Solutions of Chi-square Quantile Differential Equation

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1 Proceedigs of the World Cogress o Egieerig ad Computer Sciece 07 Vol II WCECS 07, October 5-7, 07, Sa Fracisco, USA Solutios of Chi-square Quatile Differetial Equatio Hilary I. Oagbue, Member, IAENG, Mumiu O. Adamu, Timothy A. Aae Abstract The quatile fuctio of probability distributios is ofte sought after because of their usefuless. The quatile fuctio of some distributios caot be easily obtaied by iversio method ad aroximatio is the oly alterative way. Several ways of quatile aroximatio are available, of which quatile mechaics is oe of such aroach. This paper is focused o the use of quatile mechaics aroach to obtai the quatile ordiary differetial equatio of the Chi-square distributio sice the quatile fuctio of the distributio does ot have close form represetatios except at degrees of freedom equals to two. Power series, Adomia decompositio method (ADM) ad differetial trasform method (DTM) was used to fid the solutio of the oliear Chi-square quatile differetial equatio at degrees of freedom equals to two. The aroximate solutios coverge to the closed (exact) solutio. Furthermore, power series method was used to obtai the solutios for other degrees of freedom ad series expasio was obtaied for large degrees of freedom. Idex Terms Chi-square, quatile fuctio, differetial equatio, shape parameter, Adomia decompositio method, differetial trasform method. T I. INTRODUCTION HE search for aalytic expressio of quatile fuctios has bee a subject of itese research due to the importace of quatile fuctios. Several aroximatios are available i literature which ca be categorized ito four, amely: fuctioal aroximatios, series expasios, umerical algorithms ad closed form writte i terms of a quatile fuctio of aother probability distributio which ca also be refer to quatile ormalizatio. I geeral, the otio of aroximatio of the quatile fuctios have bee discussed extesively by [-6] The quatile fuctio of the Chi-square is very importat i statistical estimatio [ 7-8]. Moreover the aim of the paper is to aly the use of quatile mechaics aroach proposed by [9] to obtai a oliear secod order ordiary differetial equatio which ca be termed as Chi-square Quatile Differetial Equatio (CQDE) usig a trasformatio of the probability desity fuctio (PDF) of the Chi-square distributio. This is a step towards the Mauscript received July 6, 07; revised July 3, 07. This wor was sposored by Coveat Uiversity, Ota, Nigeria. H. I. Oagbue ad T. A. Aae are with the Departmet of Mathematics, Coveat Uiversity, Ota, Nigeria. hilary.oagbue@coveatuiversity.edu.g timothy.aae@coveatuiversity.edu.g M.O. Adamu is with the Departmet of Mathematics, Uiversity of Lagos, Aoa, Lagos, Nigeria. madamu@uilag.edu.g ISBN: ISSN: (Prit); ISSN: (Olie) quatile aroximatio of the distributio. This is because distributios with shape parameters require two steps towards effective quatile aroximatio [0]. Chi-square is a example of such distributio. The solutio of CQDE is the major cotributio of the paper. This was doe by the use of the power series, ADM ad DTM for the case where the degrees of freedom is equal to two. This is to validate the methods for other cases ad to create room for result compariso sice the quatile fuctio of the Chi-square distributio has closed form represetatio at that istace. The power series was used to obtai solutios for the other degrees of freedom. The quatile mechaics as metioed earlier is series expasio method of quatile aroximatio ad has bee alied for ormal distributio [9], beta distributio [9], gamma [], hyperbolic [], expoetial [3] ad studet s t [4]. II. FORMULATION The probability desity fuctio of the chi-square distributio ad the cumulative distributio fuctio are give by; x f ( x) x e, 0, x[0, ) ( / ) x, x F( x, ) P, () where (.,.) icomplete gamma fuctio ad P(.,.) regularized gamma fuctio. The quatile mechaics (QM) aroach was used to obtai the secod order oliear differetial equatio. QM is alied to distributios whose CDF is mootoe icreasig ad absolutely cotiuous. Chi-square distributio is oe of such distributios. That is; F ( p) (3) Where the fuctio F ( p) is the compositioal iverse of the CDF. Suose the PDF f(x) is ow ad the differetiatio exists. The first order quatile equatio is obtaied from the differetiatio of equatio (3) to obtai; F F p f Q p ( ( )) ( ( )) Sice the probability desity fuctio is the derivative of the cumulative distributio fuctio. The solutio to equatio () (4) WCECS 07

2 Proceedigs of the World Cogress o Egieerig ad Computer Sciece 07 Vol II WCECS 07, October 5-7, 07, Sa Fracisco, USA (4) is ofte cumbersome as oted by Ulrich ad Watso [5]. This is due to the oliearity of terms itroduced by the desity fuctio f. Some algebraic operatios are required to fid the solutio of equatio (4). Moreover, equatio (4) ca be writte as; f ( ) (5) Alyig the traditioal product rule of differetiatio to obtai; V ( )( ) (6) Where the oliear term; d V ( x) (l f ( x)) dx (7) These were the results of [9]. It ca be deduced that the further differetiatio eables researchers to aly some ow techiques to fidig the solutio of equatio (6). The reciprocal of the probability desity fuctio of the chisquare distributio is trasformed as a fuctio of the quatile fuctio. d ( ( / )) e (8) Differetiate agai to obtai; d d e ( ) e Q p d ( ( / )) Factorizatio is carried out; d ( ( / )) d e d e ( ) Q p (9) (0) d d d () The secod order oliear ordiary differetial equatios is give as; ( ) ( ) () With the boudary coditios; Q(0) 0, Q(0). d Q p dq p III. POWER SERIES SOLUTION The cumulative distributio fuctio ad its iverse (quatile fuctio) of the chi- square distributio do ot have closed form. However, a aalytical formula is available for the cumulative distributio fuctio of the Chisquare distributio whe the degrees of freedom =. The formula is give as; x ( ) ( e ) F x (3) The quatile fuctio Q(p) ca be obtaied as; p p ( e ) e (4) Taig logarithm; p l (5) The quatile fuctio is give by; p Qp ( ) l (6) The exact solutio (equatio (6)) is compared with the aroximate solutio (equatio (), to compare the covergece of the aroximate solutio to the exact. This is to create a aveue for compariso betwee the exact ad aroximate values ad cosequetly examie the validity of the quatile mechaics. Whe =, equatio () becomes; d d 0 (7) Alteratively, the PDF of the chi-square distributio at = ca be used. The PDF is give as; x f( x) e (8) Alyig the Quatile Mechaics aroach to obtai; d e (9) d Q p ( ) d e d Q p (0) ( ) d Equatio () is the same with equatio (7). The geeral power series solutio of equatios () or (6) is give by; 3 4 c c p c p c p c p c p c p c p Differetiate equatio (); c c p 3c p 4c p c p 6 c p... c p Differetiate equatio (3); c 6c p c p 0c p c p 4 c p... ( ) c p () () (3) Substitute equatios (4) ad (3) ito () ad collect lie (4) ISBN: ISSN: (Prit); ISSN: (Olie) WCECS 07

3 Proceedigs of the World Cogress o Egieerig ad Computer Sciece 07 Vol II WCECS 07, October 5-7, 07, Sa Fracisco, USA terms. costat: 4c 0 c 4 p: c 4c 0 c 3 3 p : 4c 4c 6c 0 c p : 40c 8c c c 0 c p : 60c 0c 6c c 9c 0 c p : c c cc5 4c3c4 0 c p : c 4c 4c c 30c c 6c c 8 04 (5) The coefficiets are substituted ito equatio () to obtai the power series solutio. The power series solutio of equatio (7) or () is give by; p p p p p (6) p p p This is the aroximate solutio of equatio (6), IV. NUMERICAL RESULTS Adomia Decompositio Methods (ADM) ad Differetial Trasform Methods (DTM) are used to cofirm the results of the power series. This is achieved by usig the methods to solve equatio () ad compare with the exact value that is equatio (). The methods are semi-aalytic i ature ad have bee alied extesively i umerical aalysis, computatioal fluid mechaics, rigid bodies aalysis, elasticity, mathematical fiace, ris aalysis ad so o. Details o the theories, modificatios ad alicatios of ADM ad DTM ca be foud i [6], [7], [8], [9], [0], [], [], [3], [4]. Adomia Decompositio Method Writig equatio () i the itegral form gives; dq u( p) p dpdp dp 00 (7) By ADM, the ifiite series solutio is of the form; u( p) u ( p) (8) 0 Now usig (7) i (), we have; dq u ( p) p dpdp dp (9) 0 00 I view of (9), the zeroth order term ca be writte as; 0 u ( ) p p (30) while other terms ca be determied usig the recurrece relatios dq u ( p) dpdp dp (3) 0 00 The oliear terms i equatio (3) ca be represeted as dq B dp ad the Adomia polyomials are computed as follows: B dq dp dq dq, B, dp dp dq0 dq dq, (3) B (33) dp dp dp dq0 dq3 dq dq B3 dp dp dp dp Substitutig equatio (3) i equatio (3) yields u( p) BdPdP 0 00 (34) Solvig equatios (30) ad (34) yields the solutio equatio (). The series solutio of equatio () usig the ADM is; p p 4 p p p p p... (35) Differetial Trasform Method To solve the iitial value problems by DTM we first trasform equatios () as; ( r )( r ) U ( ) ( )! r0u ( ) U ( r ) (36) ad the iitial coditio as U(0) 0, U() (37) Usig equatio (37) i (36) ad resolvig the chage of variables give the solutio of equatio () The series solutio of equatio () usig the DTM is; p p p p p (38) p p p The three methods coverge favorably to the exact value as show i Table. That is the series solutios of equatios (), (35) ad (38). V. EXTENSION TO DIFFERENT DEGREES OF FREEDOM The power series method was used to obtai the series solutios of equatio () for the differet degrees of freedom up to 0. No compariso was made because of the absece of the closed form of the CDF ad QF. The result of the degrees of freedom equals to two is icluded. The coefficiets of the series are show i Table. ISBN: ISSN: (Prit); ISSN: (Olie) WCECS 07

4 Proceedigs of the World Cogress o Egieerig ad Computer Sciece 07 Vol II WCECS 07, October 5-7, 07, Sa Fracisco, USA The equatios formed a series which ca be used to predict p for ay give degree of freedom. Q p p p 4( ) ( ), (39) For very large, p (40) represetatio of both CDF ad QF at that istace. The aroximate solutios coverge to the exact solutio. The procedure serves as a validatio mode for other degrees of freedom. The series solutios for up to degrees of freedom equal to 0 ad for large cases were icluded. The methods used are efficiet i hadig oliear ODE ad is recommeded for solvig quatile differetial equatios of probability distributios ad most importatly quatile differetial equatios. VI. CONCLUDING REMARKS I this paper, the power series method, ADM ad DTM was used to obtai the aroximate solutios of the Chi-square quatile differetial equatios at degrees of freedom equals to two. Chi-square distributio has closed form ACKNOWLEDGMENT The authors are uaimous i areciatio of fiacial sposorship from Coveat Uiversity. The costructive suggestios of the reviewers are greatly areciated. Table : Numerical results of power series, ADM ad DTM p EXACT Power series ERROR ADM ERROR DTM ERROR E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-09 ISBN: ISSN: (Prit); ISSN: (Olie) WCECS 07

5 Proceedigs of the World Cogress o Egieerig ad Computer Sciece 07 Vol II WCECS 07, October 5-7, 07, Sa Fracisco, USA Table : Coefficiets of the power series solutio for differet degrees of freedom c c c3 c4 c5 c6 c ISBN: ISSN: (Prit); ISSN: (Olie) WCECS 07

6 Proceedigs of the World Cogress o Egieerig ad Computer Sciece 07 Vol II WCECS 07, October 5-7, 07, Sa Fracisco, USA REFERENCES [] J.S. Dagpuar, A easily implemeted geeralised iverse Gaussia geerator, Comm. Stat. Simul. Comput., vol. 8, o., , 989. [] N.R. Farum, A fixed poit method for fidig percetage poits, Al. Stat., vol. 40, o.,. 3-6, 99. [3] Y. Lai, Geeratig iverse Gaussia radom variates by aroximatio, Comput. Stat. Data Aaly., vol. 53, o. 0, , 009. [4] G. Derfliger, W. Hörma ad J. Leydold, Radom variate geeratio by umerical iversio whe oly the desity is ow, ACM Trasac. Model. Comp. Simul., vol. 0, o. 4, Article 8, 00. [5] G. Derfliger, W. Hörma, J. Leydold, ad H. Sa, Efficiet umerical iversio for fiacial simulatios. I Mote Carlo ad Quasi-Mote Carlo Methods, Spriger Berli Heidelberg, , 009. [6] J. Leydold ad W. Hörma, Geeratig geeralized iverse Gaussia radom variates by fast iversio, Comput. Stat. Data Aaly., vol. 55, o.,. 3-7, 0. [7] C.W. Brai ad J. Mi. "O some properties of the quatiles of the chisquare distributio ad their alicatios to iterval estimatio." Comm. Stat. Theo. Meth., vol. 30, o. 8-9, , 00. [8] T. Iglot, Iequalities for quatiles of the chi-square distributio, Prob. Math. Stat., vol. 30, o., , 00. [9] G. Steibrecher ad W.T. Shaw, Quatile mechaics, Euro. J. Al. Math., vol. 9, o.,. 87-, 008. [0] T. Luu, Fast ad accurate parallel computatio of quatile fuctios for radom umber geeratio, Doctoral thesis, Uiversity College Lodo, 06. [] T. Luu, Efficiet ad accurate parallel iversio of the gamma distributio, SIAM J. Scietific Comp., vol. 37, o., C-C4, 05. [] W.T. Shaw ad N. Bricma, Differetial Equatios for Mote Carlo Recyclig ad a GPU-Optimized Normal Quatile, arxiv preprit arxiv: , 009. [3] W.T. Shaw, Eco-mputatioal Fiace: Differetial Equatios for Mote Carlo Recyclig arxiv preprit arxiv: , 009. [4] W.T. Shaw ad A. Muir, Depedecy without copulas or ellipticity, Euro. J. Fiace, vol. 5, o. 7-8, , 009. [5] G. Ulrich ad L.T. Watso, A method for computer geeratio of variates from arbitrary cotiuous distributios, SIAM J. Scietific Comp., vol. 8, o., , 987. [6] G. Adomia, A review of the decompositio method i alied mathematics, J. Math. Aaly. Al., vol. 35, o., , 988. [7] A.M. Wazwaz, A compariso betwee Adomia decompositio method ad Taylor series method i the series solutios, Al. Math. Comput., vol. 97, o., , 998. [8] A.M. Wazwaz, A reliable modificatio of Adomia decompositio method, Al. Math. Comput., vol. 0, o., , 999. [9] A.M. Wazwaz, Adomia decompositio method for a reliable treatmet of the Bratu-type equatios, Al. Math. Comput., vol. 66, o. 3, , 005. [0] A.M. Wazwaz ad S.M. El-Sayed, A ew modificatio of the Adomia decompositio method for liear ad oliear operators, Al. Math. Comput., vol., o. 3, , 00. [] J.K. Zhou, Differetial trasformatio ad its alicatios for electrical circuits, Huazhog Uiversity, Wuha, 986. [] A.A. Opauga, S.O. Edei, H.I. Oagbue, G.O. Ailabi, A.S. Osheu ad B. Ajayi, O umerical solutios of systems of ordiary differetial equatios by umerical-aalytical method, Al. Math. Scieces, vol. 8, o. 64, , 04. [3] S.O. Edei, A.A. Opauga, H.I. Oagbue, G.O. Ailabi, S.A. Adeosu ad A.S. Osheu, A Numerical-computatioal techique for solvig trasformed Cauchy-Euler equidimesioal equatios of homogeous type. Adv. Studies Theo. Physics, vol. 9, o.,. 85 9, 05. [4] A.A. Opauga, S.O. Edei, H.I. Oagbue, S.A. Adeosu ad M.E. Adeosu, Some Methods of Numerical Solutios of Sigular System of Trasistor Circuits, J. Comp. Theo. Naosci., vol., o. 0, , 05. ISBN: ISSN: (Prit); ISSN: (Olie) WCECS 07