Applied Mathematical Sciences, Vol. 2, 2008, no. 9, Parameter Estimation of Burr Type X Distribution for Grouped Data

Transcription

1 pplied Mathematical Scieces Vol 8 o Paamete stimatio o Bu Type Distibutio o Gouped Data M ludaat M T lodat ad T T lodat 3 3 Depatmet o Statistics Yamou Uivesity Ibid Joda aludaatm@hotmailcom ad malodat@yueduo bstact I this pape we obtai Bayesia ad o-bayesia estimatos o the paamete o the Bu type distibutio ude ouped data We apply ou esults to atiicial data eywods: Bu type Bayesia estimatio Gouped data Posteio distibutio Itoductio I vaious ields o sciece such as bioloy eieei ad medicie it is ot possible to obtai the measuemets o a statistical expeimet exactly but is possible to classiy them ito itevals ectales o disoit subsets lodat ad l-saleh eita 989 Sules ad Padett Wu ad Pelo 5 Pippe ad Ritz 6 Fo example i lie testi expeimets we obseve the ailue time o a compoet to the eaest hou day o moth Data o which tue values ae ow oly up to subsets o the sample space ae called ouped data I eeal ouped data ca be omulated as ollows: Let be a adom sample om the desity x x χ Θ ad χ χ be a patitio o the sample space χ ad N the umbe o ' s that all i χ o The set o pais χ N χ N is called ouped data The ouped data poblem is to use { } these data to daw ieeces about the paamete Sice we do t have complete iomatio about the sample the thee will be a loss i the iomatio due to the oupi Schevish 995 pae 4 shows the ollowi I I { I Y } Y Y

2 46 M ludaat M T lodat ad T T lodat whee I ad I ae the Fishe's iomatio umbes obtaied om ad Y espectively ad { I } Y Y Y is the coditioal scoe uctio I we eplace N N the I IY o all which Y by the ouped sample meas that the iomatio i the sample about is educed to I because o oupi uldo 96 cosideed o-bayesia estimatio om ouped data whe the data come om omal ad expoetial distibutios lodat ad l-saleh cosideed the Bayesia estimatio om ouped data whe the udelyi distibutio is expoetial lodat et al 7 obtaied Bayesia pedictio itevals om ouped data whe the udelyi distibutio is expoetial I this pape we coside the oup data poblem whe the desity x is Bu type ie - - exp-x x > > x x exp x The coespodi distibutio uctio is x < F x exp x x The Bu type is a membe o a system o distibutios itoduced by Bu 94 It is oud that this distibutio its may pactical data Raqab 6 Fiue shows the shape o x o dieet values o Fiue: pd o Bu o dieet values o

3 Paamete estimatio o Bu type distibutio 47 This pape is oaized as ollows I Sectio we deive the lielihood desity o the ouped data I Sectio 3 we id the ML Maximum Lielihood stimato o om ouped data I Sectio 4 we deive the Fishe's iomatio umbe based o the ouped data ad we use it to costuct a lae sample coidece iteval o I Sectio 5 we obtai a Bayes estimato o usi ouped data I Sectio 6 we discuss the estimatio o the hypepaametes I Sectio 7 we obtai a PDCR ih posteio Desity Cedible eio o I Sectio 8 we coducted a simulatio study to compae the estimatos Ou coclusio is stated i Sectio 9 Lielihood o ouped data Let be a adom sample om Bu ssume that the sample space o x is patitioed ito equally-spaced itevals as ollows Let I δ δ ad I [ δ δ > I N deotes the umbe o ' s that all i I the N N Let P P P I P δ < exp δ < δ exp δ o ad P P P > δ exp δ I we let lo exp δ the P exp exp o ad P exp So the desity o N N is ive by the multiomial distibutio as ollows:! P P!! C [ exp ] [ exp exp ] 3 whee C is a omalizi costat 3 ML om ouped data I this sectio we id the ML o based o the desity 3 To do this we maximize the lo-lielihood uctio lo cos ta t loexp exp lo exp The ist deivative o the lo-lielihood is lo exp exp exp exp exp exp The ML o is the solutio o lo / so the ML is such that

4 48 M ludaat M T lodat ad T T lodat exp exp exp exp exp exp We use the otatio to deote the ML o obtaied om the ouped data Numeical calculatios ae equied to id the solutio o the last equatio Its is easy to show that the ML o based o the u-ouped sample is exp lo i i 4 Fishe's iomatio umbe To id the Fishe's iomatio umbe cotaied i the ouped sample about we id the expectatio o the secod deivative o the lo-lielihood So N N lo whee ] exp [exp exp ad ] exp [ exp I I deotes the Fishe's iomatio umbe om ouped data the lo I Sice } { N P the 7 exp exp exp exp exp N N I Usi I we ca id a lae sample % coidece iteval o as ollows: / ± I Z

5 Paamete estimatio o Bu type distibutio 49 Simple calculatios ca show that the Fishe's iomatio umbe about i a adom sample om is / I 5 Bayesia estimatio I classical statistics we use the lielihood o the data ie L to estimate the paamete I Bayesia statistics we assume that has a pio distibutio say The we combie the lielihood data with the pio distibutio to et the posteio distibutio With espect to the squaed eo loss uctio the posteio mea is used to estimate sice it miimizes the posteio expected loss ie } { I this sectio we use the ollowi pio distibutio o to deive a estimate o exp > Γ Usi the Biomial theoem we ewite the lielihood uctio othe ouped data as ollows exp exp exp exp C C whee Combii the lielihood iomatio with the pio iomatio yields the posteio distibutio o ive ie

6 4 M ludaat M T lodat ad T T lodat exp d So we et exp whee The Bayesia estimate o with espect to the squaed eo loss uctio is the posteio mea ie b The vaiace o the posteio distibutio is used as a measue o pecisio o the Bayes estimato b The posteio vaiace o b is } { b b Va whee

7 Paamete estimatio o Bu type distibutio 4 6 stimatio o ad I eeal the paametes ad ae uow mpiical Bayesia methods use the maial desity o the data ie to estimate them The desity ca be oud as ollows: d C exp Γ C whee C is a costat does ot deped o ad Sice the last desity depeds o ad we may use this maial desity to estimate the paametes ad via the stadad statistical methods 7 PDCR o Bayesia % PDCR ih Posteio Desity Cedible Reio R o is a set deied by { } a > R ad : Subect to { } R P

8 4 M ludaat M T lodat ad T T lodat I is uimodal the poblem o idi R is equivalet to id the iteval [ u v] such that u v ad P { u v } Fidi u ad v equies umeical calculatios 8 pplicatio to atiicial data I this sectio we apply ou esults to a atiicial data eeated om Bu Type We eeated a sample o size 3 om Bu Type with So we et the ollowi sample: The we put this sample ito six itevals 5 with δ Usi ad b 5 we see that the ML o om the ouped data is 466 with Vaiace equal to We also see that I 7998 I ad the Bayes estimato is b 53 with vaiace % coidece iteval o is Coclusio I this pape we obtaied Bayesia ad o-bayesia estimatos o the paamete o the Bu type distibutio whe the data ae ive i oups The estimatos have o closed oms so we eed umeical methods to id them The applicatio to the atiicial data shows that the estimatos wo well cowledemets This eseach has bee suppoted by a at om Yamou Uivesity Reeeces [] mad ad B yma Iteval estimatio o the scale paamete o Bu type distibutio based o ouped data Joual o Mode pplied Statistical Methods [] M T lodat ad M F l-saleh Bayesia estimatio usi ouped data with applicatio to the expoetial distibutio Soochow J o Mathematics

9 Paamete estimatio o Bu type distibutio 43 [3] M T lodat M ludaat ad T T lodat Bayesia pedictio itevals om ouped data: expoetial distibutio bhath l-yamou 7 ccepted [4] I W Bu Cummulative equecy uctios o Math Statist [5] D eita Ieece om ouped data: a eview with discussio Statistical scieces [6] uldo stimatio om ouped ad patially ouped samples Joh Wiley Ic New Yo 96 [7] M Z Raqab ad D udu Bu type distibutio: evisited JPSS [8] M J Schevish Theoy o Statistics Spie-Vela New Yo Ic 995 [9] J G Sules ad W J Padett Ieece o eliability ad stess-leth o a scaled Bu type distibutio Lietime Data aalysis [] Wu ad J M Pelo "Chia s icome distibutio: 985- Review o coomitics ad Statistics [] C B Pippe ad C Ritz Chei the ouped data vesio o Cox model o iteval-ouped suvival data Scadiavia Joual o Statistics Received: Jue 5 7