On Size-Biased Logarithmic Series Distribution and Its Applications

Transcription

1 7 The Ope Statistics ad Pobability Joual, 9,, 7-7 O Size-Bied Logaithmic Seies Distibutio ad Its Applicatios Ope Access Khushid Ahmad Mi * Depatmet of Statistics, Govt. College (Boys, Baamulla, Khmi, Idia Abstact: I this pape, a size-bied logaithmic seies distibutio (SBLSD, a paticula ce of the weighted logaithmic seies distibutio, taig the weights the vaiate values is defied. The momets ad ecuece elatio of (SBLSD ae obtaied. Negative momets ad ivese cedig factoial momets of the size-bied logaithmic seies distibutio have bee deived i tems of hype-geometic fuctio. Recuece elatios fo these momets have also bee deived usig popeties of hype-geometic fuctios. Diffeet estimatio methods fo the paamete of the model ae discussed. R- Softwae h bee used fo maig a compaiso amog the thee diffeet estimatio methods ad with the logaithmic seies distibutio. Key Wods: Size-bied logaithmic seies distibutio, Negative momets, Ivese cedig factoial momets, Bayes estimato, Beta distibutio, R-Softwae.. INTRODUCTION The logaithmic seies distibutio (LSD chaacteized by a paamete is give by PX ( = = log( ;,.. ( The model ( is a limitig fom of zeo-tucated egative biomial distibutio. Negative momets of discete distibutios, maily the biomial, Poisso ad egative biomial have bee ivestigated by vaious authos [Stepha [], Gab ad Savage [], Medehall ad Lehma [3], Govidaajulu [,], Tiu [6], Stacu [7], Chao ad Stawdema [8], Gupta [9,], Cessie et al. [], Cessie ad Boet [] ad Roohi[3]. Ivese cedig factoial momets have oly bee dealt with by Lepage [] ad Joes []. Best et al. [6] discussed the test of fit fo the model (. Sadile [7] lied the egative biomial distibutio with the logaithmic seies ad Shaumugam [8] studied the chaacteizatio of model (. A bief list of authos ad thei wos ca be see i Johso, Kotz ad Kemp [9]. The fist fou momets of LSD ae give. μ = (, whee ( μ = (3 3 ( ( μ = ( 3 ( ( ( ( μ = ( 7 3 ( ( 6 The vaiace of LSD is give ( 3 μ = [ + ] (6 I this pape, we have made a attempt to study poposed size-bied LSD, its momets ad ecuece elatios. Negative momets ad ivese cedig factoial momets of the size-bied logaithmic seies distibutio have bee deived i tems of hype-geometic fuctio. Recuece elatios fo these momets have also bee deived usig popeties of hype-geometic fuctios. I ode to mae a compaative aalysis amog the thee estimatio methods fo the paamete of the size-bied logaithmic seies distibutio (SBLSD, oe of the stadad softwae pacages R- Softwae is used which is meat fo data aalysis ad gaphics.. SIZE-BIASED LOGARITHMIC SERIES DISTRI- BUTION (SBLSD A size-bied logaithmic distibutio (SBLSD is obtaied by taig the weight of the LSD (. We have fom ( ad ( P( X = =, = = log( This gives the size-bied logaithmic seies distibutio (SBLSD P[ X = ] = { F[, A; A; ]} ;, (7 *Addess coespodece to this autho at the Depatmet of Statistics, Govt. College (Boys, Baamulla, Khmi, Idia; Tel: 9-968; Fa: 9-96; hshdmi@yahoo.com Whee < < ad { F[, A; A; ]} = ( 876-7/9 9 Betham Ope

2 O Size-Bied Logaithmic Seies Distibutio The Ope Statistics ad Pobability Joual, 9, Volume 7.. Momets The th momet μ ( s of SBLSD (7 about oigi is obtaied ( μ s = E( X = P[ X = ] ( s μ = ( ; =, (8 obviously μ ( s = ad fo μ ( s = = = [ = ] + P X μ ( s = μ + (9 whee μ + is the ( + th momets about oigi of LSD (. The momets of SBLSD ca be obtaied by usig elatios ( to ( i (9 μ ( s = ( Usig elatio ( i (9, we get ( s μ = ( ( ( Which gives the vaiace of SBLSD (7 μ ( s = ( ( ( ( The highe momets of SBLSD (7 about oigi ca also be obtaied by usig (9 if so desied. 3. RECURRENCE RELATION OF MOMENTS ABOUT ORIGIN OF SIZE-BIASED LSD The ecuece elatio ca be obtaied by diffeetiatig (8 μ ( s = { ( } = μ + μ μ ( ( s ( s ( s ( s μ μ + ( s = + μ ( Fo =, we get ( s (3 μ ( s =, whee μ ( s = ( The secod momet of (7 about oigi ca also be obtaied by usig the elatio (3. NEGATIVE MOMENTS AND INVERSE ASCEND- ING FACTORIAL MOMENTS Theoem I: Suppose the adom vaiable X h a sizebied logaithmic seies distibutio with paamete, the the elatio - EX ( + A = {( A+ F[, AA ; ; ] } F[, A+ ; A+ ; ] holds. Poof: Sice X h a size-bied logaithmic seies distibutio, the EX ( + A = { F[, AA ; ; ]} ( + A = {( A+ F[, A; A; ]} F[, A+ ; A+ ; ] ( This completes the poof. Theoem II: Suppose the adom vaiable X h a sizebied logaithmic seies distibutio with paamete, the the th ivese cedig factoial momet is give μ {( F [, A; A; ]} F [,; ; ],,... ad < <. = + [ ] + = μ = E[ i= X + i Poof: Hee [ ] = { F[, A; A; ]} ( + ( +...( +. = { F [, A; A; ]} [ ] (+! ( +! ( + 3! O simplificatio, we get μ [ ] = {( + F[, A; A; ]} F[,; + ; ] (. RECURRENCE RELATION FOR NEGATIVE AS- CENDING FACTORIAL MOMENTS Theoem III: Suppose the adom vaiable X h a sizebied logaithmic seies distibutio with paamete ad μ [ ] is the th ivese cedig factoial momet of X, the the elatio ( + μ [+] = [( + ( + ] μ [ ] + ( μ [], =,3...ad << holds. Poof: we ow that μ = [ ] {( + F[, A; A; ]} F[,; + ; ] μ [ + ] = {( + F[, A; A; ]} F[,; + 3; ] Usig the idetity (see Raiville [], page 7

3 73 The Ope Statistics ad Pobability Joual, 9, Volume Khushid Ahmad Mi ( a c + F [a,b;c; z] = a F [a +, b;c; z] (c F [a,b;c ; z], fo a=,b=,c=+3,z=, we get (6 F [,; + 3;] = + + F [,; + ;] + Now agai usig the idetity give by (Raiville [], page 7 F [,; + 3;] (7 ( z F [a,b;c; z] = a F [a, b;c; z] ( c b c z F [a,b;c +; z], foa=,b=,c=+,z=, we get ( ( + ( ( + F[,; + 3; ] = F[,; + ; ] F[,; + ; ] + + (8 Substitutig (8 i (7 we get F [,; + 3;] = ( + F [,; + ;]+ + ( ( ( + F [,; + ;] Agai usig (., fo a=, b=, c=+ ad z=, we have ( ( ( F [,; + ;] = ( + ( + F [,; + ;]+ + ( + ( ( F [,; +;] ( ( ( + + μ [ + ] = {( + F[, A; A; ]} [ - F[,; ; ] F[,; ; ]] ( This gives ( ( ( [ + μ = ] μ + μ + + [ + ] [ ] [ ] ( + μ [+] = [( + ( + ] μ [ ] + ( μ [], =,3... (9 Whee ( F[,; + ; ] = +! F[, A; A; ] μ ad [ ] ( F[,; + ; ] =! F[, A; A; ] μ [ ] 6. ESTIMATION METHODS I this sectio, we discuss the vaious estimatio methods fo size-bied logaithmic seies distibutio ad veify thei efficiecies. 6.. Method of Momets I the method of momets eplacig the populatio mea μ = by the coespodig sample mea ( i i = =, we get ˆ = ( 6.. Method of Maimum Lielihood Let X, X,... X be a adom sample fom size-bied logaithmic seies distibutio, the coespodig lielihood fuctio is give i L = ;, ( ( ( y =, whee y i L = ( The log lielihood fuctio ca be witte log L = log( + ( y log ad the lielihood equatio log L ( y = + = ( O solvig, we get the maimum lielihood estimate ˆ =, which coicides with the momet estimate.

4 O Size-Bied Logaithmic Seies Distibutio The Ope Statistics ad Pobability Joual, 9, Volume Bayesia Method of Estimatio Sice < <, theefoe we sume that pio ifomatio about is fom beta distibutio. Thus f ( b ( (, a = ; < <,a>, b>. (3 B a b The posteio distibutio fom ( ad (3 ca be witte ( / y = ( ( + b y+ a + b y+ a d The Bayes estimato of is give ˆ = ( /y d ˆ = ( ( + b y+ a d + b y+ a d ( y+ a = ( y + a + b Fo a=b=, we get momet ad mle estimate. Thus Bayes estimato wos a geeal estimate i compaiso to momet ad mle estimato. 7. COMPUTER SIMULATION AND CONCLUSIONS It is vey difficult to compae the theoetical pefomaces of diffeet estimatos poposed i the pevious sectio. Theefoe, we pefom etesive simulatios to compae the pefomaces of diffeet methods of estimatio maily with espect to thei bies ad the mea squaed eos (MSE's, fo diffeet sample sizes ad diffeet paametic values. Regadig the choice of values of (a, b i Bayes estimato ˆ, thee w o ifomatio about thei values ecept that they ae eal ad positive umbes. Theefoe, combiatios of values of (a, b wee cosideed fo a, b=,,3,, ad those values of a, b wee selected fo which the Bayes estimato h miimum vaiace. It w foud that fo a=b=, the Bayes estimato h miimum vaiace ad values betwee the simulated sample fequecies ad the estimated Bayes fequecies wee let. I Table., we have fitted the LSD (, SBLSD (7 to some zeo-tucated distibutio of 3 biologists accodig to the umbe of eseach papes to thei cedit i the eview of Applied Etomology, vol, 936 (see Williams []. Data give i Table. is o species fequecy distibutio of isect catches fom Kempto []. As evidet fom the data, the umbe of moths pe species is 8+, hece the data h a vey log tail. Cosequetly, the fit by simple logaithmic seies distibutio is poo is evidet fom the value of the Peo's chi-squae. Howeve, the fit give by size-bied logaithmic seies distibutio eflects that the size-bied pheomeo is woig i the sese that if species with moe ad moe moths ae icluded i the study, the those species will have a highe pobability of beig epeseted i the sample. Table.. Distibutio of 3 Biologists Accodig to the Numbe of Reseach Papes to thei Cedit i The Review of Applied Etomology, Vol., 936 No. of Papes Pe Autho No. of Authos Epected Fequecy LSD SBLSD MLE Bayes Total ˆ

5 7 The Ope Statistics ad Pobability Joual, 9, Volume Khushid Ahmad Mi Table.. Species Fequecy Distibutio of Isect Catches fom the Tap at Rathamsted Epected Fequecy Moths pe Species Obseved Fequecy LSD SBLSD MLE Bayes Total ˆ ACKNOWLEDGEMENT The autho is highly thaful to the edito ad the two aoymous efeees fo thei valuable suggestios. REFERENCES [] F.F. Stepha, The epected value ad vaiace of the ecipocal ad othe egative powes of a positive Beoullia vaiate, Aals of Mathematical Statistics, vol. 6, pp. -6, 9. [] E.L. Gab ad I.R. Savage, Tables of the epected value of /X fo positive Beoulli ad Poisso vaiables, Joual of Ameica Statistical Associatio, vol. 9, pp.69-77, 9. [3] W. Medehall, ad E.H. J. Lehma, A appoimatio to the egative momets of the positive biomial useful i life testig, Techometics, vol., pp. 7-, 96. [] Z. Govidaajulu, The ecipocal of the decapitated egative biomial vaiable, Joual of Ameica Statistical Associatio, vol. 7, pp , 96. [] Z. Govidaajulu, Recuece elatios fo the ivese momets of the positive biomial vaiable, Joual of Ameica Statistical Associatio, vol. 8, pp , 963. [6] M.L. Tiu, A ote o the egative momets of a tucated Poisso vaiate, Joual of Ameica Statistical Associatio, vol. 9, pp. -, 96. [7] D.D. Stacu, O the momets of egative ode of the positive Beoulli ad Poisso vaiables, Studia Uivesitis Babes Bolyai Seies, Mathematics ad Physics, vol., pp. 9-3, 968. [8] M.T. Chao ad W.E. Stawdema, Negative momets of positive adom vaiables. Joual of Ameica Statistical Associatio, vol. 67, pp. 9-3, 97. [9] R.C.Gupta, Modified powe seies distibutio ad some of its applicatios, Sahya, vol. 36, pp , 97. [] R.C.Gupta, Estimatig the pobability of wiig (losig i a gambles ui poblem with applicatios, Joual of Statistical Plaig ad Ifeece, vol. 9, pp. -6, 98. [] N. Cessie, A.S. Davids, J.L. Fols ad G.E. Policello, The momet geeatig fuctio ad egative itege momets. Ameica Statisticia, vol. 3, pp.8-, 98. [] N. Cessie ad M. Boet, The momet geeatig fuctio h its momets. Joual of Ameica Statistical Associatio, vol. 3, pp , 986. [3] A. Roohi, Upublished M.Phil dissetatio, Uivesity of Lahoe, Lahoe,. [] Y. Lepage, Negative factoial momets of positive adom vaiables, Idustial Mathematics, vol. 8, pp. 9-, 978. [] M.C. Joes, Ivese factoial momets, Statistics ad Pobability Lettes, vol. 6, pp. 37-, 987. [6] D.J. Best, J.C.W. Raye ad O. Th, Test of fit fo the logaithmic distibutios. Joual of Applied Mathematics ad Decisio Scieces, vol. 8, pp. -8, 8. [7] M. Sadile, Liig the egative biomial ad logaithmic seies distibutios via thei sociated seies. Revista Colombiaa de Estadistica, vol. 3, pp. 3-39, 8. [8] R. Shaumugam ad J. Sigh, A chaacteizatio of the logaithmic seies distibutio ad its applicatios. Commuicatios i Statistics, vol. 3, pp , 98. [9] N.L. Johso, S. Kotz ad A. W. Kemp, Uivaiate discete distibutios, 3 d ed. Joh Wiley & Sos ic; Hoboe, New Jesey,. [] E.D. Raiville, Special Fuctios, Chelsa publishig compay, Bo New Yo, 96. [] C.B. William, The umbe of publicatios witte by biologists, Aals of Eugeics, vol., pp. 3-6, 9. [] R.A. Kempto, A geealized fom of Fishe's logaithmic seies. Biometia, vol. 6, pp. 9-38, 97. Received: Jue, 9 Revised: August 3, 9 Accepted: August 9, 9 Khushid Ahmad Mi; Licesee Betham Ope. This is a ope access aticle licesed ude the tems of the Ceative Commos Attibutio No-Commecial Licese ( which pemits uesticted, o-commecial use, distibutio ad epoductio i ay medium, povided the wo is popely cited.