Study of Correlation using Bayes Approach under bivariate Distributions
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1 Iteratoal Joural of Scece Egeerg ad Techolog Research IJSETR Volume Issue Februar 4 Stud of Correlato usg Baes Approach uder bvarate Dstrbutos N.S.Padharkar* ad. M.N.Deshpade** *Govt.Vdarbha Isttute of Scece Ad Humates Amravat.M.S **E -Drector Isttute of Scece Nagpur M.S. Abstract Coeffcet of correlato plas a major role statstcal aalss. It s used as a measure of etet of assocato betwee the two radom varables uderstud. Preset artcle cotas the stud of correlato Baesa framework for dscrete bvarate dstrbutos where oe varable s treated as a parameter. Estmators for lear ad o-lear fuctos of the parameter are derved ad correlatos betwee them are studed.. Itroducto I statstcal methodolog Pearso s coeffcet s oe the most mportat cocept. Ths cocept s frst troduced b Sr R. A. Fsher. It s frequetl used as a measure of assocato betwee the varables regresso aalss prcpal compoet aalss ad observatoal studes. Ths cocept has bee cosdered Baesa framework probabl for the frst tme b Dasgupta et al..the emploed the cocept of correlato Baesa framework ad studed the correlatos betwee the fuctos of radom varables ad parameter θ for uvarate dstrbuto havg a parameter θ. I Baesa framework parameter θ s treated as a radom varable havg some pror dstrbuto. The studed the correlato betwee two arbtrar fuctos of the data ad parameter θ. Takg oe of the varables as θ ad other varable as estmator of θ correlatos betwee parameter ad varous estmators are also studed. The also studed correlato betwee two estmators of the same parameter. Ths work for uvarate dstrbutos gave us a ew sght to stud the correlato ad motvated us to thk for the bvarate dstrbutos. Ths stud s carred out for dscrete bvarate dstrbutos. We have cosdered those dscrete bvarate dstrbutos whose margals are Bomal Negatve Bomal hece partcular Geometrc Hpergeometrc ad Posso dstrbutos. S tpes of dscrete bvarate dstrbutos are cosdered for ths stud. For these dstrbutos the cocept of correlato betwee parameter ad ts estmator Baesa framework s emploed the followg wa. For the bvarate dstrbuto of radom varable s cosdered as parameter ad t has a pror dstrbuto whch volves a parameter θ sgle parameter case.we assume that θ s kow. Takg the loss fucto as squared error loss Baes estmator of ad that of are obtaed. The the followg coeffcets of correlato are computed ad studed. ISSN: All Rghts Reserved 4 IJSETR 86
2 Iteratoal Joural of Scece Egeerg ad Techolog Research IJSETR Volume Issue Februar 4 Coeffcet of correlato betwee ad Baes estmator of. Coeffcet of correlato betwee ubased estmator of ad Baes estmator of. Coeffcet of correlato betwee ubased estmator of ad Baes estmator of.. S Bvarate Dstrbutos The stud of correlato s carred out for s tpes of bvarate dstrbutos of the tpe gve below: I. Bvarate dstrbutos havg sgle parameter ad θ ad << <<. Here rages of varables are depedet of each other. We refer to ths as bvarate dstrbuto of tpe I. II. Bvarate dstrbuto havg sgle parameter θ ad rage of oe varable depeds o other varable. We refer to ths as bvarate dstrbuto of tpe II. III. Bvarate dstrbuto havg two parameters ad << << that s rage of radom varables are depedet of each other. We refer to ths as bvarate dstrbuto of tpe III. IV. Dscrete bvarate dstrbuto havg two parameters ad rage of oe varable depeds o the other varables. We refer to ths as bvarate dstrbuto of tpe IV. V. Bvarate dstrbuto havg three parameters ad rage of oe varable s depedet of other varable. We refer to ths as bvarate dstrbuto of tpe V. VI. Bvarate dstrbuto havg three parameters rage of oe varable s depedet of other varable ad varables are sum of the two mutuall depedet radom varables. We refer to ths as bvarate dstrbuto of tpe VI. These s bvarate dstrbuto alog wth ther probablt mass fuctos ad rages of radom varables are gve the followg table. ISSN: All Rghts Reserved 4 IJSETR 87
3 Iteratoal Joural of Scece Egeerg ad Techolog Research IJSETR Volume Issue Februar 4 88 ISSN: All Rghts Reserved 4 IJSETR Table.: S tpes of bvarate dstrbuto Tpe of dstrbuto Probablt mass fucto Rage of ad I Otherwse II Otherwse III = =- - < < + < = Otherwse IV + > ad < < ad teger Otherwse V N =.m =.m Ma- Otherwse VI m!!! e Otherwse.!!!.!
4 Iteratoal Joural of Scece Egeerg ad Techolog Research IJSETR Volume Issue Februar 4. Correlato Baesa framework Now we stud correlato Baesa stle for the above dscrete bvarate dstrbutos where we treat radom varable as a parameter havg a pror dstrbuto whch volves parameter. Hece the jot probablt fucto of ca be epressed as p = p/.h where p/ s the probablt mass fucto of gve ad h s pror dstrbuto of whch volves parameter. Here we derve ubased estmator of Baes estmator of ubased estmator of ad Baes estmator of ad the stud ther correlatos. We cosder squared error as loss fucto. We have derved codtoal dstrbuto of gve p / ad posteror dstrbuto of that s p /=.Those are gve the followg Table. Tpe of Dstrbuto Table.: p / ad p /= Dstrbuto of gve p / Posteror Dstrbuto of p /= I NB+.θ NB+-θ+ θ II NB+θ B θ /+θ+ θ III NB+ - θ NB+ - θ IV Bθ - θ /+θ Bθ - θ /+θ V HGθ +θ θ - HGθ +θ θ - VI B θ /θ +θ +Pθ B θ /θ +θ +Pθ Ubased estmator ad Baes estmator of ad for all the s dstrbutos ca be derved easl. Those are dsplaed the table. ISSN: All Rghts Reserved 4 IJSETR 89
5 Iteratoal Joural of Scece Egeerg ad Techolog Research IJSETR Volume Issue Februar 4 9 ISSN: All Rghts Reserved 4 IJSETR Table.: Ubased estmator of ad Baes estmator of p /=Tpe of dstrbuto Ubased estmator of Ŷ Baes estmator of B Ŷ I II III IV V VI Table.: Ubased estmator of ad Baes estmator of Tpe of dstr buto Ubased estmator of Ŷ Baes estmator of B Ŷ I II
6 Iteratoal Joural of Scece Egeerg ad Techolog Research IJSETR Volume Issue Februar 4 9 ISSN: All Rghts Reserved 4 IJSETR III IV V VI ] [ The coeffcets of correlato betwee parameter ts ubased estmator Ŷ ad ts Baes estmator B ˆ are derved for all the dstrbuto dscussed earler.those are lsted the followg table. Table.4: Coeffcets of correlato Tpe of dstrbuto coeffcet of correlato ˆ ˆ B ˆ ˆ B I II III
7 Iteratoal Joural of Scece Egeerg ad Techolog Research IJSETR Volume Issue Februar 4 IV V N N N N VI 4. Cocluso From all the dervatos ad dscussos prevous chapters we ote that for such tpe bvarate dstrbutos correlato betwee ubased estmator ad Baes estmator of s alwas ut. But correlato betwee ubased estmator ad Baes estmator of whch s a o lear fucto of s ot ut.we also coclude that the coeffcet of correlato betwee the parameter ad the estmator s postve. But f correlato betwee the parameter ad estmator δ s ut t does ot mpl that δ s a approprate estmator of the parameter θ but ts lear fucto [aδ+b] would be a approprate for some value of a ad b. Refereces Aderso T. W. 984: A troducto to multvarate aalss secod edto Wle easter prvate lmted. Berger J. O. 98: Statstcal decso theor ad Baesa aalss secod edto Sprger verlag. Campbell J. T. 98:The Posso correlato fucto Proceedgs of the Edburgh Mathematcal Socetseres vol Casella G. ad Berger R.L. : Statstcal Iferece secod edto Thomso Dubur. 5 Dasgupta A. Casella Dalampad M. Gaest C. Rub H. ad Strawderma W. E. : Correlato Baesa framework The Caada Joural of Statstcs vol.8.o Deshpade M.N. ad Welukar R.M. 6: A co tossg epermet ad etee dstrbutos Teachg Statstcs vol.8 o ISSN: All Rghts Reserved 4 IJSETR
8 Iteratoal Joural of Scece Egeerg ad Techolog Research IJSETR Volume Issue Februar 4 7 Johso N.L. ad Kotz S. 969: Dscrete DstrbutosAwle terscece publcatos. 8 JohsoN.L. KotzS. ad BalakrshaN. 997:Dscrete Multvarate dstrbutos Wle seres Probablt ad Statstcs. 9 Kocherlakota S.ad KocherlakotaK.. 99 : Bvarate dscrete dstrbutos Marcel Dekkar; Ic. Phatak A.G. ad Sreehar M. 98:Some Characterzatos of a bvarate geometrc dstrbuto Joural of Ida Statstcal Assocatovol Sreehar M. 5: Characterzatos va codtoal dstrbutos Joural of Ida Statstcal Assocato vol SreeharM.:Bvarate Geometrc Dstrbutos revsted Sc. Lett. Vol.7-6. ISSN: All Rghts Reserved 4 IJSETR 9
ρ < 1 be five real numbers. The
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