Efficient Estimator for Population Variance Using Auxiliary Variable
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1 Amec Joul o Opetol eech 6 6): 9-5 DOI:.59/j.jo.66. Ecet Etmto o Populto Vce Ug Aul Vble ubhh Kum Yv heel M.. Mh * Deptmet o Mthemtc tttc A Cete o Ecellece) D. ML Avh Uvet Fzb U.P. I Deptmet o tttc Uvet o Luckow Luckow U.P. I Abtct Populto vce oe o the mpott meue o peo. Fo emple oe teete kowg the etmte o vce o ptcul cop bloo peue tempetue etc. Th ppe el wth the etmto o populto vce ug ul omto ue mple om mplg cheme. I the peet ppe we hve popoe mpove etmto though well kow kpp techque ug Yv et l ) ppe. The lge mple popete o the etmto hve bee tue up to the t oe o ppomto tht t b me que eo hve bee obte up to the t oe o ppomto. The optmum vlue o the chctezg cl kpp h bee obte o th optmum vlue o the kpp the mmum me que eo h bee obte. A compo h bee me wth the etg etmto o populto vce ug eco t. A mpovemet o the popoe etmto h bee how ove ll etg metoe etmto t h lee me que eo compe to othe etmto. Kewo to etmto utle B Me que eo Ecec. Itoucto I the theo o uve mplg the ul omto pl pmout ole evelopg echg mpove etmto o populto pmete o the tu vble. The ul omto ue t both the tge o egg etmto. Hee we hve ue th omto t etmto tge ol. The ul vble X) the m vble Y) ue tu e hghl cloel elte wth ech othe. Whe thee cloe potve octo betwee the tu vble the ul vble the le o egeo o the tu vble Y o the ul vble X pe though og the the to tpe etmto ue o mpovemet ove the pmete o the populto ue coeto. O the othe h the pouct tpe etmto e ue o mpove etmto o pmete whe the ul vble X the tu vble Y hve egtve coelto betwee them. Whle the egeo tpe etmto e ue o the mpove etmto o populto pmete whe the le o egeo oe ot p though the og. Let the populto ue vetgto te t cot o tct etble ut. Let )... be om mple o ze om bove bvte populto X Y) o ze ug * Coepog utho: t_@hoo.co... Mh) Publhe ole t Copght 6 cetc & Acemc Publhg. All ght eeve WO cheme. Let X Y epectvel e the populto me o the ul the tu vble let e the coepog mple me whch e ube etmto o populto me X Y epectvel. Let ρ eote the coelto coecet betwee the vble X Y the te-qutle ge o the ul vble X. I th mucpt we hve popoe mpove to tpe etmto o populto vce o tu vble b utbl ug the coelto coecet ρ betwee the two vble te-qutle ge o the ul vble X. Futhe we ume tht elble etmte o the coelto coecet ρ vlble vce om plot uve etc.. Vce Etmto Ltetue The mple vce the mot ppopte etmto o populto vce gve b: t.) Th etmto o populto vce ube t h the vce up to the t egee o ppomto : V t γ.) ) Ik 98) popoe the ollowg to etmto o populto vce ug ul omto : )
2 ubhh Kum Yv et l.: Ecet Etmto o Populto Vce Ug Aul Vble whee ) t.) ) X X) X Y Y. X The t oe o ppomto o the B Me que Eo ME) epectvel e gve b whee t ) γ [ ) )] B.) ME t ) γ [ ) + ) )].5) µ µ Y Y) X X) µ µ γ. evel utho popoe eet etmto b utlzg ul omto eet om. The ue t the om o eet pmete o ul vble o etmtg the populto vce o the m vble ue tu. ome o them om the ltetue e ollow Uph gh 999) utlze coecet o kuto ) o ul vble popoe the ollowg etmto o populto vce ˆ The b Me que Eo o bove etmto up to the t oe o ppomto epectvel e ˆ ) γ ) ) Whee ) ) ) B [ ) + ) ) ] ˆ ME ) γ.7) Kl Cg 6) popoe the ollowg etmto ug eet pmete o ul omto ˆ ˆ ) ) ˆ + C ) C ) The b Me que Eo o bove etmto up to the t oe o ppomto epectvel e ˆ B ) γ ) ) Whee [ ) + ) ) ] ˆ ME ) γ.8) ) ) C C ubm Kump ) utlzg vou populto pmete o ul vble popoe the ollowg etmto o populto vce ).6)
3 Amec Joul o Opetol eech 6 6): 9-5 ˆ 5 ˆ 6 ˆ ˆ 7 8 ˆ 9 The epeo o the b Me que Eo o bove etmto up to the t oe o ppomto epectvel e ˆ B ) γ ) ) Whee 5 [ ) + ) ) ] ˆ ME ) γ ) Whee ) e the qutle the thee pot vg the whole tbuto to ou equl pt. Futhe the ucto o qutle e the te qutle ge the em-qutle ge the qutle vege. Kh hbb ) popoe the ollowg etmto ug coelto coecet the th qutle o the ul vble ˆ ρ ρ The epeo o the b Me que Eo o the etmto up to the t oe o ppomto epectvel e ˆ ) γ ) ) Whee ρ ρ B [ ) + ) ) ] ˆ ME ) γ.) Yv et l. ) utlzg the coelto coecet o the te-qutle ge o ul vble popoe the ollowg etmto ˆ ρ ρ The b Me que Eo o the bove etmto up to the t oe o ppomto epectvel e ˆ ) γ ) ) Whee ρ. ρ B [ ) + ) ) ] ˆ ME ) γ.). Popoe Etmto Hee mpove to etmto o populto vce beg uggete the lght o etmto popoe b Yv et l. ) P 989)
4 ubhh Kum Yv et l.: Ecet Etmto o Populto Vce Ug Aul Vble t ρ ρ κ.) whee κ chctezg cl to be eteme uch tht the ME o the popoe etmto t mmze. To obt the b Me que eo o the popoe etmto we wh to ee + e ) + e ) uch tht E e ) o ) E e ) ) E e ) ) E ee ) ). The popoe etmto t c be wtte tem oε ) t κ + e ) + e ) Epg the ght h e o bove equto coeg the tem we get: Ate ubtctg the populto vce t κ + e e e e + ε up to the t egee o ppomto o tu vble o both the e o bove equto we hve + e e ee + e ) t κ.) The b o popoe etmto t obte b tkg epectto o both e o.) puttg the vlue o eet epectto : B.) t) κ [ ) )] + κ ) ) whee. The me que eo o the popoe etmto t obte b qug both e o.) mplg tkg epectto o both e up to the t oe o ppomto t) [ κ ) + κ κ ) ) κ κ) ) + κ ) ME.) ME t) mmum o whee + ) A B + ) + ) ) The mmum ME o the etmto t o th optmum vlue o κ : ME e ) A κ.5) B ) A t).6) B m ]. Ecec Compo The popoe etmto t peom bette th the etmto t the ee hvg lee me que eo ue the coto:
5 Amec Joul o Opetol eech 6 6): 9-5 A A MEm t) V t ) ) < + ) >.) B B The popoe etmto.) wll peom bette th the etmto.) ue the coto : ME A t) ME t ) { ) + ) ) } < B m A B { ) + ) ) } > + The popoe etmto t h moe ecec compe to ll othe etmto mucpt ue the coto : ˆ....) ) metoe th ) ˆ A MEm t ME ) { ) + ) ) } <... B A B { ) } + ) + ) > ).) 5. umecl Illutto Followg populto hve bee coee to eme the peomce o eet etmto o populto vce Populto I: Itl bueu o the evomet potecto-apat Wte Y: Totl mout to) o ecclble-wte collecto Itl. X: Totl mout to) o ecclble-wte collecto Itl. Y 66. X ρ C C Populto II: Itl bueu o the evomet potecto-apat Wte Y: Totl mout to) o ecclble-wte collecto Itl. X: umbe o hbtt. Y 6. 6 X ρ C C Populto III: Muth 967) Y: Output o 8 ctoe ego. X: Fe cptl. 8 Y X. 66 ρ C C Populto IV: gh Chuh The populto cot o 7 whet m 7 vllge cet ego o I the vble ue coeto e ee : Y e ue whet cop ce) ug 97 X e ue whet cop ce) ug Y X ρ C C
6 ubhh Kum Yv et l.: Ecet Etmto o Populto Vce Ug Aul Vble Tble. Compo o B Me que eo o eet etmto Etmto B ME Populto I II III IV I II III IV Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ Ŝ t Popoe) eult Cocluo Th ppe el wth the etmto o populto vce ug mpove to tpe etmto. A ecet etmto o populto vce ug coecet o coelto the te qutle ge o the ul vble h bee popoe. Up to the t egee o ppomto the epeo o the b me que eo o the popoe etmto hve bee obte. The optmum vlue o the chctezg cl kpp whch mmze the me que eo o the popoe etmto lo obte. Futhe the mmum vlue o the me que eo o th optmum vlue o kpp h lo bee obte. It h bee pove theoetcll well empcll tht the popoe etmto peom much bette th ll o the othe metoe etmto o populto vce the ee o hvg lee B ME. It o woth to be metoe tht the kowlege egg the coelto coecet ρ houl be vlble vce. Th kowlege o coelto coecet ethe vlble vce geell) o t obte om po tue lke plot uve etc. I ce we o ot hve po kowlege o coelto coecet the t the epeo o etmto eplce b t etmte thee o eect o the me que eo the etmto. Theeoe t togl ecommee tht the popoe etmto houl be peee ove the etmto metoe th mucpt o the etmto o populto vce ue mple om mplg cheme. ACKOWLEDGEMET The utho e ve much thkul to the eto che o AJO the omou lee eeee o ctcll emg the mucpt gvg the vluble uggeto o uthe mpovemet the ele t. EFEECE [] Ik C T. Vce etmto ug ul omto Joul o Amec tttcl Aocto ). [] Kl C. Cg H. Impovemet vce etmto ug ul omto Hcettepe Joul o mthemtc tttc 5-5 6). [] Kl C. Cg H. to etmto o populto vce mple tte mplg Apple Mthemtc Computto ).
7 Amec Joul o Opetol eech 6 6): [] Kh M. hbb J. A to Tpe Etmto o the Etmto o Populto Vce ug utle o Aul Vble Joul o tttc Applcto & Pobblt 9-5 ). [5] Muth M.. mplg Theo Metho tttcl Publhg ocet Clcutt I 967). [6] gh D. Chuh F.. Theo l o mple uve eg ew-age Itetol Publhe 986). [7] ubm J. Kump G. Vce etmto ug qutle the ucto o ul vble Itetol Joul o tttc Applcto 67-7 ). [8] Uph L.. gh H. P. Ue o ul omto the etmto o populto vce mthemtcl oum -6 98). [9] TpoDoc6. [] Yv.K. Mh.. Gupt. Impove Vce Etmto Utlzg Coelto Coecet utle o Aul Vble Commucte to Amec Joul o Mthemtc Mthemtcl cece ).
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