New Difference Estimator in Two-phase Sampling using Arbitrary Probabilities

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1 IN Joural o tatstcs olume 15, 008, Abstract New Derece Estmator Two-hase amlg usg Arbtrar Probabltes Asa Kamal 1 ad Muhammad Qaser hahbaz A ew derece estmator has bee costructed two-hase samlg usg two auxlar varables w ad x.the rst hase samlg ut has bee selected wth robablt roortoal to measure o sze ad secod hase samle s selected wth equal robablt wthout relacemet. The roosed estmator has bee oud to be more ecet as comared to Raj (1965) whch sgle auxlar characterstc has bee used. The varace exresso ca be easl exteded to - auxlar varates. Kewords Two-hase samlg, Auxlar varable, Measure o sze, Arbtrar robabltes 1. Itroducto Auxlar ormato ca be used to crease the recso o regresso estmators. Ote ormato o ma asects o stud s ether avalable or ca be collected rom samle surve. Ths ormato ca be used to mrove the ecec o regresso estmators. Furthermore, aother techque o mrovg the recso s to use uequal robabltes or selecto o samle. A ubased estmator has bee develoed b extedg the umber o auxlar varables regresso te estmator whereas samle s selected wth arbtrar robabltes. The uavalablt o ror kowledge o auxlar ormato leads to the two-hase samlg. 1 Deartmet o tatstcs, Lahore College or Wome Uverst, Paksta Emal: asa_kamal@hotmal.com Deartmet o Mathematcs, COMAT Isttute o Iormato Techolog, Lahore, Paksta Emal: qshahbaz@gmal.com

2 8 Kamal ad hahbaz I ormato o a measure o sze z s avalable, the tal samle s o sze s draw wth robablt roortoal to measure o sze z wth relacemet ad ormato about w ad x s collected rom s. The secod hase samle o sze s draw usg smle radom samlg wthout relacemet rom s to collect ormato about w, x ad. Raj (1965) used the robablt roortoal to sze techque ad develoed the derece estmator usg sgle auxlar characterstc. Raj (1965) derece estmator o oulato total s: x x 1 k (1.1) N z where ad Z z Z The arace o 1 s: k x k( ) x (1.) A estmator o 1 s: d d v where d kx ad Y 1 X x Cov x X, Y rvekataramaa ad Trac (1989) develoed a varace exresso usg the otmum value o k whch s gve as:

3 New Derece Estmator Two-hase amlg Usg Arbtrar Probabltes where Y Y 1 (1.3) (1.4) Mukherjee et al. (1987) have develoed seres o regresso estmators. Followg regresso estmator was roosed wth two auxlar varables usg equal robablt wthout relacemet at both hases: x x w ( ) w (1.5) where the otmum values o ad are gve as: ce x wx. 1 wx wx. 1 w wx ad are oulato quattes ther cosstet estmators b ad b are comuted ad estmator (1.5) becomes b ( x x) b w w The exresso o ME gve b Mukherjee et al. (1987) or (1.5) s ME.. wx 1 wx where wx. wx 1 wx s multle correlato coecet.. The Proosed New Regresso Estmator Usg Arbtrar Probabltes

4 10 Kamal ad hahbaz A ubased estmator o oulato total Y usg two auxlar characterstcs s roosed as: w w x x (.1) where ad are costats. The varace o 3 s derved as: w x w x wx w x (.) The otmum value o ad s obtaed b mmzg (.). Hece deretatg (.) wth resect to ad resectvel ad equatg the resultg dervatve equal to zero: w w w x 0 wx x x w x 0 wx Ater smlcato, ollowg exressos o x wx ad w 1 wx wx 1 ubsttutg the values o wx ad are obtaed as: ad (.), the resultg exresso s:

5 New Derece Estmator Two-hase amlg Usg Arbtrar Probabltes 11 wx wx w x w 1 x 1 wx wx wx w w 1 wx wx 3 x x 1 wx wx wx w 1 wx x wx w x 1 wx Ater smlcato 3 where s gve as: 3 1. wx. wx 1 wx 1. wx wx (.3) (.4) The result derved (.3) ca easl be geeralzed to q auxlar varables where =1, q. x 4 1.1,,... q.1,,... q (.5) Comarso o ew estmator has bee made wth varace exresso or sgle auxlar characterstc (1.3) derved b rvekataramaa ad Trac (1989). Takg the derece o varace exresso (.3) ad (1.3) wx wx Ater smlcato

6 1 Kamal ad hahbaz wx. wx It s eas to coclude rom the above exresso that 3 erorms better tha 1 because rst hase samle s alwas greater tha secod hase samle.e. whch makes above exresso egatve. Hece crease the auxlar varables brgs more recso the regresso estmator. 3. ecal Result or arace Derved or New Estmator a. There s vercato o varace exresso. I 1 P s substtuted N ad (.1) ad (.3) the these two exressos become equvalet to ME resectvel. Takg (.1) w w x x ubstutg 1 P N ad resectvel ollowg results are obtaed: x wx wx w 1 1 wx wx Hece the exresso (.1) becomes ~ w w x x 3 N N N N N (3.1)

7 New Derece Estmator Two-hase amlg Usg Arbtrar Probabltes 13 1 mlarl, substtutg P (1.4) ( ) ad.4). wx( ) the ollowg N results are obtaed: N ad. wx. wx Hece (.3) becomes equvalet to ME ( ) ME 3.. wx N 1 wx 3 (3.) whch s exresso o estmator o oulato total o (5) ad ts ME usg smle radom samlg wth relacemet at tal hase ad smle radom samlg wthout relacemet at al hase. b. These are the codtos uder whch varace o 3 s more recse tha cometet estmator smle radom samlg. Comarg 3 (.3) ad ME (3.) 1 ME 3 srs 1 1 srs. wx. wx From (3.3) t s observed that (3.) s greater as comared to (.3) ollowg both equaltes hold that s 0 srs (3.3) (3.4) ad srs wx wx. 0. (3.5)

8 14 Kamal ad hahbaz N Y The equalt (3.4) s greater tha zero Z Z 0 that s 1 Z estmated s closel related to measure o sze. Ad also (3.5) holds true N Y both Z Z 0 ad. wx. wx hold. 1 Z 4. Emrcal tud Emrcal stud has bee coducted to llustrate the erormace o the ew estmator wth the same estmator usg sgle auxlar characterstc ad also wth the cometet estmator smle radom samlg whch two auxlar characterstcs have bee used. Four oulatos have bee selected or ths urose rom agrculture sector o Pujab (005). z=area ow (thousad hectares) = Producto (Thousad Metrc Tos /Thousad Bales) x= Producto (Thousad Metrc Tos /Thousad Bales) w= Number o Tractors (Prvate ad Govermet) 004 _cesus (March) Percet relatve ecec o ew derece estmator (.1) wth Raj (1965) ad Mukherjee et al.(1987) s calculated Table 1 ad Table, resectvel, where ad N

9 New Derece Estmator Two-hase amlg Usg Arbtrar Probabltes 15 Table 1: Percet Relatve Ececes o New Derece Estmator 3 (.1) wth 1 (1.1) Estmator Po.I Po.II Po.III Po.I Table : Percet Relatve Ececes o New Derece Estmator 3 (.1) wth (3.1) Estmator Po.I Po.II Po.III Po.I Cocluso From emrcal stud t s cocluded that varace or roosed estmators 3.e. (.1) s more recse tha (3.1) ad Raj (1965) regresso estmator 1 (1.1). Comarg the ercet relatve ececes t s observed that there s substatal crease the ercet relatve ecec o roosed estmator 3 or all oulatos. From Table t ca be easl cocluded that ercet relatve ececes alwas crease as creases whle case o N the crease s ot alwas there.

10 16 Kamal ad hahbaz It s thereore cocluded that crease auxlar varable creases the ececes. Also usg arbtrar robabltes mroves the ecec o estmator. Ackowledgmets The authors ackowledge the reerees ad Che Edtor or ther valuable commets ad suggestos. Reereces 1. Mukherjee, R., Rao, T. J. ad jaak, K. (1987). Regresso te estmators usg multle auxlar ormato. Australa Joural o tatstcs, 9, Pujab Develomet tatstcs, Bureau o tatstcs, Govermet o Pujab, Lahore, Raj, D. (1965). O samlg over two occasos wth robablt roortoate to sze. Aals o Mathematcal tatstcs, 36(1), rvekataramaa, T. ad Trac, D.. (1989). Two-hase samlg or selecto wth robablt roortoal to sze samle surve. Bometrka, 76(4),