COMBINED-TYPE FAMILY OF ESTIMATORS OF POPULATION MEAN IN STRATIFIED RANDOM SAMPLING UNDER NON-RESPONSE

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1 Joural of elablty ad tatstcal tudes; I (Prt): , (Ole): Vol. 5, Issue (): 33-4 COMBIED-YPE FAMILY OF EIMAO OF POPULAIO MEA I AIFIED ADOM AMPLIG UDE O-EPOE Maoj K. Chaudhary, V. K. gh ad. K. hula Deartmet of tatstcs, Baaras Hdu Uversty, Varaas-5, Ida Deartmet of tatstcs, Cetral Uversty of Bhar, B.I.. Camus, P.O.- B.V. College Pata-84, Ida (eceved July 8, ) Abstract he reset aer focuses o e study of combed-tye famly of estmators of oulato mea stratfed radom samlg e resece of o-resose. I s aer, we have suggested a famly of factor-tye estmators of oulato mea stratfed radom samlg uder o-resose usg a auxlary varable. he roertes of e famly have bee dscussed detal. he eoretcal results are also suorted by a emrcal study. Key ords: Combed-tye famly of estmators, Factor-tye estmators, tratfed radom samlg, Auxlary varable, o-resose.. Itroducto Auxlary formato recorded from e oulato elemets ca be successfully used to desg a maageable ad effcet samlg desg ad after samle selecto, to furer mrove e effcecy of estmators. If a cotuous auxlary varable s avalable.e. strogly correlated w e study varable, t s ossble to mrove e effcecy by usg rato meod of estmato or regresso meod of estmato. I ese meods, auxlary formato s cororated to e estmato rocedure usg statstcal models. he use of ese techques ca cosderably mrove e accuracy of estmates,.e. roduce estmates at are close to e corresodg oulato values ad addto, decrease e desg varaces of e estmators. here are several auors who have suggested estmators usg some ow oulato arameters of auxlary varable(s). Uadhyaya ad gh (999) have suggested e class of estmators smle radom samlg. Kadlar ad Cg (3) ad habbr ad Guta (5) exteded ese estmators for e stratfed radom samlg. gh et al () have suggested e geeral famly of rato-tye estmators systematc samlg. alor et al () have recetly roosed dual to rato-cumroduct estmator usg ow arameters of auxlary varables. Kadlar ad Cg (5) ad habbr ad Guta (6) have suggested ew rato estmators stratfed samlg to mrove e effcecy of e estmators. Koyucu ad Kadlar (8) have roosed famles of estmators for estmatg oulato mea stratfed radom samlg by cosderg e estmators roosed earls (964) ad Khoshevsa et al (7). gh ad Vshwaarma (8) have suggested a famly of estmators usg trasformato e stratfed radom samlg. Koyucu ad Kadlar (9) have

2 34 Joural of elablty ad tatstcal tudes, December, Vol. 5 () roosed a geeral famly of estmators, whch uses e formato of two auxlary varables e stratfed radom samlg to estmate e oulato mea of e study varable. I e reset aer, we have suggested a famly of combed-tye estmators of oulato mea stratfed radom samlg e resece of o-resose usg e formato avalable o a auxlary varable. he otmum roerty of e famly has bee dscussed. Choce of arorate estmator e famly order to get a desred level of accuracy eve f o-resose s hgh, s also dscussed.. amlg trategy ad Estmato Procedure Let us cosder a oulato cosstg of uts whch s dvded to strata. Let e sze of stratum be, (,,..., ) ad we select a samle of sze from e etre oulato such a way at uts are selected from e stratum. hus, we have. Let us assume at e stuato whch e oresose s observed o study varable ad auxlary varable s free from o-resose. It s observed at ere are resodet uts ad o-resodet uts e samle of uts for e stratum regardg e study varable. Usg Hase ad Hurwtz (946) rocedure, we select a sub-samle of sze m uts out of o- resodet uts w e hel of smle radom samlg wout relacemet (O) scheme such at Lm, L ad e formato are observed o all e m uts. Let X ad X be e study ad auxlary varables w resectve oulato meas X ad X. for e he Hase-Hurwtz estmator of oulato mea stratum, s gve by where x ad x + xm, (,,...,) x m are e samle meas based o o-resodet uts resectvely e X of study varable X (.) resodet uts ad m stratum for e study varable. Obvously s a ubased estmator ofx. Combg e estmators over all e strata, we get e ubased estmator of oulato meax of study varable, gve by where. st (.) ow, we defe e estmator of oulato mea X of auxlary varable as

3 Combed-ye Famly of Estmators of Poulato 35 st x (.3) wherex s e samle mea based o uts e stratum for e auxlary varable. It ca easly be see at st s a ubased estmator ofx because x gves ubased estmate of e oulato mea stratum. X of auxlary varable for e. uggested Famly of Estmators Motvated by gh ad hula (987), we suggest a famly of factor-tye estmators for estmatg e oulato mea X stratfed radom samlg uder o-resose as A+ C X + fbst st (..) ( A+ fb) X + Cst where A, B ad Care e fuctos of, gve by A ( )( ), B ( )( 4), C ( )( 3)( 4) ; > ad f... Partcular Cases of Case-: For, we have X ( ) st (..) st whch s usual combed rato estmator uder o-resose. Case-: If, e st st (..3) X whch s usual combed roduct estmator uder o-resose. Case-3: If 3, e X fst ( 3) st (..4) ( f) X whch s combed dual to rato-tye estmator uder o-resose. he dual to rato tye estmator was roosed by rveataramaa (98). Case-4: If 4, ( 4) (..5) st whch s usual mea estmator of stratfed oulato uder o-resose.

4 36 Joural of elablty ad tatstcal tudes, December, Vol. 5 ().. Proertes of Usg large samle aroxmato, e bas ad mea square error (ME) of u to e frst order of aroxmato ca be obtaed by e equatos (..6) ad (..7) resectvely. B [ X ] [ ( )] E X C C CC A+ fb+ C (..6) CfB where, C, C A+ fb+ C X X, ad are e oulato mea squares of study ad auxlary varables resectvely e stratum. s e oulato correlato coeffcet betwee X ad X e stratum. where M X [ X ] [ ( )] E V X ( ) st + ϕ V ( ) X st ϕ Cov (, ) X st [ ( ) +ϕ V( ) ϕ Cov ( )] X st V st st st, st X. X V( ) st ce V( ) st + ad Cov where ( ) Cov (, x) st L, st s e oulato mea square of e o-resose grou e ad s e o-resose rate of e have M +, stratum stratum e oulato. herefore, we [ ] [ ]

5 Combed-ye Famly of Estmators of Poulato 37 L +. (..7)..3 Otmum Choce of w resect to ad equatg e Dfferetatg ME of dervatve to zero, we get whch yelds ce [ ] V (..8) (say). (..9) s a cubc equato. o at for gve value ofv, equato (..9) may be solved so as to obta at e most ree real ad ostve values of for M would be mmum. [ ] whch..4 educg ME hrough Arorate Choce of By utlzg factor-tye estmators (FE) suggested by gh ad hula (987), order to roose combed- tye estmators e reset wor, we are able to cotrol e recso of e estmator to a desred level oly by mag a arorate choce of. Let e o-resose rate ad mea-square of e o- resose grou e he, for a choce of stratum at a tme be ad, e ME of e estmator would be [ ] + ( ) ( ) M resectvely. [ ] / L + (..) Let us ow suose at e o-resose rate creased over tme ad t s such at >. Obvously, w chage o-resose rate, oly e arameter wll chage. Let t becomes. he we have

6 38 Joural of elablty ad tatstcal tudes, December, Vol. 5 () [ ] [ ] + M / + L (..) Clearly, f ad > e [ ] [ ] F F M M >. herefore, we have to select a sutable value, such at eve f > ad >, exresso (..) becomes equal to equato (..) at s, e ME of s reduced to a desred level gve by (..). Equatg (..) to (..) ad solvg for, e arorate choce of e arameter, so as to reduce e mea square error of e combed-tye estmator to a desred level, may be obtaed from e equato ± { } + L (..)

7 Combed-ye Famly of Estmators of Poulato 39 I order at e roots are real, e codtos o e value of are gve by + > L (..3) or < L (..4) 3. Emrcal tudy e have cosdered MU84 oulato avalable ardal et al (99, age 65, Aedx B). It s assumed at e oulato e year 985 s study varable ad at e year 975 s auxlary varable. here are 84 mucaltes whch have bee dvded radomly to four strata havg szes 73, 7, 97 ad 44. able shows e values of e arameters of e oulato uder cosderato for e four strata whch are eeded comutatoal rocedure. tratum Mea X Mea X able : Parameters of e Poulato

8 4 Joural of elablty ad tatstcal tudes, December, Vol. 5 ().9. X /X he value of e fx e samle sze to be 6. he e allocato of samles to dfferet strata uder roortoal ad eyma allocatos are show e followg table tratum ( ) Proortoal Allocato ze of amles uder eyma Allocato able : Allocato of amle he equato (..9) yelds otmum values of roortoal allocato as.9573, ot ( ,.697,.5) ad uder eyma allocato as.9466, ot (3.6986,.637,.5) ad uder he followg table llustrates e ercetage relatve effceces (PE) of e for corresodg ot, ad 4resectvely w resect to estmators 4.e. studer roortoal ad eyma allocatos. A comarso of PE of ad w at at 4 w ot reveals e fact at e utlzato of auxlary formato at e estmato stage certaly mroves e effcecy of e estmator as comared to e usual mea estmator. st M ME [ ] ot M [ ] M [ ( 4) ] V [ ] st Allocato Proortoal eyma able 3: Percetage elatve Effcecy (PE) Comarso ( L, % for all )

9 Combed-ye Famly of Estmators of Poulato 4 educg ME rough Arorate Choce of e shall ow demostrates e arorate choce of for at e ME of ca be reduced to a desred level eve f e o-resose rate e estmators s creased. 4 ad ( ) Let us tael,.,. 3 for all. 3 Codtos for real roots of ( ) uder roortoal allocato are < If ( ) >.588 ad ( ). e [ ] M.957 ad M[ ( ) ] for ( ) From (..), we have ( ).9 ad.879 ad M[ ( ) ].957 for ( ).9. ow codtos for real roots of ( ) ( ) >.75 ad ( ) <.73 If ( ). e [ ] M[ ( ) ] 4.7 for.. e get ( ).6 ad.845 ad M ( ).48 for.6. [ ]. uder eyma allocato are M.48 ad 4. Cocluso I e reset aer, we have suggested a geeral famly of combed-tye estmators of oulato mea stratfed radom samlg e resece of oresose. I s cotext, we have made a attemt to use a auxlary varable order to crease e effcecy of e estmator. he otmum roerty of e famly has bee to a dscussed. e have also llustrated e reducto of ME of e famly desred extet by a arorate choce of e arameter eve f e o-resose rate s hgh e oulato. he eoretcal results have bee llustrated e emrcal study. he table 3 reveals at e otmum estmator ad combed rato estmator certaly rovde e better estmates a e usual mea estmator. efereces. Hase, M. H. ad Hurwtz,.. (946). he roblem of o-resose samle surveys, Joural of he Amerca tatstcal Assocato, 4, Kadlar, C. ad Cg, H. (3). ato estmators stratfed radom samlg, Bometrcal joural. 45 (), Kadlar, C. ad Cg, H. (5). A ew rato estmator stratfed samlg, Commucatos tatstcs heory ad Meods, 34, st

10 4 Joural of elablty ad tatstcal tudes, December, Vol. 5 () 4. Khoshevsa, M., gh,., Chauha, P., awa,. ad maradache, F. (7). A geeral famly of estmators for estmatg oulato mea usg ow value of some oulato arameter(s), Far East J. heor. tatst.,, Koyucu,. ad Kadlar, C. (8). ato ad roduct estmators stratfed radom samlg, Joural of statstcal lag ad ferece, 38, Koyucu,. ad Kadlar, C. (9). Famly of estmators of oulato mea usg two auxlary varables stratfed radom samlg, Commucatos tatstcs heory ad Meods, 38, ardal, C. E., wesso, B. ad retma, J. (99). Model Asssted urvey amlg, rger-verlag, ew Yor, Ic. 8. earls, D.. (964). he utlzato of a ow coeffcet of varato e estmato rocedure. Joural of he Amerca tatstcal Assocato, 59, habbr, J. ad Guta,. (5). Imroved rato estmators stratfed samlg. Amerca Joural of Maematcal ad Maagemet ceces, 5 (3-4), habbr, J. ad Guta,. (6). A ew estmator of oulato mea stratfed amlg, Commucatos tatstcs heory ad Meods, 35,. 9.. gh, H. P. ad Vshwaarma, G. K. (8). A Famly of Estmators of Poulato Mea Usg Auxlary Iformato tratfed amlg, Commucatos tatstcs heory ad Meods, 37, gh,., Mal,., Chaudhary, M. K., Verma, H. ad Adewara, A. A. (). A geeral famly of rato tye- estmators systematc samlg, Jour. elab. ad tat. tud. 5(), 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 (), alor,., alor,., Parmar,. ad Kumar, M. (). Dual to ato-cum- Product estmator usg ow arameters of auxlary varables, Joural of elablty ad tatstcal tudes 5(), Uadhyaya, L.. ad gh, H. P. (999). Use of trasformed auxlary varable estmatg e fte oulato mea, Bometrcal Joural, 4 (5),