of which are used to rove the mai result. f ad (.) f C, Lemma.. Let {,..., } ad {,..., }, the the followig iequalities hold: ad > if, f ( ) < if fc ( ) < if > > if,. Proof. Firstly, we have to show that (.) holds. For, it follows from (.) that For, we have to show (.), for >, f ( ) (.) (.) f ( ). > (.) f ( ) < whe > ad + ( ) < as follows. From q + + ( ) ( ) For Proof Read oly

8 Soglaaari Joural of Sciece ad Techology SJST--.R Teeraabolar Page of + + Let ξ( ) + ( ), the + + ξ ( ) ( + ) + ( + )( ) + + ( + ) ( + ) + ( ) <. Thus for >, For <, f ( ) + + For Review Oly [ ( ) ] + ( ) f ( ). < (.) q + + ( ) ( ) + + Let ξ( ) ( ), the ξ ( ) ( + ) + ( + )( ) + + ( + ) ( + ) + ( ) [ ( ) ] For Proof Read oly

9 Page of Soglaaari Joural of Sciece ad Techology SJST--.R Teeraabolar <. Thus for <, Followig (.) ad (.), whe, it yields Hece, from (.) ad (.), the iequalities i (.) hold. f ( ). < (.) f ( ). < (.) For Review Oly Net, we shall show that (.) holds. For >, it follows from (.) that j q j j j fc ( ) ( ) + + ( ) j j + + j j ( ) >. For <, by (.), we obtai f C [ ( ) ] + ( ) ( ) j q j j j + + ( ) ( ) j j + + j j ( ) <. For Proof Read oly [ ( ) ] (.) (.)

10 Soglaaari Joural of Sciece ad Techology SJST--.R Teeraabolar Page of Hece, from (.) ad (.), the iequalities i (.) hold. Lemma.. For {,..., }, f C is a icreasig fuctio i {,..., }. C C C Proof. Let f ( ) f ( + ) f ( ). We shall show that {,..., }. It follows from (.) that fc ( ) + j j j j + + ( ) ( + ) q + For Review Oly j ( ) j j j + fc j j + + j j ( + ) ( ) + j j + + j + j [ ( ) ] ( ) ( ) > for [( )( ) ] j q j j j ( + )( + ) + ( )( ) + j q j j + j ( + ) ( + ) + ( )( + ) For Proof Read oly

11 Page of Soglaaari Joural of Sciece ad Techology SJST--.R Teeraabolar j q j j j ( + )( + ) + ( )( + ) ( + ) ( + ) + + j q j j j ( + )( + ) + ( )( + ) For Review Oly ( + ) ( ) + j q j j j ( + ) ( )( + ) ( + ) + + >. Therefore f ( ) is a icreasig fuctio i {,..., }. C Lemma.. Let {,..., }, the we have the followig iequalities hold: ad q su fc ( ) (Teeraabolar ad Wogasem, ) (.) + + q su fc ( ) mi,, q ( + ) q where {,..., }. (.) Proof. We shall show that the iequality (.) holds. Teeraabolar ad Wogasem () showed that For Proof Read oly

12 Soglaaari Joural of Sciece ad Techology SJST--.R Teeraabolar Page of From (.), we have + + q f ( ) mi,. q ( + ) q f ( ) + + ( ) Substitutig by i the roof of. For Review Oly Wogasem (), we also obtai + + q f ( ) mi,. q ( + ) q I order to rove that { ( ) } + + fc q + ( ) mi {, ( ) } + + fc for every {,..., }, q +,. For, we have which gives For <, we obtai (.) f ( ) detailed i Teeraabolar ad (.) su ( ) mi,, it suffices to show < f ( ) (by (.)) C C because f ( ) (by Lemma.) f ( ) f ( ) + L + f ( ) f ( ) (by (.)) + + q mi,, q ( + ) q + + q fc ( ) mi,. q ( + ) q For Proof Read oly f (by (.)) C ( ) as (.)

13 Page of Soglaaari Joural of Sciece ad Techology SJST--.R Teeraabolar which gives < f ( ) (by (.)) C f ( ) f ( ) + f ( ) + + f ( ) + f ( ) + L + f ( ) + f ( ) + f ( ) + L + f ( ) C + f ( ) (by (.) ad (.)) For Review Oly + + q mi, q ( + ) q + + q mi,, q ( + ) q + + q fc ( ) mi,. q ( + ) q Hece, by (.) ad (.), the iequality (.) holds.. The characterizatio of beta biomial radom variable (by (.)) (.) For the characterizatio associated with the beta biomial radom variable Y, by alyig Lemma. i Cacoullos ad Paathaasiou (), the covariace of Y ad f C where. ( Y ) ca be eressed as α µ, ( ) y ( ) ( ) ( ) Cov Y fc Y, fc y µ bb y y (.) ( µ ) bb( ) Lemma.. Let ϕ ( y), y,...,, be the characterizatio associated with bb( y) the beta biomial radom variable Y, the we have the followig. For Proof Read oly

14 Soglaaari Joural of Sciece ad Techology SJST--.R Teeraabolar Page of ad ( y)( y) ϕ( y), y,...,, (.) Cov Y, f ( Y ) E f ( Y ). ( Y )( Y ) C C ( y)( y) For Review Oly (.) Proof. We shall show that ϕ( y) by mathematical iductio as follows. It α ca be see that ϕ () µ ad m ( µ ) bb( ) ϕ m bb( m) ( m)( m) ( µ ) bb( ) ϕ bb() ( )( ) (). For m<, let m+ m+ ( ), we have to show that ϕ ( m+ ) ( ( ))( ( )). Sice ϕ ( m+ ) m+ ( µ ) bb( ) bb( m+ ) m ( µ ) bb( ) bb( m) + µ ( m+ ) bb( m+ ) bb( m) ( m+ )( + m) + µ ( m+ ) ( ( m+ ))( ( m+ )), thus by mathematical iductio, (.) is obtaied. Substitutig (.) ito (.), it becomes this yields (.). y (by m ( µ ) bb( ) bb( m) Cov( Y, fc ( Y )) ( y)( y) f ( ) ( ) C y bb y ( Y )( Y ) E f ( ), C Y For Proof Read oly ( m)( m) )

15 Page of Soglaaari Joural of Sciece ad Techology SJST--.R Teeraabolar. Result The mai oit of this study is to determie a o-uiform boud for the distace betwee the biomial ad beta biomial cumulative distributio fuctios, C (, ) d Y X as {,..., }. The followig theorem resets this desired result. For Review Oly α Theorem.. Let ad {,..., }, the we have the followig. d C ( Y, X) ( q )( ) q if, q ( ) mi {, ( ) } if. + + For Proof Read oly (.) Proof. Usig the same argumets detailed as i the roof of Theorem. i Teeraabolar (), it follows that d C Because ( Y, X) E µ f ( Y ) E Y f ( Y ) Cov Y, f ( Y ) C C C E Y f Y E f Y ( Y )( Y ) ( µ ) C ( ) ( ) C { ( Y )( Y ) µ C ( ) } E Y f Y ( Y )( Y ) E µ Y fc ( Y ) ( Y) Y E f ( Y ) C (by (.)) ( Y ) Y E fc ( Y ) q ( Y ) Y E if, (by Lemma.) + + q ( Y ) Y mi {, ( ) } E if. q + q ( Y ) Y µ σ µ ( ) α ( )( + ), E we have

16 Soglaaari Joural of Sciece ad Techology SJST--.R Teeraabolar Page of d C ( Y, X) Hece, the iequality (.) is obtaied. Remar. Sice q + + q ( + ) q q ( ) α if ( )( ), q ( ) α mi {, ( ) } if ( )( ). q + q + < ad mi {, } For Review Oly + + q + q + q ( + ) q ( + ) q the boud i Theorem. is better tha that metioed i (.).. Numerical eamles whe {,..., }, Teeraabolar () suggested the result i (.) to give a good aroimatio whe α ad are small. So, we rovide two eamles to illustrate the result i Theorem. by settig arameters, α ad to satisfy this suggestio. Eamle.. Let, α ad, the the umerical result i Theorem. is of the form d C ( Y, X). if,. if,. if,...,. It is better tha the umerical result i (.), C (, ) d Y X.,,,...,. Eamle.. Let, α ad, the the umerical result i Theorem. is of the form d C ( Y, X). if,. if,,,,. if,...,. It is better tha the umerical result i (.), For Proof Read oly

17 Page of Soglaaari Joural of Sciece ad Techology SJST--.R Teeraabolar C (, ) d Y X.,,,...,. The two eamles are idicated that the result i Theorem. gives a good aroimatio whe ad α are small, esecially, whe α For Review Oly is small. Furthermore, these eamles are oited out that the boud i Theorem. is better tha that show i (.).. Coclusio The boud i this study, o-uiform boud, was determied by usig Stei s method ad the characterizatio of beta biomial radom variable. It is aroriate to aroimate the distace betwee the beta biomial cumulative distributio fuctio with arameters, α > ad > ad the biomial cumulative distributio fuctio with arameters ad, because it chages alog + {,..., }. I additio, by α α theoretical ad umerical comariso, the result i this study is better tha that metioed i (.), ad it gives a good biomial aroimatio whe ad α are small. Acowledgemets The authors would lie to tha the aoymous referees for their useful commets ad suggestios. Refereces Barbour, A. D., Holst, L., & Jaso, S. (). Poisso aroimatio (Oford Studies i Probability ). Oford: Claredo Press. For Proof Read oly

18 Soglaaari Joural of Sciece ad Techology SJST--.R Teeraabolar Page of Cacoullos, T., & Paathaasiou, V. (). Characterizatio of distributios by variace bouds. Statistics & Probability Letters, (), -. Dig, C., Rusmevichietog, P., & Toaloglu, H. (). Balacig reveues ad reair Costs uder artial iformatio about roduct reliability. Productio ad Oeratios Maagemet, (), -. Guta, A. K., & Nadarajah, S. (). Hadboo of beta distributio ad its For Review Oly alicatios. New Yor: Marcel Deer. Salem, S. A., & Abu El Azm, W. S. (). The samlig distributio of the maimum lielihood estimators for the arameters of beta-biomial distributio. Iteratioal Mathematical Forum, (), -. Sat Aa, Â. M. O., & Cate, C. S. (). Beta cotrol charts for moitorig fractio data. Eert Systems with Alicatios, (), -. Stei, C. M. (). A boud for the error i ormal aroimatio to the distributio of a sum of deedet radom variables. I L. M. Le Cam, J. Neyma, & E. L. Scott (Eds), Proceedigs of the Sith Bereley Symosium o Mathematical Statistics ad Probability (. -), Bereley: Uiversity of Califoria Press. Stei, C. M. (). Aroimate comutatio of eectatios (Lecture Notes- Moograh Series ). Istitute of Mathematical Statistics: Hayward Califoria. Teeraabolar, K. (). A boud o the biomial aroimatio to the beta biomial distributio. Iteratioal Mathematical Forum, (), -. Teeraabolar K., & Wogasem, P. (). O oitwise biomial aroimatio by w-fuctios. Iteratioal Joural of Pure ad Alied Mathematics, (), -. For Proof Read oly

19 Page of Soglaaari Joural of Sciece ad Techology SJST--.R Teeraabolar For Review Oly For Proof Read oly