New Product-Type and Ratio-Type Exponential Estimators of the Population Mean Using Auxiliary Information in Sample Surveys

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Dorthy Patterson
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1 ISSN 68-8 Jourl of Sisics olum, 6. pp Absrc Nw roduc-tp d io-tp Epoil Esimors of h opulio M Usig Auilir Iformio i Smpl Survs Housil. Sigh, r Lshkri d Sur K. l This ppr ddrsss h problm of simig h populio m of h sud vribl usig iformio o uilir vribl. A clss of Epoil- Tp Esimors hs b suggsd log wih is propris udr lrg smpl pproimio. I is idifid h h usul Ubisd Esimor, roduc-tp d io-tp Epoil Esimors r mmbrs of h proposd clss of Epoil-Tp Esimors. I hs b show h h proposd clss of Epoil-Tp Esimors is mor ffici h h usul Ubisd Esimor d som isig Esimors. A mpiricl sud is crrid ou i suppor of h prs sud. Kwords Auilir vribl, Sud vribl, Bis, M squrd rror. Iroducio I is commo o us h uilir iformio h simio sg i ordr o obi improvd sims of h populio m of h sud vri. Ou of m, io, roduc d grssio mhods r good mpls i his co. Wh h uilir vribl is posiivl high corrld wih h sud vribl, h io mhod of simio is qui ffciv. O h ohr hd, if his corrlio is giv, h roduc mhod of simio c b mplod. School of Sudid i Sisics, ikrm Uivrsi, Ujji-56, Mdh rdsh, Idi School of Sudid i Sisics, ikrm Uivrsi, Ujji-56, Mdh rdsh, Idi School of Sudid i Sisics, ikrm Uivrsi, Ujji-56, Mdh rdsh, Idi

2 68 Housil. Sigh, r Lshkri d Sur K. l osidr fii populio U u, u,..., u of siz N. L, b h sud N d uilir vribls, rspcivl, kig vlus, o h i i h i ui U i i,,..., N of h populio. L, X b h populio ms of,, rspcivl. I is ssumd h h populio m X of h uilir vribl is kow. For simig h populio m of h sud vribl, simpl rdom smpl of siz is slcd wihou rplcm from h populio U. Th h lssicl io Esimor for h populio m is dfid b. whr,, is h sim of h io of h populio ms. i i d i i r h u-wighd smpl ms of d, rspcivl. This Esimor is ol ffici if h vribl d r srogl posiivl corrld. Th ordir roduc Esimor for is dfid b. X whr,. is h sim of h roduc of h populio ms, will of b usd if h wo vribls r supposd o b srogl givl corrld. I is o b od h h Esimor is du o obso 957 d rvisid b Murh 96. Bhl d Tuj 99 suggsd io d roduc-tp Epoil Esimors for h populio m rspcivl s X p. X d X p. X Th simpl Epsio Esimor for h populio m is i i.5

3 Nw roduc-tp d io-tp Epoil Esimors of h opulio M Usig Auilir Iformio i Smpl Survs is usd ohrwis. Murh 967, p. 7 poid ou h h vribili of h smpl m is usull lss h h of smpl m. If dos h coffici of vriio of likwis h of, h f d f whr, f / N is h smplig frcio, S X d S r h / populio cofficis of vriio for h wo vribls. N S N X N d S N. i i I follows h if, w hv, S S i i / ;.6 W suppos h h obsrvio o d r ll o-giv, so h h smpl d populio ms r ll posiiv. Murh 96 suggsd h us of if >,.7 if,.8 if <,.9 whr, S is h corrlio coffici bw h sud vribl d h uilir i N vribl d S N X Hr, of cours. i d s I his ppr moivd b Shi 979, w hv suggsd vri of h roduc d io-tp Epoil Esimors, wih h iio o improv i 69

4 7 Housil. Sigh, r Lshkri d Sur K. l hir fficic. W hv obid h Bis d M Squrd Error of h roposd Esimor o h firs dgr of pproimio d comprd wih hos of h wll kow mhods Simpl Epsio, io d roduc. A mpiricl sud is giv i suppor of h prs sud.. Th Suggsd Esimor Moivd b Shi 979, w driv h followig modifid Epoil-Tp Esimor for h populio m s X X M p X X X p X X p X X X M p. whr, is sclr usd s dsig prmr. I is worh mio h, for M d h for, M. Morovr, if is vr lrg M is lmos h sm s h i.. lim M lim p X X X p. X Somims, good guss of h vlu of / is vilbl from pilo smpl ps d, pric or ohrwis. I ohr prcicl siuios, h vlu of / m b kow or gussd o b i cri irvl. Usig such kowldg, o c giv suibl vlu o, h dsig prmr i ordr h h roposd Modifid Esimor will hv smllr M Squrd Error h h usul io, roduc, Epoil-Tp io, Epoil-Tp roduc d Simpl Epsio Esimor, rspcivl.

5 Nw roduc-tp d io-tp Epoil Esimors of h opulio M Usig Auilir Iformio i Smpl Survs. Smplig Bis d M Squrd Error of h Esimors: Followig Murh 967, w wri,, X such h E E d f, f E f E, E. W c rsobl suppos h h smpl siz is lrg o mk d <. Furhr, o vlid h firs ordr lrg smpl pproimio w r goig o obi, w suppos h h smpl siz is lrg ough o obi d smll so h h rms ivolvig d / or i dgr grr h wo will b gligibl; ssumpio which is usull o urlisic. Now prssig q.. i rms of d, w hv, M p p p.. whr,. Epdig h righ hd sid of q..., muliplig d glcig rms of s hvig powr grr h wo, w hv, 8 M d M.. 8 Tkig pcio o boh sids of q..., w g h Bis of h firs dgr of pproimio s 7

6 7 Housil. Sigh, r Lshkri d Sur K. l f B M B 8.. f whr, B d. Squrig boh sids of q... d glcig rms of s hvig powr grr h wo, w hv, M... Tkig pcio of boh sids of q..., w g h of M o h firs dgr of pproimio s f M M I..5 whr, f S r..6 Th M is miimum wh..7 Thus, h rsulig miimum of M is giv b mi. M..8 which quls o h pproim vric/ of h usul grssio Esimor ˆ X...9 lr. Efficic ompriso oicids wih is Tru Opimum lu: I is wll kow udr Simpl dom Smplig WihOu plcm SSWO schm h. Wh h Sclr or

7 Nw roduc-tp d io-tp Epoil Esimors of h 7 opulio M Usig Auilir Iformio i Smpl Survs f f r S.. To h firs dgr of pproimio, Biss d M Squrd Errors of h Esimors,,, d r rspcivl giv b B B.. B.. B B B.. 8 B B whr, f B From q...8,..,..5,..6,..7 d..8, w hv, mi. M.. mi. M mi... M.. mi. M.. mi. M.. I is obsrvd from q... o q... h h roposd clss of Esimors M is mor ffici h Th usul Ubisd Esimor ulss h corrlio bw h sud vribl d h uilir vribl is zro. W o h wh

8 7 Housil. Sigh, r Lshkri d Sur K. l i.. h wo vribls d r ucorrld boh h Esimors d M r qull ffici. Th usul io Esimor cp wh, h cs whr boh h Esimors d M r qull ffici. Th usul roduc Esimor cp wh h Esimors d M r qull ffici., h cs whr boh Th Bhl d Tuj 99 io-tp Epoil Esimor cp wh, h cs whr boh h Esimors d M r qull ffici. Th Bhl d Tuj 99 roduc-tp Epoil Esimor cp wh, h cs whr boh h Esimors d M r qull ffici. dos o oicid wih Is Tru Opimum lu: I cosquc of formul q..6, w hv, M r. Wh h Sclr or which is lss h zro if i.. if.. or quivll mi., m., From q...5 d q...6, w hv,

9 Nw roduc-tp d io-tp Epoil Esimors of h opulio M Usig Auilir Iformio i Smpl Survs 75 M which is lss h zro if i.. if.. or quivll, m.,. mi From q...5 d q...7, w hv, p M which is lss h zro if i.. if.. or quivll, m.,. mi From q...5 d q...8, w hv,

10 Housil. Sigh, r Lshkri d Sur K. l 76 M which is lss h zro if i.. if.. or quivll, m.,. mi From q...5 d q...9, w hv, M which is lss h zro if i.. if..5 or quivll, m.,. mi

11 Nw roduc-tp d io-tp Epoil Esimors of h opulio M Usig Auilir Iformio i Smpl Survs I follows from q...,.....,.. d..5 h h roposd Modifid Epoil Esimor M is mor ffici h Th usul Ubisd Esimor if or quivll mi., m., Th usul io Esimor if or quivll mi., m., Th usul roduc Esimor if or quivll mi., m., Th io-tp Epoil Esimor du o Bhl d Tuj 99 if or quivll mi., m., Th roduc-tp Epoil Esimor du o Bhl d Tuj 99 if or quivll mi., m.,. Accurc of Firs Ordr Approimios o s W hv lrd comprd h s of h roposd Esimor d h ohr Esimors, subjc o h firs ordr of pproimios. Hr, w id o mi h ccurc of hs pproimios b obiig h scod ordr pproimios o h s. W ssum h, s d h h smpl coms from lrg Bivri Norml populio s ohrwis vr complicd prssios r obid, s Shi 979, p.. For his cs, w hv, 77

12 Housil. Sigh, r Lshkri d Sur K. l 78 whr, b ij j i E,,,,, j i d b for j i Thus, o h scod ordr of pproimios, w hv, p M W do b, 8, d 8 6. h... M... Nglcig rm of s ' hvig powr grr h four, w hv, ] [ M or ] [ M. Tkig pcio of boh sids of q.., w g h bis of M s B M. Squrig boh sids of q.. d glcig rm of s ' hvig powr grr h wo, w hv, M.

13 Nw roduc-tp d io-tp Epoil Esimors of h opulio M Usig Auilir Iformio i Smpl Survs Tkig pcio of boh sids of q.., w g h M Squrd Error o h scod ordr of pproimio riig h rm up o fourh dgr i d/or, w hv, 6 M M I M I 6 I cs good guss of is vilbl, i q.., w hv, 79. wih =. Th puig M M I M I.6 Hr, w o h M I lr.8 I Isrig d i q.., w g h prssios for io- Tp d roduc-tp Epoil Esimors o h scod ordr of pproimio, rspcivl s 8 8 Ad whr, I I

14 8 Housil. Sigh, r Lshkri d Sur K. l I 5. d I 5. From q..6 d q..9, w hv, M which is posiiv if Furhr, from q..7 d q.., w hv, M which is posiiv if To compr M wih rgrssio Esimor lr w hv from q..9 d q..7 h for lrg smpl h formr r pproiml s ffici s h lr, if good guss is vilbl. I cs h smpl o vr lrg is drw from lrg Bivri Norml populio, h M Squrd Error of h usul grssio Esimor lr ; o h scod ordr pproimioi.. rms up o.[ochr 967, 7., p.98,, for Norml Disribuio] giv s lr.5 Also, for his cs, if good guss of is vilbl, w hv from q..6 d q..5 lr M 5 5 >

15 Nw roduc-tp d io-tp Epoil Esimors of h opulio M Usig Auilir Iformio i Smpl Survs 8 if Thus h supposiio ; s = d h h smpl coms from lrg Bivri Norml populio, h M Squrd Error of h usul io Esimor X d h roduc Esimor X o h scod dgr of pproimio r rspcivl giv b 6 I 6.7 d I.8 From q..9 d q..7, w hv, which is grr h zro if N, from q..9 d q..8, w hv, which is grr h zro if

16 8 Housil. Sigh, r Lshkri d Sur K. l Thus h io-tp d roduc-tp Epoil Esimors r br h h usul io d roduc Esimors, rspcivl, s log s h codiios q..9 d q.. r sisfid. 5. Empiricl Sud To mi h mris of h suggsd Esimor w hv cosidrd fiv url populio d ss. Th dscripios of h populio r giv blow opulio : [Murh 967] : Oupu, : Fid pil, N 8,, 5. 86, X. 66,. 5,.757,. 9,., f. 5. opulio : [Murh 967] : Oupu, : Numbr of workrs, N 8,, 5. 86, X. 85,. 5,.98,. 95,. 7, f. 5. opulio : [Ds 988] : Numbr of griculurl lborrs for 97, : Numbr of griculurl lborrs for 96, N 78,, 9. 68, X 5.,. 5,.698,. 7,. 65, f. 79. opulio : [Sl d Torri 96] : Log of lf bur i scs, : hlori prcg, N, 6,. 686, X. 877,. 7,.79,. 996,., f..

17 Nw roduc-tp d io-tp Epoil Esimors of h opulio M Usig Auilir Iformio i Smpl Survs opulio 5: [Mddl 977] : osumpio pr cpi, : Dfld pric of vl, N 6,, , X 75.,. 78,.986,. 68,. 576, f. 5. W hv compud h rg of d prc rliv fficicis of diffr Esimors,,,, d M of h populio m wih rspc o. Fidigs r compild i Tbls, d. I is obsrvd from Tbl, d h hr is ough scop of slcig h vlu of sclr or i ordr o g Esimors br h smpl m,,, d. W lso o h h roposd clss of Esimors M is br h oviol Esimors v if h sclr or slidr w from h ru opimum vlu of or. Lrgr gi i fficic is obsrvd if h sclr or movs roud h vicii of h ru opimum vlu. Ackowldgms Th uhors r grful o h rfrs d ssoci diors for hir comms h hlpd o improv h ppr. Tbl : g of i which roposd Esimor M is br h,, opulio g of i which roposd Esimor, d. 8 M is br h ommo rg of i which is br M h,,, d , -, , -, , -, , -, , -, , -, , -, , -.57,- -.57, -.57,-.57,.867 -, , -, ,.867, 5,6.59 -,8.59,.5 -,7.59,5.59 -,.5

18 8 Housil. Sigh, r Lshkri d Sur K. l Tbl : Es of Esimors,,, d wih rspc o. E, Esimor opulio Tbl : Es of M wih rspc o for diffr vlus of E M, opulio

19 Nw roduc-tp d io-tp Epoil Esimors of h 85 opulio M Usig Auilir Iformio i Smpl Survs frcs. Bhl, S. d Tuj,.K. 99. io d roduc-tp Epoil Esimors. Jourl Iformio Opimizio Scic,, Ds, A. K oribuio o h hor of smplig srgis bsd o uilir iformio, h.d. hsis submid o BK; Mohpur, Ndi, Ws Bgl, Idi.. Mddl,. S Ecoomrics. Mcrw, Nw ork, USA.. Murh, M. N. 96. roduc mhod of simio. Skh, 6, Murh, M. N Smplig Thor d Mhods. Sisicl ublishig Soci, lcu, Idi. 6. obso, D. S Applicios of Mulipr k-sisics o h hor of Ubisd io-tp Esimio. Jourl of h Amric Sisicl Associio, 5, Shi, A A ffici vri of h roduc d io Esimors. Sisic Nrldic,, Sl,.. D. d Torri, J. H. 96. ricipls d rocdur of Sisics. Mcrw, Nw ork, USA.

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