Improvemen n Esmang Populaon Mean usng Two Auxlar Varables n Two-Phase amplng Rajesh ngh Deparmen of ascs, Banaras Hndu Unvers(U.P.), Inda (rsnghsa@ahoo.com) Pankaj Chauhan and Nrmala awan chool of ascs, DAVV, Indore (M.P.), Inda Florenn marandache Deparmen of Mahemacs, Unvers of New Mexco, Gallup, UA (smarand@unm.edu) Absrac Ths sud proposes mproved chan-rao pe esmaor for esmang populaon mean usng some known values of populaon parameer(s) of he second auxlar characer. The proposed esmaors have been compared wh wo-phase rao esmaor and some oher chan pe esmaors. The performances of he proposed esmaors have been supposed wh a numercal llusraon. Ke words: Auxlar varables, chan rao-pe esmaor, bas, mean squared error.. Inroducon The rao mehod of esmaon s generall used when he sud varable Y s posvel correlaed wh an auxlar varable X whose populaon mean s known n advance. In he absence of he knowledge on he populaon mean of he auxlar characer we go for wo-phase (double) samplng. The wo-phase samplng happens o be a powerful and cos
effecve (economcal) procedure for fndng he relable esmae n frs phase sample for he unknown parameers of he auxlar varable x and hence has emnen role o pla n surve samplng, for nsance, see Hdroglou and arndal (998). Consder a fne populaon U (U, U,..., U ). Le and x be he sud and auxlar varable, akng values and x respecvel for he h un U. Allowng RWOR (mple Random amplng whou Replacemen) desgn n each phase, he wo-phase samplng scheme s as follows: () () he frs phase sample s n ( s n U) of a fxed se n s drawn o measure onl x n order o formulae a good esmae of a populaon mean X, Gvens n, he second phase sample s n ( sn sn ) of a fxed se n s drawn o measure onl. Le x x, and x x n n. n s n s n The classcal rao esmaor for Y s defned as s n r X (.) x If X s no known, we esmae Y b wo-phase rao esmaor rd x x (.) ome mes even f X s no known, nformaon on a cheapl asceranable varable, closel relaed o x bu compared o x remoel relaed o, s avalable on all uns of he populaon. For nsance, whle esmang he oal eld of whea n a vllage, he eld and area under he N
crop are lkel o be unknown, bu he oal area of each farm ma be known from vllage records or ma be obaned a a low cos. Then, x and are respecvel eld, area under whea and area under culvaon see ngh e.al.(004). Assumng ha he populaon mean Z of he varable s known, Chand (975) proposed a chan pe rao esmaor as Z x (.3) everal auhors have used pror value of ceran populaon parameer(s) o fnd more precse esmaes. ngh and Upadhaa (995) used coeffcen of varaon of for defnng modfed chan pe rao esmaor. In man suaon he value of he auxlar varable ma be avalable for each un n he populaon, for nsance, see Das and Trpah (98). In such suaons knowledge on Z, C, ( ) (coeffcen of skewness), ( ) (coeffcen of kuross) and possbl on some oher parameers ma be uled. Regardng he avalabl of nformaon on C, ( ) and ( ), he researchers ma be referred o earls(964), en(978), ngh e.al.(973), earls and Inarapanch(990) and ngh e.al.(007). Usng he known coeffcen of varaon C and known coeffcen of kuross ( ) of he second auxlar characer Upadhaa and ngh (00) proposed some esmaors for Y. If he populaon mean and coeffcen of varaon of he second auxlar characer s known, he sandard devaon σ s auomacall known and s more meanngful o use he σ n addon o C, see rvasava and Jhajj (980). Furher, C, ( ) and ( ) are he un free consans, her use n addve form s no much jusfed. Movaed wh
he above jusfcaons and ulng he known values of σ, ( ) and ( ), ngh (00) suggesed some modfed esmaors for Y. In hs paper, under smple random samplng whou replacemen (RWOR), we have suggesed mproved chan rao pe esmaor for esmang populaon mean usng some known values of populaon parameer(s).. The suggesed esmaor The work of auhors dscussed n secon can be summared b usng followng esmaor az + b (.) a + b where a ( 0), b are eher real numbers or he funcons of he known parameers of he second auxlar varable such as sandard devaon ( σ ), coeffcen of varaon ( C ), skewness ( ( ) ) and kuross ( ( )). The followng scheme presens some of he mporan known esmaors of he populaon mean whch can be obaned b suable choce of consans a and b. Esmaor Z Chand (975) chan rao pe esmaor a Values of 0 b Z + C + C ngh and Upadhaa C
3 (995) esmaor ()Z + C () + C Upadhaa and ngh (00) esmaor () C 4 CZ + C + () () Upadhaa and ngh (00) esmaor 5 Z + σ + σ ngh (00) esmaor C ( ) σ 6 ()Z + σ () + σ ngh (00) esmaor () σ 7 ()Z + σ () + σ () σ In addon o hese esmaors a large number of esmaors can also be generaed from he esmaor a (.) b pung suable values of a and b. Followng Kadlar and Cng (006), we propose modfed esmaor combnng and (,3,...,7) as follows α + ( α), ( 3,...,7) (.) where α s a real consan o be deermned such ha ME of s mnmum and (,3,...,7) are esmaors lsed above. To oban he bas and ME of, we wre
Y( ), x X( + ), x X( + ), Z( + ) + such ha ( e 0 ) and where e 0 e E E ( ) E ( ) ( ) e e E 0 E ( e0 ) fc, E ( e ) fcx, E ( e ) f Cx E ( e ) f E(e e ) f C E(e e ) f E(e e C, 0 x xc, 0 xcxc E(e, E (e e ) f, E(ee ) fxcxc 0e ) f C C e) fxcxc Cx f, f, n N n N Y x C x x x x ( N ) C C X Z x x x ( N ) ( ) Y U ( ) ( ) Z U x ( x X)( Z) N, U x x Expressng n erms of e s, we have N e ( ) N ( ) ( x ) X U e ( x X)( Y) U ( N ) U ( Y)( Z) [ ( + e ) α( + e )( + e ) ( + e ) + ( α)( + e )( + e ) ( + e ) Y (.3) 0 az where (.4) az + b Expandng he rgh hand sde of (.3) and reanng erms up o second power of e s, we have.
or [ + e e + e e ( α + α) Y 0 (.5) [ e + e e ( α + α) Y Y e0 (.6) quarng boh sdes of (.6) and hen akng expecaon, we ge he ME of he esmaor, up o he frs order of approxmaon, as ME( ) Y [ f C + f C + ( α + α) f C f C C ( α + α)f C C 3 x 3 x where (.7) f 3. n n Mnmaon of (.7) wh respec o α eld s opmum value as K αop (.8) where K C. C ubsuon of (.8) n (.7) elds he mnmum value of ME ( ) as mn.me( 3. Effcenc comparsons ) [ f C + f (C C C ) f C Mo Y 3 x x x (.9) In hs secon, he condons for whch he proposed esmaor s beer han (,...7) have been obaned. The ME s of hese esmaors up o he order o(n) are derved as ME( rd [ fc + f3(cx xccx ) [ f C + f (C C C ) + f (C C C ) ) Y (3.) ME() Y 3 x x x (3.)
and where ME( ME( ME( ME( ME( ME( [ fc + f( C CC ) + f3(cx xccx) [ fc + f( 3C 3CC ) + f3(cx xccx) [ fc + f( 4C 4CC ) + f3(cx xccx) [ fc + f( 5C 5CC ) + f3(cx xccx) [ f C + f ( C C C ) + f (C C Cx) ) Y (3.3) 3) Y (3.4) 4) Y (3.5) 5) Y (3.6) 6) Y 6 6 3 x x (3.7) [ f C + f ( C C C ) + f (C C Cx) 7 ) Y 7 7 3 x x (3.8) Z Z + C ()Z 3 ()Z + C CZ 4 C Z + () Z 5 Z + σ ()Z 6 ()Z + σ ()Z 7. ()Z + σ From (.9) and (3.), we have ME( rd ) M f C 0 (3.9) o Also from (.9) and (3.)-(3.8), we have ( C C ) 0 ME( ) M f, ( 3,...,7) (3.0) o Thus follows from (3.9) and (3.0) ha he suggesed esmaor under opmum condon s alwas beer han he esmaor (,...7). 4. Emprcal sud To llusrae he performance of varous esmaors of Y, we consder he daa used b Anderson (958). The varaes are : Head lengh of second son x : Head lengh of frs son : Head breadh of frs son
N 5, Y 83. 84, X 85. 7, Z 5., σ 7. 4, C 0. 0546 C x 0.056, C 0. 0488, x 0. 708, 0. 693, 0. x 7346, () 0. 00, (). 659. Consder n 0 and n 7. We have compued he percen relave effcenc (PRE) of dfferen esmaors of Y wh respec o usual esmaor and compled n he able 4.: Table 4.: PRE of dfferen esmaors of Y wh respec o esmaor PRE 00 rd.5393 78.889 78.8405 3 78.877 4 86.39 5 8.605 6.5473 7 79.9636 86.655 5. Concluson We have suggesed modfed esmaors ( 3,...,7). From able 4., we conclude ha he proposed esmaors are beer han usual wo-phase rao esmaor rd, Chand (975) chan pe rao esmaor, esmaor proposed b ngh and Upadhaa (995), esmaors ( 3,4) and han ha
of ngh (00) esmaors ( 5,6,7). For praccal purposes he choce of he esmaor depends upon he avalabl of he populaon parameer(s). References Anderson, T. W. (958), An Inroducon o Mulvarae ascal Analss. John Wle & ons, Inc., New York. Chand, L. (975): ome rao pe esmaors based on wo or more auxlar varables. Unpublshed Ph. D. hess, Iowa ae Unvers, Ames, Iowa (UA). Das, A. K. and Trpah, T. P. (98): a class of samplng sraeges for populaon mean usng nformaon on mean and varance of an auxlar characer. Proc. of he Indan ascal Insue Golden Jublee Inernaonal Conference on ascs. Applcaons and New Drecons, Calcua, 6-9, December 98, 74-8. Hdroglou, M. A. and arndal, C.E. (998): Use of auxlar nformaon for wo-phase samplng. urve Mehodolog, 4(), -0. earls, D.T. (964): The ulaon of known coeffcen of varaon n he esmaon procedure. Journal of Amercan ascal Assocaon, 59, 5-6. earls, D.T. and Inarapanch, R. (990): A noe on an esmaor for varance ha ules he kuross. Amer. a., 44(4), 95-96. en, A.R. (978): Esmaon of he populaon mean when he coeffcen of varaon s known. Comm. a.-theor Mehods, A7, 657-67. ngh, G. N. (00): On he use of ransformed auxlar varable n he esmaon of populaon mean n wo-phase samplng. ascs n Transon, 5(3), 405-46. ngh, G. N. and Upadhaa, L. N. (995): A class of modfed chan pe esmaors usng wo auxlar varables n wo-phase samplng. Meron, LIII, 7-5.
ngh, H. P, Upadhaa, L. N. and Chandra, P. (004): A general faml of esmaors for esmang populaon mean usng wo auxlar varables n wo-phase samplng. ascs n ranson, 6(7), 055-077. ngh, J., Pande, B.N. and Hrano, K. (973): On he ulaon of a known coeffcen of kuross n he esmaon procedure of varance. Ann. Ins. as. Mah., 5, 5-55. ngh, R. Chauhan, P. awan, N. and marandache, F. (007): Auxlar nformaon and a pror values n consrucon of mproved esmaors. Renassance hgh press, UA. rvasava,.k. and Jhajj, H.. (980): A class of esmaors usng auxlar nformaon for esmang fne populaon varance. ankha, C, 4, 87-96. Upadhaa, L. N. and ngh, G. N. (00): Chan pe esmaors usng ransformed auxlar varable n wo-phase samplng. Advances n Modelng and Analss, 38, (-), -0.