Olufadi Yuusa Departmet of tatistics ad Mathematical cieces Kwara tate Uiversit.M.B 53 Malete Nigeria ajesh igh Departmet of tatistics Baaras Hidu Uiversit Varaasi (U..) Idia Floreti maradache Uiversit of New Meico Gallup UA Improvemet i Estimatig The opulatio Mea Usig Dual To atio-cum-roduct Estimator i imple adom amplig ublished i: ajesh igh Floreti maradache (Editors) ON IMOVEMENT IN ETIMATING OULATION AAMETE() UING AUXILIAY INFOMATION Educatioal ublishig (Columbus) & Joural of Matter egularit (Beijig) UA - Chia 3 IBN: 978--59973-3-5 pp. 4-53
ABTACT I this paper we propose a ew estimator for estimatig the fiite populatio mea usig two auiliar variables. The epressios for the bias ad mea square error of the suggested estimator have bee obtaied to the first degree of approimatio ad some estimators are show to be a particular member of this estimator. Furthermore compariso of the suggested estimator with the usual ubiased estimator ad other estimators cosidered i this paper is carried out. I additio a empirical stud with two atural data from literature is used to epoud the performace of the proposed estimator with respect to others. Kewords: Dual-to-ratio estimator; fiite populatio mea; mea square error; multiauiliar variable; percet relative efficiec; ratio-cum-product estimator. INTODUCTION It is well kow that the use of auiliar iformatio i sample surve desig results i efficiet estimate of populatio parameters (e.g. mea) uder some realistic coditios. This iformatio ma be used at the desig stage (leadig for istace to stratificatio 4
sstematic or probabilit proportioal to sie samplig desigs) at the estimatio stage or at both stages. The literature o surve samplig describes a great variet of techiques for usig auiliar iformatio b meas of ratio product ad regressio methods. atio ad product tpe estimators take advatage of the correlatio betwee the auiliar variable ad the stud variable. For eample whe iformatio is available o the auiliar variable that is positivel (high) correlated with the stud variable the ratio method of estimatio is a suitable estimator to estimate the populatio mea ad whe the correlatio is egative the product method of estimatio as evisaged b obso (957) ad Murth (964) is appropriate. Quite ofte iformatio o ma auiliar variables is available i the surve which ca be utilied to icrease the precisio of the estimate. I this situatio Olki (958) was the first author to deal with the problem of estimatig the mea of a surve variable whe auiliar variables are made available. He suggested the use of iformatio o more tha oe supplemetar characteristic positivel correlated with the stud variable cosiderig a liear combiatio of ratio estimators based o each auiliar variable separatel. The coefficiets of the liear combiatio were determied so as to miimie the variace of the estimator. Aalogousl to Olki igh (967) gave a multivariate epressio of Murth s (964) product estimator while aj (965) suggested a method for usig multi-auiliar variables through a liear combiatio of sigle differece estimators. More recetl Abu- Daeh et al. (3) Kadilar ad Cigi (4 5) erri (4 5) Diaa ad erri (7) Malik ad igh () amog others have suggested estimators for Y usig iformatio o several auiliar variables. Motivated b rivekataramaa (98) Badopadha (98) ad igh et al. (5) ad with the aim of providig a more efficiet estimator; we propose i this paper a ew estimator for Y whe two auiliar variables are available. 43
. BACKGOUND TO THE UGGETED ETIMATO Cosider a fiite populatio ( ) =... N of N uits. Let a sample s of sie be draw from this populatio b simple radom samplig without replacemets (WO). Let ad ) represets the value of a respose variable ad two auiliar variables i ( i i ( ) are available. The uits of this fiite populatio are idetifiable i the sese that the are uiquel labeled from to N ad the label o each uit is kow. Further suppose i a surve problem we are iterested i estimatig the populatio mea Y of assumig that the populatio meas ( Z ) for Y are give as X of ( ) are kow. The traditioal ratio ad product estimators X = ad = Z respectivel where = i i= = i i= ad = i i= are the sample meas of ad respectivel. igh (969) improved the ratio ad product method of estimatio give above ad suggested the ratio-cum-product estimator for Y as X = I literature it has bee show b various authors; see for eample edd (974) ad Z rivekataramaa (978) that the bias ad the mea square error of the ratio estimator ca be reduced with the applicatio of trasformatio o the auiliar variable. Thus authors like rivekataramaa (98) Badopadha (98) Trac et al. (996) igh et al. (998) igh et al. (5) igh et al. (7) Bartkus ad likusas (9) ad igh et al. () have improved o the ratio product ad ratio-cum-product method of estimatio usig the trasformatio o the auiliar iformatio. We give below the trasformatios emploed b these authors: 44
= ( + g) X ad i = ( + g) Z gi for i =... N () i g i where g =. N The clearl = ( + g) X g ad = ( + g) Z g are also ubiased estimate of X ad Z respectivel ad Corr ( ) = ρ ad Corr ( ) = ρ. It is to be oted that b usig the trasformatio above the costructio of the estimators for Y requires the kowledge of ukow parameters which restrict the applicabilit of these estimators. To overcome this restrictio i practice iformatio o these parameters ca be obtaied approimatel from either past eperiece or pilot sample surve iepesivel. The followig estimators ad E are referred to as dual to ratio product ad ratio-cum-product estimators ad are due to rivekataramaa (98) Badopadha (98) ad igh et al. (5) respectivel. The are as preseted: Z = = X ad E = X Z It is well kow that the variace of the simple mea estimator uder WO desig is V ( ) = λ ad to the first order of approimatio the Mea quare Errors (ME) of ad E are respectivel give b ( ) = λ ( + ) ME ( ) = λ ( + + ) ME ME ( ) = λ [ D C] + ( ) = λ ( + g g ) ME ( ) = λ ( + g + g ) ME 45
ME ( ) = λ ( + g C gd) E where f λ = N N N f = = ( i Y ) = ( i Y )( i X ) N i= N i= ρ = Y = X Y = Z = D C + = ad j for ( j = ) represets the variaces of ad respectivel; while ad deote the covariace betwee ad ad ad ad respectivel. Note that ρ ρ ad are defied aalogousl ad respective to the subscripts used. More recetl harma ad Tailor () proposed a ew ratio-cum-dual to ratio estimator of fiite populatio mea i simple radom samplig their estimator with its ME are respectivel give as T X = α + ( α ) ( ) λ ( ρ ) X ME T =. 3. OOED DUAL TO ATIO-CUM-ODUCT ETIMATO Usig the trasformatio give i () we suggest a ew estimator for Y as follows: Z = θ + X X ( θ ) Z We ote that whe iformatio o the auiliar variable is ot used (or variable takes the value `uit') ad θ = the suggested estimator reduces to the `dual to ratio' estimator proposed b rivekataramaa (98). Also reduces to the `dual to product' estimator proposed b Badopadha (98) estimator if the iformatio o the auiliar variate is ot used ad θ =. Furthermore the suggested estimator reduces 46
to the dual to ratio-cum-product estimator suggested b igh et al. (5) whe θ = ad iformatio o the two auiliar variables ad are bee utilied. I order to stud the properties of the suggested estimator (e.g. ME) we write = Y ( ); = X ( + ); Z ( + ) + k with E ( k ) = E( k ) = E( k ) ad E E = λ Y k = k ; λ λ ( k ) = ; E( k ) = ; E( k ) ( k k ) λ = XZ Now epressig i terms of k' s we have = Y X = ; E( k k ) = ; E( k k ) Z λ YX [ ] ( + k ) ( gk )( gk ) + ( θ )( gk ) ( gk ) λ = YZ ; θ () We assume that gk < ad gk < so that the right had side of () is epadable. Now epadig the right had side of () to the first degree of approimatio we have Y = Y [ k + ( ) g( k k + k k k k ) + g ( k k k α( k k )] α (3) Takig epectatios o both sides of (3) we get the bias of to the first degree of approimatio as where B [ gda + g ( ( )] ( ) = λ Y θ A = θ two we have quarig both sides of (3) ad eglectig terms of ( Y ) = Y [ k + Agk Agk ] [ k + Agk k Agk k A g k k + A g k A g k ] Y + k' s ivolvig power greater tha = (4) 47
Takig epectatios o both sides of (4) we get the ME of to the first order of approimatio as ME ( ) [ + AgD + A g C] = λ (5) The ME of the proposed estimator give i (5) ca be re-writte i terms of coefficiet of variatio as ME ( ) = [ + ] λ Y C AgC D + A g C where C = C + C ρ C C ad D = ρ C ρ C C = Y C = X C = Z The ME equatio give i (5) is miimied for D + Cg θ = = θ (sa) Cg We ca obtai the miimum ME of the suggested estimator b usig the [ ] optimal equatio of θ i (5) as follows: mi. ME( ) = λ + F( D CF ) + where F = g E ad E = D + Cg C 3. EFFICIENCY COMAION I this sectio the efficiec of the suggested estimator over the followig estimator E ad T are ivestigated. We will have the coditios as follows: D + gc ME if θ > gc (a) ( ) V ( ) < (b) ( ) ME( ) < ME if ( D AgC) < ( ) Ag + provided (c) ( ) ME( ) < ME if < 48
( D AgC) < ( ) Ag + + provided < D ME if C < provided Ag (d) ( ) ME( ) < (e) ( ) ME( ) < ME if ( ) g < A ( θgc gc D) < g + g 4θ 4 (f) ( ) ME( ) < ME if ( θgc gc D) < g + 4 g 4θ (g) ( ) ME( ) < ME if E D g < provided A < C ( A ) (h) ME ( ) ME( ) < if ( ) 4. NUMEICAL ILLUTATION T Ag D + AgC < ρ To aale the performace of the suggested estimator i compariso to other estimators cosidered i this paper two atural populatio data sets from the literature are beig cosidered. The descriptios of these populatios are give below. () opulatio I [igh (969 p. 377]; a detailed descriptio ca be see i igh (965) : Number of females emploed : Number of females i service : Number of educated females N = 6 = Y = 7. 46 X = 5. 3 Z = 79 = 8. 88 = 6. 76 = 8.953 ρ =. 7737 ρ =. 7 ρ =. 33 () opulatio II [ource: Johsto 97 p. 7]; A detailed descriptio of these variables is show i Table. : ercetage of hives affected b disease : Mea Jauar temperature 49
: Date of flowerig of a particular summer species (umber of das from Jauar ) N = = 4 Y = 5 X = 4 Z = = 65. 9776 = 9. 988 = 84 ρ =.8 ρ =. 94 ρ =. 73 For these comparisos the ercet elative Efficiecies (Es) of the differet estimators are computed with respect to the usual ubiased estimator usig the formula ( ) ( ) (). V E. = ME ad the are as preseted i Table. Table : Descriptio of opulatio II. 49 35 4 35 4 38 46 4 5 4 3 59 4 94 53 44 94 6 46 88 55 5 96 64 5 9 Table shows clearl that the proposed dual to ratio-cum-product estimator has the highest E tha other estimators; therefore we ca coclude based o the stud populatios that the suggested estimator is more efficiet tha the usual ubiased estimators the traditioal ratio ad product estimator ratio-cum-product estimator b igh (969) 5
rivekataramaa (98) estimator Badopadha (98) estimator igh et al. (5) estimator ad harma ad Tailor (). Table : E of the differet estimators with respect to Estimators opulatio I opulatio II 5 77 87 4 395 5 39 5 5 36 4 E 5 78 T 79 457 5. CONCLUION We have developed a ew estimator for estimatig the fiite populatio mea which is foud to be more efficiet tha the usual ubiased estimator the traditioal ratio ad product estimators ad the estimators proposed b igh (969) rivekataramaa (98) Badopadha (98) igh et al. (5) ad harma ad Tailor (). This theoretical iferece is also satisfied b the result of a applicatio with origial data. I future we hope to eted the estimators suggested here for the developmet of a ew estimator i stratified radom samplig. 5
EFEENCE. ABU-DAYYEH W. A. AHMED M.. AHMED. A. ad MUTTLAK H. A. (3): ome estimators of fiite populatio mea usig auiliar iformatio Applied Mathematics ad Computatio 39 87 98.. BANDYOADHYAY. (98): Improved ratio ad product estimators akha series C 4 45-49. 3. DIANA G. ad EI.F. (7): Estimatio of fiite populatio mea usig multiauiliar iformatio. Metro Vol. LXV Number 99-4. JOHNTON J. (97): Ecoometric methods (d ed) McGraw-Hill Toko. 5. KADILA C. ad CINGI H. (4): Estimator of a populatio mea usig two auiliar variables i simple radom samplig Iteratioal Mathematical Joural 5 357 367. 6. KADILA C. ad CINGI H. (5): A ew estimator usig two auiliar variables Applied Mathematics ad Computatio 6 9 98. 7. Malik. ad igh. () : ome improved multivariate ratio tpe estimators usig geometric ad harmoic meas i stratified radom samplig. IN rob. ad tat. Article ID 5986 doi:.54//5986. 8. MUTHY M. N. (964): roduct method of estimatio. akha A 6 94-37. 9. OLKIN I. (958): Multivariate ratio estimatio for fiite populatios Biometrika 45 54 65.. EI. F. (4): Alcue cosideraioi sull efficiea degli stimatori rapporto-cumprodotto tatistica & pplicaioi o. 59 75.. EI. F. (5): Combiig two auiliar variables i ratio-cum-product tpe estimators. roceedigs of Italia tatistical ociet Itermediate Meetig o tatistics ad Eviromet Messia -3 eptember 5 93 96.. AJ D. (965): O a method of usig multi-auiliar iformatio i sample surves Joural of the America tatistical Associatio 6 54 65. 3. EDDY V. N. (974): O a trasformed ratio method of estimatio akha C 36 59 7. 4. HAMA B. ad TAILO. (): A New atio-cum-dual to atio Estimator of Fiite opulatio Mea i imple adom amplig. Global Joural of ciece Frotier esearch Vol. Issue 7-3 5. INGH M.. (965): O the estimatio of ratio ad product of the populatio parameters akha series B 7 3-38. 6. INGH M.. (967): Multivariate product method of estimatio for fiite populatios Joural of the Idia ociet of Agricultural tatistics 3 375 378. 7. INGH M.. (969): Compariso of some ratio-cum-product estimators akha series B 3 375-378. 5
8. INGH H.. INGH. EEJO M.. INEDA M.D ad NADAAJAH. (5): O the efficiec of a dual to ratio-cum-product estimator i sample surves. Mathematical roceedigs of the oal Irish Academ 5A () 5-56. 9. INGH. INGH H. AND EEJO M.. (998): The efficiec of a alterative to ratio estimator uder a super populatio model. J...I. 7 87-3.. INGH. CHAUHAN. AND AWAN N. (7): O the bias reductio i liear variet of alterative to ratio-cum-product estimator. tatistics i Trasitio8()93-3.. INGH. KUMA M. CHAUHAN. AWAN N. AND MAANDACHE F. (): A geeral famil of dual to ratio-cum-product estimator i sample surves. IT_ewseries (3) 587-594.. IVENKATAAMANA T. (978): Chage of origi ad scale i ratio ad differece methods of estimatio i samplig Caad. J. tat. 6 79 86. 3. IVENKATAAMANA T. (98): A dual to ratio estimator i sample surves Biometrika 67() 99 4. 4. TACY D.. INGH H.. AND INGH. (996): A alterative to ratio-cum-product estimator i sample surves. J...I.53375-387. 53