Estimating the Population Mean in Stratified Population using Auxiliary Information under Non-Response

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1 Maoj Kr. Chaudhary, V. K. gh, ajesh gh Deartmet of tatstcs, Baaras Hdu Uversty Varaas-5, IDIA Floret maradache Deartmet of Mathematcs, Uversty of ew Mexco, Gallu, UA Estmatg the Poulato Mea tratfed Poulato usg Auxlary Iformato uder o-esose Publshed : ajesh gh, F. maradache (Edtors) UDIE I AMPLIG ECHIQUE AD IME EIE AALYI Z Publshg, Columbus, UA, IB

2 Abstract he reset chater deals wth the study of geeral famly of factor-tye estmators for estmatg oulato mea of stratfed oulato the resece of oresose wheever formato o a auxlary varable are avalable. he roosed famly cludes searate rato, roduct, dual to rato ad usual samle mea estmators as ts artcular cases ad exhbts some ce roertes as regards to locate the otmum estmator belogg to the famly. Choce of arorate estmator the famly order to get a desred level of accuracy eve f o-resose s hgh, s also dscussed. he emrcal study has bee carred out suort of the results. Keywords: Factor-tye estmators, tratfed oulato, o-resose, Otmum estmator, Emrcal study.. Itroducto I samlg theory the use of sutable auxlary formato results cosderable reducto varace of the estmator. For ths reaso, may authors used the auxlary formato at the estmato stage. Cochra (94) was the frst who used the auxlary formato at the estmato stage estmatg the oulato arameters. He roosed the rato estmator to estmate the oulato mea or total of a character uder study. Hase et. al. (953) suggested the dfferece estmator whch was subsequetly modfed to gve the lear regresso estmator for the oulato mea or ts total. Murthy (964) have studed the roduct estmator to estmate the oulato mea or total whe the character uder study ad the auxlary character are egatvely 4

3 correlated. hese estmators ca be used more effcetly tha the mea er ut estmator. here are several authors who have suggested estmators usg some ow oulato arameters of a auxlary varable. Uadhyaya ad gh (999) have suggested the class of estmators smle radom samlg. Kadlar ad Cg (3) ad habbr ad Guta (5) exteded these estmators for the stratfed radom samlg. gh et. al. (8) suggested class of estmators usg ower trasformato based o the estmators develoed by Kadlar ad Cg (3). Kadlar ad Cg (5) ad habbr ad Guta (6) have suggested ew rato estmators stratfed samlg to mrove the effcecy of the estmators. Koyucu ad Kadlar (8) have roosed famles of estmators for estmatg oulato mea stratfed radom samlg by cosderg the estmators roosed earls (964) ad Khoshevsa et. al. (7). gh ad Vshwaarma (8) have suggested a famly of estmators usg trasformato the stratfed radom samlg. ecetly, Koyucu ad Kadlar (9) have roosed a geeral famly of estmators, whch uses the formato of two auxlary varables the stratfed radom samlg to estmate the oulato mea of the varable uder study. he wors whch have bee metoed above are based o the assumto that both the study ad auxlary varables are free from ay d of o-samlg error. But, ractce, however the roblem of o-resose ofte arses samle surveys. I such stuatos whle sgle survey varable s uder vestgato, the roblem of estmatg oulato mea usg sub-samlg scheme was frst cosdered by Hase ad Hurwtz (946). If we have comlete formato o study varable ad comlete formato o auxlary varable, other words f the study varable s affected by o-resose error but the auxlary varable s free from o-resose. he utlzg the Hase-Hurwtz (946) techque of sub-samlg of the o-resodets, the covetoal rato ad roduct estmators the resece of o-resose are resectvely gve by ( x) (.) HH / 5

4 P HH. x /. (.) ad ( ) he urose of the reset chater s to suggest searate-tye estmators stratfed oulato for estmatg oulato mea usg the cocet of sub-samlg of o-resodets the resece of o-resose study varable the oulato. I ths cotext, the formato o a auxlary characterstc closely related to the study varable, has bee utlzed assumg that t s free from o-resose. I order to suggest searate-tye estmators, we have made use of Factor-ye Estmators (FE) roosed by gh ad hula (987). FE defe a class of estmators volvg usual samle mea estmator, usual rato ad roduct estmators ad some other estmators exstg lterature. hs class of estmators exhbts some ce roertes whch have bee dscussed subsequet sectos.. amlg trategy ad Estmato Procedure Let us cosder a oulato cosstg of uts dvded to strata. Let the sze of th stratum s, (,,..., ) ad we decde to select a samle of sze from the etre oulato such a way that uts are selected from the th stratum. hus, we have. Let the o-resose occurs each stratum. he usg Hase ad Hurwtz (946) rocedure we select a samle of sze m uts out of o-resodet uts the th (O) scheme such that the stratum wth the hel of smle radom samlg wthout relacemet m uts by tervew method. L m, L ad the formato are observed o all he Hase-Hurwtz estmator of oulato mea of study varable for the th stratum wll be x + x m, (,,..., ) (.) 6

5 where x ad x m are the samle meas based o resodet uts ad m oresodet uts resectvely the th stratum for the study varable. Obvously s a ubased estmator of. Combg the estmators over all the strata we get the estmator of oulato mea of study varable, gve by st (.) where. whch s a ubased estmator of. ow, we defe the estmator of oulato mea of auxlary varable as st x (.3) where x s the samle mea based o uts the th stratum for the auxlary varable. It ca easly be see that estmates of the oulato mea 3. uggested Famly of Estmators st s a ubased estmator of because of auxlary varable for the th stratum. x gves ubased Let us ow cosder the stuato whch the study varable s subjected to oresose ad the auxlary varable s free from o-resose. Motvated by gh ad hula (987), we defe the searate-tye famly of estmators of oulato mea usg factor-tye estmators as ( ) ( ) (3.) F F 7

6 where ( ) ( A + C) ( A + fb) + fbx F (3.) + Cx ad C ; >. f, A ( )( ), B ( )( 4), ( )( 3)( 4) 3. Partcular Cases of ( ) Case-: If the A B, C 6 F so that () F x ad hece F (). (3.3) x hus, () s the usual searate rato estmator uder o-resose. F Case-: If the A C, B so that ( ) F x ad hece ( ) F x (3.4) whch s the usual searate roduct estmator uder o-resose. Case-3: If 3 the A, B, C so that () 3 F f x ( f ) ad hece () 3 () 3 F (3.5) F 8

7 whch s the searate dual to rato-tye estmator uder o-resose. he dual to rato tye estmator was roosed by rveataramaa (98). Case-4: If 4 the A 6, B, C so that ( ) F 4 F 4 st ad hece ( ) (3.6) whch s usual mea estmator defed stratfed oulato uder o-resose. 3. Proertes of ( ) F Usg large samle aroxmato, the bas of the estmator ( ) order of aroxmato was obtaed followg gh ad hula (987) as B [ ] [ ( )] E ( ) F F F, u to the frst C ( ) C CC (3.7) A + fb + C where ( ) C fb, A + fb + C C, C, ad are the oulato mea squares of study ad auxlary varables resectvely the th stratum. s the oulato correlato coeffcet betwee ad the th stratum. he Mea quare Error (ME) u to the frst order of aroxmato was derved as M [ ] [ ( )] E ( ) F F ME F [ ( )] 9

8 3 ( ) ( ) ( ) ( ) ( ) + x Cov x V V,. ce ( ) L V +, ( ) x V ad ( ) x Cov, [ due to gh (998)]. where s the oulato mea square of the o-resose grou the th stratum ad s the o-resose rate of the th stratum the oulato. herefore, we have ( ) [ ] ( ) ( ) [ ] + F M ϕ ϕ + L (3.8) where. 3.3 Otmum Choce of I order to obta mmum ME of ( ) F, we dfferetate the ME wth resect to ad equate the dervatve to zero ( ) ( ) ( ) [ ], (3.9) where ( ) stads for frst dervatve of ( ).From the above exresso, we have

9 ( ) V (say). (3.) It s easy to observe that ( ) s a cubc equato the arameter. herefore, the equato (3.) wll have at the most three real roots at whch the ME of the estmator ( ) attas ts mmum. F Let the equato (3.) yelds solutos as, ad [ ] such that ( ) M s same. A crtero of mag a choce betwee, ad s that comute the bas of the estmator at, ad ad select ovel roerty of the FE. 3.4 educg ME through Arorate Choce of F ot at whch bas s the least. hs s a By usg FE for defg the searate-tye estmators ths chater, we have a advatage terms of the reducto of the value of ME of the estmator to a desred extet by a arorate choce of the arameter eve f the o-resose rate s hgh the oulato. he rocedure s descrbed below: ce ME s of the roosed strateges are fuctos of the uow arameter as well as fuctos of o-resose rates, t s obvous that f s tae to be costat, ME s crease wth creasg o-resose rate, f other characterstcs of the oulato rema uchaged, alog wth the rato to be sub samled the o-resose class, that s, L. It s also true that more the o-resose rate, greater would be the sze of the o-resose grou the samle ad, therefore, order to lowerg dow the ME of the estmator, the sze of sub samled uts should be creased so as to ee the value of L the vcty of ; but ths would, term, cost more because more effort ad moey would be requred to obta formato o sub samled uts through ersoal tervew method. hus, creasg the sze of the sub samled uts order to 3

10 reduce the ME s ot a feasble soluto f o-resose rate s suosed to be large eough. he classcal estmators such as HH,, P, dscussed earler lterature resece of o-resose are ot helful the reducto of ME to a desred level. I all these estmators, the oly cotrollg factor for lowerg dow the ME s L, f oe desres so. By utlzg FE order to roose searate- tye estmators the reset wor, we are able to cotrol the recso of the estmator to a desred level oly by mag a arorate choce of. Let the o-resose rate ad mea-square of the o-resose grou the th stratum at a tme be ad ME of the estmator would be resectvely. he, for a choce of, the [ ] [ ( ) ] + ( ) ( ) M F / + L (3.) Let us ow suose that the o-resose rate creased over tme ad t s such that >. Obvously, wth chage o-resose rate, oly the arameter wll chage. Let t becomes. he we have [ ] [ ( ) ] + ( ) ( ) M F / + L (3.) 3

11 33 Clearly, f ad > the ( ) [ ] ( ) [ ] F F M M >. herefore, we have to select a sutable value, such that eve f > ad >, exresso (3.) becomes equal to equato (3.) that s, the ME of ( ) F s reduced to a desred level gve by (3.). Equatg (3.) to (3.) ad solvg for ( ), we get ( ) ( ) ( ) ( ) { } ( ) + L, (3.3) whch s quadratc equato ( ). O solvg the above equato, the roots are obtaed as ( ) ± + ( ) ( ) { } ( ) L (3.4) he above equato rovdes the value of o whch oe ca obta the recso to a desred level. ometmes the roots gve by the above equato may be magary. o, order that the roots are real, the codtos o the value of are gve by

12 34 ( ) ( ) + > L (3.5) ad ( ) ( ) < L (3.6) 4. Emrcal tudy I ths secto, therefore, we have llustrated the results, derved above, o the bass of some emrcal data. For ths urose, a data set has bee tae to cosderato. Here the oulato s MU84 oulato avalable ardal et. al. (99, age 65, Aedx B). e have cosdered the oulato the year 985 as study varable ad that the year 975 as auxlary varable. here are 84 mucaltes whch have bee dvded radomly to four strata havg szes 73, 7, 97 ad 44. able shows the values of the arameters of the oulato uder cosderato for the four strata whch are eeded comutatoal rocedure. able : Parameters of the Poulato tratum () tratum ze ( ) Mea ( ) Mea ( ) ( ) ( ) ( )

13 he value of / comes out to be.9. e fx the samle sze to be 6. he the allocato of samles to dfferet strata uder roortoal ad eyma allocatos are show the followg table able : Allocato of amle tratum () Proortoal Allocato ze of amles uder eyma Allocato O the bass of the equato (3.), we obtaed the otmum values of : Uder Proortoal Allocato ( ).949, ot (3.9975,.68,.) ad Uder eyma Allocato ot ( ).957, ot (34.435,.64,.3). he followg table dects the values of the ME s of the estmators ( ) for, ad 4 uder roortoal ad eyma allocatos. A comarso of ME of F ( ) wth ot ad wth that at 4 reveals the fact that the utlzato of auxlary formato at the estmato stage certaly mroves the effcecy of the estmator as comared to the usual mea estmator. st F 35

14 able 3: ME Comarso ( L, % for all ) ME Allocato Proortoal eyma M [ ( ) ] F ot [ () ] M F M [ ( 4) ] [ ] F V st e shall ow llustrate how by a arorate choce of, the ME of the estmators ( ) creased. ca be reduced to a desred level eve f the o-resose rate s F Let us tae L,.,. 3 Uder Proortoal Allocato as 4 ad ( ) for all 3 From the codto (3.5) ad (3.6), we have codtos for real roots of ( ) ( ) >.57 ad ( ) < herefore, f we tae ( )., the for ths choce of ( ) M [ ( ) ] 3.7 ad M ( ) F [ ] F, we get hus, there s about 5 ercet crease the ME of the estmator f oresose rate s trled. ow usg (3.4), we get ( ).957 ad.85. At ths value of ( ), M [ F ( )] reduces to 3.7 eve f o-resose rate s 3 ercet. hus a ossble choce of may be made order to reduce the ME to a desred level. 36

15 Uder eyma Allocato Codtos for real roots of ( ) ( ) >.746 ad ( ) <.739. If ( ). the we have M [ ( ) ].4885 ad ( ) F [ ] M F 4.7. Further, we get from (3.4), ( ).6 ad.8435, so that M [ ( ) ].4885 for ( ) Cocluso F e have suggested a geeral famly of factor-tye estmators for estmatg the oulato mea stratfed radom samlg uder o-resose usg a auxlary varable. he otmum roerty of the famly has bee dscussed. It has also bee dscussed about the choce of arorate estmator of the famly order to get a desred level of accuracy eve f o-resose s hgh. he able 3 reveals that the otmum estmator of the suggested famly has greater recso tha searate rato ad samle mea estmators. Besdes t, the reducto of ME of the estmators ( ) 37 to a desred extet by a arorate choce of the arameter eve f the o-resose rate s hgh the oulato, has also bee llustrated. efereces Cochra,. G. (94): he estmato of the yelds of cereal exermets by samlg for the rato of gra total roduce. Jour. of he Agr. c., 3, Hase, M. H. ad Hurwtz,.. (946): he roblem of o-resose samle surveys. Jour. of he Amer. tats. Assoc., 4, Hase, M. H., Hurwtz,.. ad Madow,,. G. (953): amle urvey Methods ad heory, Volume I, Joh ley ad os, Ic., ew Yor. Kadlar, C. ad Cg, H. (3): ato estmators stratfed radom samlg, Bom. Jour. 45 (), 8 5. F

16 Kadlar, C. ad Cg, H. (5): A ew rato estmator stratfed samlg, Comm. tat. heor. ad Meth., 34, Khoshevsa, M., gh,., Chauha, P., awa,., maradache, F. (7): A geeral famly of estmators for estmatg oulato mea usg ow value of some oulato arameter(s), Far East Jour. of heor. tats.,, 8 9. Koyucu,. ad Kadlar, C. (8): ato ad roduct estmators stratfed radom samlg. Jour. of tats. Pla. ad If., 38, -7. Koyucu,. ad Kadlar, C. (9): Famly of estmators of oulato mea usg two auxlary varables stratfed radom samlg. Comm. tat. heor. ad Meth., 38, Murthy, M.. (964): Product method of estmato. ahya, 6A, ardal, C. E., wesso, B. ad retma, J. (99): Model Asssted urvey amlg, rger-verlag, ew Yor, Ic. earls, D.. (964): he utlzato of a ow coeffcet of varato the estmato rocedure. Jour. of he Amer. tats. Assoc., 59, 5-6. habbr, J. ad Guta,. (5): Imroved rato estmators stratfed samlg. Amer. Jour. of Math. ad Maag. c., 5 (3-4), habbr, J. ad Guta,. (6): A ew estmator of oulato mea stratfed samlg. Comm. tat. heor. ad Meth., 35, 9. gh, H. P., alor,., gh,. ad Km, J. (8): A modfed estmator of oulato mea usg ower trasformato. tat. Paer., 49, gh, H. P. ad Vshwaarma, G. K. (8): A Famly of Estmators of Poulato Mea Usg Auxlary Iformato tratfed amlg, Comm. tat. heor. ad Meth., 37, gh, L. B. (998): ome Classes of Estmators for Fte Poulato Mea Presece of o-resose. Uublshed Ph. D. hess submtted to Baaras Hdu Uversty, Varaas, Ida. 38

17 gh, V. K. ad hula, D. (987): Oe arameter famly of factor-tye rato estmators. Metro, 45 (-), rveataramaa,. (98): A dual to rato estmator samle surveys, Bometra, 67 (), Uadhyaya, L.. ad gh, H. P. (999): Use of trasformed auxlary varable estmatg the fte oulato mea, Bom. Jour., 4 (5),