ON THE COMBINATION OF ESTIMATORS OF FINITE POPULATION MEAN USING INCOMPLETE MULTI- AUXILIARY INFORMATION

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1 Journal of Relablty and tatstcal tudes; I (Prnt): , (Onlne): Vol. 9, Issue (06): 99- O THE COMBIATIO OF ETIMATOR OF FIITE POPULATIO MEA UIG ICOMPLETE MULTI- AUILIAR IFORMATIO eha Garg and Meenaksh rvastava* chool of cences, Indra Gandh atonal Oen Unversty, ew Delh, Inda. * Deartment of tatstcs, Insttute of ocal cences, Dr. B. R. Ambedkar Unversty, Agra, Inda. E Mal: nehagarg@gmal.com; *msrvastava_ss@hotmal.com Receved October, 06 Modfed ovember 3, 06 Acceted December 6, 06 ABTRACT The resent study s concerned wth the estmaton of a fnte oulaton mean when we have nformaton on several auxlary varables only for some art of the oulaton. The maxmum utlzaton of ncomlete mult-auxlary nformaton s carred out n such cases by stratfyng the oulaton on the bass of avalable mult-auxlary nformaton at hand. Ths aer deals wth the stuaton where some of the auxlary varables are ostvely and some of them are negatvely correlated wth the man varable. For ths urose, n ths aer rato-cum-roduct and regresson-cum-roduct tye estmators are consdered for estmatng the mean of the fnte oulaton utlzng avalable ncomlete mult-auxlary nformaton. The aroxmate exressons for bas and mean square error of the suggested estmators have also been derved and theoretcal results are numercally suorted. Key Words: Bas; Mean quare Error; Incomlete Mult-Auxlary Informaton.. Introducton The use of sulementary nformaton s wdely dscussed n samlng theory. Auxlary varables are commonly used n samle survey ractces to obtan mroved desgns and to acheve hgher recson n the estmates of some oulaton arameters such as the mean or the total of a study varable. In ths context t s well known that when the auxlary nformaton s to be used at the estmaton stage, the rato, roduct and regresson methods are wdely emloyed n many stuatons. It should be recalled that rato and regresson estmators are used when study varable s hghly ostvely correlated wth the auxlary varable and roduct estmator s deloyed when the study varable s negatvely correlated wth the auxlary varable. There may arse stuatons where we ossess nformaton about several auxlary varables but each varable may be known for some art of oulaton only. To handle such tyes of stuatons, ngh (977), rvastava and Garg (009; 03) have consdered the concet of stratfcaton for weghtng the gven ncomlete auxlary nformaton, whch enables the nvestgator to make maxmum use of such nformaton. In ths technque the oulaton may be consdered as consstng of

2 00 Journal of Relablty and tatstcal tudes, December 06, Vol. 9() dfferent strata accordng to the number of known auxlary varables. Then a stratfed samle of sze n can be selected from each stratum of the oulaton. It may haen that some of the auxlary varables are ostvely and some are negatvely correlated wth the study varable. Thus, n order to obtan a more recse estmate of the oulaton mean, the am of ths aer s to develo rato-cum-roduct and regresson-cum-roduct tye estmators by makng maxmum use of avalable ncomlete mult-auxlary nformaton. It s seen that t s always better to use addtonal auxlary varables, whch are correlated wth. These estmators are constructed when frame s known for each stratum.e. nformaton on number of auxlary varables that can be collected for each and every oulaton unt should be known n advance. Generally the stratfcaton s done on the bass of heterogenety n the oulaton wth resect to the study varable. But we vew stratfcaton n the other way. In our case, the heterogenety of the oulaton s consdered wth resect to the unequal number of auxlary varables. We stratfy the oulaton n terms of the nformaton rovded by the auxlary varables. Thus, there wll be a stratum for whch no auxlary nformaton s avalable, strata for whch only one auxlary varable out of auxlary varables s known. mlarly there wll be C strata for whch the two auxlary varables are known, C 3 strata for whch three auxlary varables are known, and so on. Ultmately, we wll have a stratum for whch all the auxlary varables are known. It s seen that the gven method, consderng stratfcaton of oulaton on the bass of unequal number of auxlary varable s caable of gvng more recse results than smle mean er unt. In ecton 3 of the aer the constructon of strata s exlaned whle the suggested rato-cum-roduct tye estmator and regresson-cum-roduct tye estmator are dscussed n secton 4 and 5 resectvely. In secton 6, an emrcal study s carred out.. otatons Let us consder a fnte oulaton U (U, U,, U ) of dentfable unts takng values on a study varable and auxlary varables,,,, whch are correlated wth. Auxlary varables,,, are known for total M, M,, M unts of the oulaton resectvely. For the maxmum utlzaton of avalable ncomlete auxlary nformaton, the oulaton s dvded nto dfferent strata accordng to the known number of auxlary varables and a random samle of n unts s drawn from these grous wth smle random samlng wthout relacement. C : umber of strata for whch j auxlary varables are known; : n : 0 : : j : j j 0,,,,. Poulaton sze amle sze ze of the stratum for whch no auxlary varable s known ze of the stratum for whch auxlary varable s known;,,, sze of the stratum for whch auxlary varable and j are known; < j;, j,,,

3 On the combnaton of estmators of fnte oulaton mean 0 jk :,,.., : sze of the stratum for whch 3 auxlary varable, j and k are known; < j < k;, j, k,,, sze of the strata for whch all auxlary varable,,, s known; < j < k;, j, k,,, Where + j + jk + +,,, M : Total number of strata,.e. : 0 C oulaton sze of the th stratum;,,..., such that n : samle sze of the th stratum;,,, such that n n k : Value of the k th observaton on varable under study n th stratum;,,, C j; j 0,,,, ; k,,..., jk : : y : j : x j : : j : C : C : j j: jh : Value of the k th observaton on j th auxlary varable n th stratum,,, C j; j 0,,,, ; k,,..., Poulaton mean of the varable n th stratum amle mean of the varable n th stratum Poulaton mean of the j th auxlary varable n th stratum amle mean of the j th auxlary varable n th stratum Poulaton mean square error of varable n th stratum Poulaton mean square error of j th auxlary varable n th stratum Coeffcent of varaton of the varable under study n th stratum,.e. C Coeffcent of varaton of the j th auxlary varable n th stratum,.e. C j j j Correlaton coeffcent between the varables, varable under study and j, j th auxlary varable n th stratum Correlaton coeffcent between the varables j and h (j h) n th stratum.

4 0 Journal of Relablty and tatstcal tudes, December 06, Vol. 9() b j : W : f : f : α j : Defne y e o and uch that E( e 0) E( e ) 0, j Regresson coeffcent of the varables, varable under study and j, j th auxlary varable n th stratum Proorton of unts n the th stratum,.e. amlng fracton,.e. f n amlng fracton n the th stratum,.e. W f Weghts, attached to the j th auxlary varable n th stratum addng u to unty.e. α e j j f E(eo) Cy n, E(ej) f E(eoej) jcycxj n f E(ejek ) jkcxjcxk n C Kj j ; j,,, C j xj j j j,,, n ; < j;, j,,, f C xj n, j,,,, j,,,, j,,, j 3. The constructon of strata The followng fgure shows the constructon of strata ( 3 8) on the bass of avalable ncomlete three-auxlary nformaton (, and 3 ) and then selecton of samle:

5 On the combnaton of estmators of fnte oulaton mean 03 If some of the j s are ostve; say, for j,,..., and some j s are negatve; say, for j +,..., then our estmator wll be weghted rato-cum-roduct tye or regresson-cum-roduct tye. ow, we have obtaned the followng weghted rato-cum-roduct tye or regresson-cum-roduct tye estmators usng ncomlete mult-auxlary nformaton: n n n n 3 n 4 n 5 n 6 n 7 n 8 4. Rato-Cum-Product Tye Estmator ' + yr.ratc W αjgj.rat y,xj αjg j j ' + ( ) ( y, x ) j.rod j Where, g ( y, x ) y and g ( y, x ) j. rat j j j. rod j j xj j y x 4. Its Bas and ME To the frst-degree aroxmaton, the bas and mean square error are resectvely gven by

6 04 Journal of Relablty and tatstcal tudes, December 06, Vol. 9() E ( ) ' yr.ratc E W α jgj.rat( y,xj) + αjgj.rod( y, xj) j j ' + + f α jcjkj W α jcj n j j Where K K, for j,,, j j K, for j +, +,, j K j Bas ME ( y ) E( y ) r.ratc r. ratc f W α jcj n j j α C j j K j ( ) yr.ratc ME W αjgj.rat ( y,xj) + αjgj.rod ( y,xj) + 0 j j + W ME α jgj. rat ( y, xj) + αjgj. rod ( y, xj) + 0 j j + The covarance terms wll vansh because all strata are ndeendent of each other. W j + h α α j h j j + h + n f W n j cov α α + ( g, g ) h j. rat cov ( g, g ) j. rod j j h + j. rat α α Where, v Cov( g, g ) jh h α α v j h j jh h h j. rod cov ( g, g ) j. rat j. rod where,

7 On the combnaton of estmators of fnte oulaton mean 05 v jh v jh.rat [ C + C C CC CC ] jh For j,,..., and h,,...., v jh v jh.rod j h j [ C + C C + CC + CC ] jh j h For j +, +,..., and h +, +,...., v jh v jh.ratc [ C C C CC + CC ] jh j For j,,..., and h +, +,...., h Remark : The otmum values of α j for j,,, are obtaned by adotng the rocedure gven by Olkn(958). Otmum Values of α j for j,,, It s farly smle to establsh that the otmum α j s gven by th umof theelementsof the j column of V αj umof allthe elementsnv Where the matrx V (v jh ) and α ( α α..., α ) ~ j j j j j h h h,,..., h h h α beng the transose of α. V s the matrx nverse to V. Usng the otmum weghts, the mean ~ square error s found to be ME ( yr.ratc) W umof allthe elements n V n ~ 5. Regresson-Cum-Product Tye Estmator ' + yr.regc W αjgj.reg y,xj αjg j j ' + ( ) ( y, x ) j.rod Where, g ( y, x ) y + b ( x ) and g ( y, x ) j. rod j y j x j j. reg 5. Its Bas and ME To the frst-degree aroxmaton, the bas and mean square error are resectvely gven by ( ) ' E yr.regc E W α jgj.reg( y,xj) + αjgj.rod( y, xj) j j ' + j j j j j

8 06 Journal of Relablty and tatstcal tudes, December 06, Vol. 9() Bas f + W α jc n j + ( y ) E( y ) r.regc r. regc f W α jc n j + MEy j j + W ME αjgj. reg ( y, xj) + αjgj. rod ( y, xj) + 0 j j + The covarance terms wll vansh because all strata are ndeendent of each other. f W αjαhvjh n j h ( ) r.regc ME W α jgj.reg( y,xj) + αjgj.rod( y,xj) + 0 j K j j K j Where, v jh v jh.reg [ + b b b b ] j h For j,,..., and h,,...., v jh v jh.rod jh j h j [ C + C C + CC + CC ] jh j h For j +, +,..., and h +, +,...., v jh v jh.regc j j j [ C b C C b CC + CC ] j j For j,,..., and h +, +,...., jh j h j j j h h j h h j h h h Remark : It s to be noted that y r could be used f the arameters b, C y, C x and yx are known. As remarked by Murthy (967, 96), aha and aha (985) and Tracy and ngh (999), these arameters are stable quanttes, therefore, these can be known ether from the ast studes or from the exerence gathered n due course of tme. 6. An Emrcal tudy We consder the oulaton of sze 90 for comarng the roosed ratocum-roduct and regresson-cum-roduct estmators wth mean er unt estmator. The oulaton set conssts of the data where some of the auxlary varables are ostvely correlated and others are negatvely related wth the study varable. uose a samle of sze n43 s drawn by RWOR from ths oulaton.

9 On the combnaton of estmators of fnte oulaton mean 07 below: The comuted samle sze for each stratum by roortonal allocaton s gven trat a I II III IV V VI VII VIII Total n Table : The comuted strata samle szes usng roortonal allocaton Wthout tratfcaton , 8.06, y tratum I II K b III IV V K K b K K b VI K K b b VII

10 08 Journal of Relablty and tatstcal tudes, December 06, Vol. 9() K K b VIII K K K b b Table : The oulaton arameters for oulaton gven n Aendx II tratum o. of used Auxlary varable rat cum rod reg cum rod Bas ME Bas ME I II III IV V VI VII VIII Table 3: The bases and mean square errors for rato cum roduct and regresson cum roduct method of estmaton n each stratum

11 On the combnaton of estmators of fnte oulaton mean 09 are resented n the followng table: Estmators Bas ME Relatve Effcency y y r.ratc y r.regc Table 4: The bases, mean square errors and the relatve effcences of all the estmators of the suggested class 7. Dscusson and Concluson The roosed estmators usng ncomlete mult-auxlary nformaton has been comared wth smle mean er unt estmator n whch auxlary nformaton has not been used. It s seen that both roosed estmators are more effcent for mean estmaton. () A crtcal revew of table 4 reveals that, though both the suggested estmators are based but the amount of bas s almost neglgble n both the cases. () When we comare the V (y) wth all the roosed estmators, we fnd that V(y).7695, whch s consderably hgher than the ME (.636 and ME ( y r. regc) () The relatve effcency of mean er unt. yr. regc y r. ratc) s hghest.e %. as comared to It s evdent from the above results that the roosed estmators establsh the suremacy over y n the estmaton of mean usng stratfcaton on the bass of avalable ncomlete auxlary nformaton. Thus, we see that how the maxmum use of avalable ncomlete mult-auxlary nformaton can ncrease the effcency of the estmators. References. Cochran, W. G. (977). amlng Technque, 3rd edton, ew ork, Wley and ons.. Olkn, I. (958). Multvarate rato estmaton for fnte oulaton, Bometrka, 45,

12 0 Journal of Relablty and tatstcal tudes, December 06, Vol. 9() 3. ukhatme P.V. and ukhatme, B.V. (997). amlng Theory of urveys wth Alcatons, Ames, Iowa, Iowa tate Unversty Press, UA. 4. ngh, R. (977). A ote On The Use Of Incomlete Mult-Auxlary Informaton In amle urveys, Australan Journal of tatstcs, 9 (), rvastava, M. and Garg,. (009). A class of estmators of fnte oulaton mean usng ncomlete mult-auxlary nformaton when frame s unknown, Algarh Journal of tatstcs, 9, rvastava, M and Garg,.(03) - The class of estmators of fnte oulaton mean usng ncomlete auxlary nformaton, tatstcs In Transton - ew eres 4(),.0 6.

13 On the combnaton of estmators of fnte oulaton mean Aendx I Data et tratum I tratum II tratum III tratum IV tratum V tratum VI tratum VII

14 Journal of Relablty and tatstcal tudes, December 06, Vol. 9() tratum VIII