Bivariate Zero-Inflated Power Series Distribution

Transcription

1 Appled athematcs do:.436/am..7 ublshed Ole July ( Bvarate Zero-Iflated ower Seres Dstrbuto Abstract atl arut Krsha Shrke Dgambar Tukaram Departmet of Statstcs. V.. ahavdyalaya Kavathe ahakal Sagl Ida Departmet of Statstcs Shvaj Uversty Kolhapur Ida E-mal: Receved December 9 ; revsed ay ; accepted ay 5 ay researchers have dscussed zero-flated uvarate dstrbutos. These uvarate models are ot sutable for modelg evets that volve dfferet types of couts or defects. To model several types of defects multvarate osso model s oe of the approprate models. Ths ca further be modfed to corporate flato at zero ad we ca have multvarate zero-flated osso dstrbuto. I the preset artcle we troduce a ew Bvarate Zero Iflated ower Seres Dstrbuto ad dscuss ferece related to the parameters volved the model. We also dscuss the ferece related to Bvarate Zero Iflated osso Dstrbuto. The model has bee appled to a real lfe data. Exteso to k-varate zero flated power seres dstrbuto s also dscussed. Keywords: Bvarate Zero-Iflated ower Seres Dstrbuto Bvarate Zero-Iflated osso Dstrbuto K-Varate Zero-Iflated ower Seres Dstrbuto. Itroducto I a maufacturg process there may exst several types of (say m) defects for example solder short crcuts solder vods absece of solder etc. o oe prted crcut board. These defects cause dfferet types of product falure ad geerate dfferet types of equpmet problems. I the above example there ca be oly oe type of defect whch occurs more frequetly ad the other defects occurs very rarely. Aother stuato could be both types of defects occur rarely ad so o. To model several types of defects multvarate osso model s oe of the approprate models to use. Ths ca further be modfed to corporate flato at zero ad we ca have multvarate zero-flated osso (ZI) dstrbuto. There are several ways to costruct ZI dstrbutos. I the lterature Ch-Shag et al. [] have dscussed varous types of ZI models ad vestgated ther dstrbutoal propertes. Deshmukh ad Kasture [] have studed bvarate dstrbuto wth trucated osso margal dstrbutos. Gupta et al. [3] have cosdered flated dstrbutos at the pot zero ad studed the structural propertes of the flated dstrbuto. Gupta et al. [4] have dscussed score test for zero-flated geeralzed osso regresso model. Holgate [5] descrbed the estmato of covarace parameter of bvarate osso dstrbuto by teratve method. Lambert [6] cosdered zero-flated osso regresso model. Laxmarayaa et al. [7] have studed bvarate osso dstrbuto ad the dstrbutoal propertes of the model. atl ad Shrke [8] studed testg parameter of the power seres dstrbuto of a zero-flated power seres model. atl ad Shrke [9] also studed equalty of flato parameters of two zero-flated power seres dstrbutos. It appears that majorty of the study the lterature s restrcted to osso dstrbuto ad ts exteso to multvarate set up. Relatvely less has bee reported for the famly of dstrbutos cotag other dstrbutos. I ths artcle we troduce a ew Bvarate Zero-Iflated ower Seres Dstrbuto (BZISD) ad dscuss ferece related to the parameters volved the same. The rest of the paper s orgazed as follows. Secto troduces the BZISD alog wth momets of the same. Secto 3 deals wth ferece related to the parameters volved the BZISD. I Secto 4 we dscuss ferece related to Bvarate Zero-Iflated osso Dstrbuto (BZID). The data set reported by Arbous ad Kerrch [] s modeled by Bvarate Zero Iflated osso Dstrbuto. The paper cocludes wth geeralzato to multvarate setup. Copyrght ScRes.

2 .. KRISHNA ET AL. 85. Bvarate Zero-Iflated ower Seres Dstrbuto Let ad Y be two radom varables wth probablty mass fuctos x x y T. x ax ad y f f b y where T s the commo support of ad Y x a (.) (.) f a x y by. f b y Defe x y x y Y g x E g gy E gy where g x ad g ( ) We ote that Y y are bouded fucto o (.). x y s a proper bvarate dstrbuto for a sutable choce of. Based o the dstrbuto (.) the followg we troduce three types of BZISD. Type-I BZISD: Whe there s a flato oly at y we defe the BZISD as x π πy x y π x yπ π Y x yπ x y (.) Type-II BZISD: Whe there s flato at compoet oly we defe the BZISD as π πy y x y π x yπ π Y x yπ x y (.3) Type-III BZISD: Whe there s flato at Y - compoet oly we defe the BZISD as ππ Y x x y x x x yπ π Y x yπ x y 3 π (.4) I the preset dscusso we focus oly o Type-IBZISD omet Geeratg Fucto results o the remag two ca be obtaed aalogously. The momet geeratg fucto of (Y) s t ty t t E e Y t t Y ty t ππ t t E e g t E g E e g Y t E g Y Therefore we have t Y t ππt Y t Y t ππ t t ad t (.6) where Y are the momet geeratg fuctos of radom varables ad Y of zeroflated power seres dstrbuto ad t ad t are the momet geeratg fuctos of radom varables havg power seres dstrbuto wth parameters ad respectvely. (.5) f j Supp ose f j ad f j deote ad f j respectvely for j. Ths gves us E π π E Y Y f f f f π f Var f( ) f f f Copyrght ScRes.

3 86.. KRISHNA ET AL. f π Var f( ) f f f ad the correlato coeffcet s f f f e f e f f e f e f e f e f πf πf f f f f f f (.7) 3. Estmato of the arameters of BZISD Let Y 3 be a radom sample observed from BZISD π. The lkelhood fucto for the observed radom sample s gve by. L π ; x y ππy a (3.) π x y Y where a f x y ad a otherwse. The correspodg log lkelhood fucto s gve by a lo g L π ; x y log π π Y (3.) a log π a log x y Y log L log L log L log L ad π gve the followg equatos. a Y (π) π π Y π Y ππ Y Y x y a Y x y Y π π π ( ) Y Y x y a Y x y (3.3) (3.4) (3.5) Y π π π ( ) Y Y x y a Y x y (3.6) where Y (.) deote Y (.) Solvg Equatos (3.3) to (3.6) smultaeously we get maxmum lkelhood estmators of the desred four parameters. We ote that all the four lkelhood equatos are o-lear ature ad do ot have closed form soof BZISD luto. Now we dscuss a partcular case amely BZID. 4. Bvarate Zero-Iflated osso Dstrbuto Let us set a x x! by y! g y e y f e f e g x e x the model (.). The we get BZID wth probablty mass fucto. x Y y c π πe e e xy ( ) x y πe x c y c e e e e xy!! where c e c xy The momet geeratg fucto of (Y) s tx e x e y x y t y π π g x E g. g y E g Y (4. ) (4.) Copyrght ScRes.

4 It s clear from the expressos of momet geeratg fuctos of ad Y that the margal dstrbutos of ad Y are uvarate zero-flated power seres dstrbutos wth parameters π ad π respectvely. Further we have π ad EY π ππ Var Y E.. KRISHNA ET AL. 87 Var Estmato of the arameters of BZID ad π π Suppose x y; s a radom sample observed The correlato coeffcet s turs out to be from BZID π ;. The the lkelhood fucto s gve by c c e (4.3) Lπ ; x y π π a ( ) c c ( ) πe e e Remark : Whe there s o flato π the cora c relato coeffcet s gve by c e x y πe x c y c e e e e x! y! whch cocdes wth the correlato coeffcet gve by Laxmaraya et al. [8] (4.4) Remark : If we choose g (.) to be ay ot her sut- where a f x y ad a otherwse. The correspodg log lkelhood s gve by able bouded fucto we wll have dfferet form of BZID. Some other possble fuctos ca be z g z a a ; g( z) e z etc. Remark 3: If we get Bvarate Zero-Iflated osso dstrbuto based o two depedet radom varables. c c e log L π ; x y log π π e e a log(π) a x c y c ax log ay log a log x! a log y! a log e e e e (4.5) The mles of the parameters ca be obtaed by solvg equatos log L log L log L ad π c c e c c e e e e π πe log L smultaeously. These equatos are gve the followg: a π (4.6) ( ) c c c c c c ππe e e π e c e e e e c y c e e e c c e e e e ax c a x y (4.7) ( ) c c c e c e c c ππe e e π e c e e c x c e e e c c e e e e ay c a x y (4.8) Copyrght ScRes.

5 88 ( ) c c c c e e e x c y c e e e e c c e e e e π e e e ππ a x y.. KRISHNA ET AL. (4.9) From the above equatos t s clear that Equatos (4.6) to (4.9) are o-lear ature. Solvg these equatos s computatoally cumbersome. Laxmaraya et al. [7] adopt method of momets for the model wthout flato parameter (.e. π ). I ther model they have used estmates based o ethod of omet Estmators (E) whch cocde wth axmum Lkelhood Estmators (LE) of the margal dstrbutos. Ths s ot the case for the jot dstrbuto. We have to solve four equatos smultaeously order to get the LEs. I the followg we obta maxmum lkelhood estmators for the followg example ad test for goodess of ft. 5. A Applcato The data set Table reported by Arbous ad Kerrch [] represets accdets sustaed by ralway me cosecutve perods of 6 ad 5 years. s the accdet dstrbuto of ralway me durg ad Y s the accdet dstrbuto of ralway me durg By assumg margal dstrbutos of s ZID π. The LEs of data are ˆπ.8938 ad ˆ.564 Smlarly assumg margal dstrbuto of Y s ZID π. The LEs of Y data are ˆπ.8494 ˆ.3. Usg these mles we ft the data of the margal dstrbuto of to ZID we get Ch square statstc = ad value = If we ft the data Table. Bvarate accdet dstrbuto of ralway me durg two perods. Y Total of the margal dstrbuto of Y to ZID we get Ch square statstc =.665 ad value =.436. The table value of (.5) = Therefore ZID fts well for ad Y data. Thus ow we ca test whether data s comg from BZID π. axmzg the log lkelhood the Equato (4.5) usg ATLAB R software we get maxmum lkelhood estmators of the parameters as π Wth these parameters we ft Bvarate Zero-Iflated osso Dstrbuto to the above data. The expected frequeces are as show the Table. From the ch-square goodess of ft we observed that calculated 4.6 s less tha the table value of (4.5) The value s Hece we coclude that Bvarate Zero-Iflated osso Dstrbuto fts well for the data. Remark 4: There ca be may ways to defe k-varate ZISD by extedg the above defed BZISD. Oe of the ways s gve below. A k-varate Zero-Iflated ower Seres Dstrbuto ca be defed as where x π x f x π π f x x π x k k k x x k g E g x s probablty ma ss fuct o of ower Se-. res Dstrbuto Iferece related to the parameters volved ths model ca be attempted smlarly. I the preset work we troduced a ew bvarate zero-flated power seres dstrbuto. Ths dstrbuto ca accommodate umber of zero-flated bvarate dscrete dstrbutos. Further work uder cosderato s testg of depedece for BZISD. Applcato of the proposed model for some other dstrbutos lke Bvarate Zero-Iflated Negatve Bomal Dstrbuto or k-varate zero flated osso dstrbuto ca also be Table. Expected frequeces usg BZID. Y Total Total Total Copyrght ScRes.

6 .. KRISHNA ET AL. 89 cosdered. These models are useful to model zero-flated do:.8/sta-6576 bvarate data. [5]. Holgate Estmato for the Bvarate osso Dstr- buto Bometrka Vol. 5 No pp Refereces [6] D. Lambert Zero-Iflated osso Regresso wth a Applcato to Defects aufacturg Techometrcs [] L. Ch-Shag K. Kyugmoo J.. eterso ad. A. Vol. 34 No. 99 pp. -4. do:. 37/69547 Brkley ultvarate Zero-Iflated osso odels [7] J. Lakshmarayaa S. N. N. adt ad K. Srvasa ad Ther Applcatos Techometrcs Vol. 4 No. Rao O a Bvarate osso dstrbuto Commucato Statstcs: Theory ad etords Vol. 8 No. 999 pp do:.37/799 [] S. R. Deshmukh ad. S. Kasture Bvarate Dstrbu- 999 pp Vol. ower Seres Dstrbuto of a Zero-Iflated ower Seres to wth Trucated osso argal Dstrbutos [8]. K. atl ad D. T. Shrke Testg arameter of the Commucato Statstcs: Theory ad etords 3 No. 4 pp odel Statstcal ethodology Vol. 4 No. 4 7 pp. do:.8/sta do:.6/j.stamet.6.. [3]. L. Gupta ad R. C. Trpath Iflated odfed ower [9]. K. atl ad D. T. Shrke Tests for Equalty of Iflato arameters of Two Zero-Iflated ower Seres Ds- Seres Dstrbutos wth Applcatos Commucato Statstcs: Theory ad etords Vol. 4 No trbutos Commucatos Statstcs: Theory ad pp do:.8/ ethods Vol. 4 No. 4 pp [4] R. L. Gupta ad R. C. Trpath Score Test for Zero- do:.8/ Iflated Geeralzed osso Regresso odel Commucato Statstcs: Theory ad etords Vol. 33 the Cocept of Accdet roeess Bometrcs Vol. 7 [] A. G. Arbous ad J. E. Kerrch Accdet Statstcs ad No. 4 pp No. 95 pp do:.37/3656 Copyrght ScRes.