Some Chain Type Estimators for Population Variance in Two Phase Sampling

Transcription

1 Ieraal Jural Rece ad Iva Treds Cmpu ad Cmmuca I: me Cha Tpe Esmars fr Ppula Varace Tw Phase ampl A. Badpadha, P. Parchha ad Pambar Das. Deparme f Mahemacs, Asasl Eeer Cllee, Asasl 7335, Ida. Emal: arabbadpadha4@mal.cm, parhaparchha989@mal.cm. Deparme f Mahemacs, eaj aar Da Cllee, Klkaa- 79, Ida. Emal: pambardas.@mal.cm Absrac :- The prese vesa deals wh he prblem f esma he ppula varace w-phase sampl. me mdfed cha pe esmars have bee prpsed whch make use ceffce f skewess, ceffce f kurss, ad ceffce f vara f secd aular varables. Furher eeralas have bee made ad prperes are dscussed. The prpsed esmars have bee cmpared wh w-phase varace esmar ad sme her cha-pe esmars. The perfrmaces f he prpsed esmars have bee suppred wh a umercal llusra. Mahemacs ubjec Classfca: 6D5 Kewrds wh phase: Duble sampl, sud varable, aular varable, cha-pe, ppula varace,ceffce f skewess, ceffce f kurss, ceffce f vara, bas, mea squared errr. *****. Irduc Ifrma varables crrelaed wh he ma varables uder sud s ppularl kw as aular frma. The frma used a esma sae, hereb lead a beer chce f esmars. Ou f ma, ra ad reress mehds f esma are d eample hs ce. I he absece f he kwlede he ppula varace f he aular varable we fr w-phase (duble) sampl. The w-phase sampl happes be a pwerful ad cs effecve (ecmcal) echque fr ba he relable esmae frs phase sample fr he ukw parameers f he aular varable. Csder a fe ppula U = U,U,...,U f us. Le ad sad fr he varable uder sud ad a aular varable respecvel. I s assumed ha ad are hhl psvel crrelaed. Le, values f he ppula us fr, respecvel. Furher le,, =,,,, dee he, =,,,, dee he values f he us cluded a sample s f se draw b smple radm sampl whu replaceme (RWOR). Assum ha he ppula varace = = -X / - wh s kw, Isak (983) prpsed a ra esmar fr he ppula varace = X= /, = -Y / - = wh Y= /, as = = s r IJRITCC Ma 7, hp:// 879

2 Ieraal Jural Rece ad Iva Treds Cmpu ad Cmmuca I: where s = - / - f wh = ad respecvel., ad s = - / - = = = / Whe frma s avalable he ppula varace wh f, we seek esmae hruh a w-phase sampl. Allw RWOR each phase, he w phase sampl wll be as fllws: (a) The frs phase sample s s U (b) Gve s, he secd-phase sample ss measure l. s = - /( -), = /, Le s s ad s = - /( -), = /, s s s = - /( -), = /, s s s = - /( -), = /, s s = /, are ubased esmar = f frm a sample s, baed f fed se s draw measure l rder ba a d esmae. f fed se s seleced Mvaed b Chad (975) e ma sues a cha ra-pe esmar fr () as s r s s = s Furher hs wrk was eeded b Chadra, h ad Mahur (9) b us β () (ceffce f kurss) as s r = s +β s s +β I hs paper he have als preseed w eeral class f esmars fr ppula varace as ad α s +β = s s s +β α α α α 3 () s +β s +β = s s s +β +β where α,α,α,α ad α 3 are suable chse csas. Mvaed b abve ps, hs wrk, a aemp has bee made ule he frma ceffce f skewess, ceffce f kurss, ad ceffce f vara f secd aular varables Fur ew cha-pe esmars fr he ppula vara f he sud varable have bee prpsed. Furher eeralas f he prpsed esmars are als preseed. (3) (4) (5) IJRITCC Ma 7, hp:// 88

3 Ieraal Jural Rece ad Iva Treds Cmpu ad Cmmuca I: Prpsed Esmars We assume ha he frma abu he secd aular varable s avalable fr all he us. Thus, he values f cera parameers f he varable ca be made avalable. Thus ul he kwlede f ad C ( ceffce f vara), β () (ceffce f skewess) ad β ( ceffce f kurss ) we sues sme ra-prduc pe cha esmars as s + s + a = s k + (- k) s s + + where α s s deermed s as mmed M..E f a,( =,,,4) ad = fr = = C ( ceffce f vara) fr =. = β (ceffce f skewess ) fr =3. = β () ( ceffce f kurss ) fr = 4. α fr =,,,4 (6) We have see ha k = ve several ra esmars fr ppula varace based values f ad fr =, reduces he usual ra esmar. If k = reduces several prduc esmars f based values f usual prduc esmar. ad fr =, reduces he 3. Bas ad Mea quared Errrs (M..E) f he prpsed Esmar a, (=,,,4) As he prpsed sequece f esmars a, ( =,,..,4) (6) are based fr have bee baed up he frs rder f apprmas s = (+e ), s = (+e ), s = (+e ), s = (+e ) 3 Ad E(e ) = fr j=,,,4. j The prpsed sequece f esmars a, (=,,,4) he becme where θ = a 3 3, her bases ad mea squared errrs (m.s.e. s) = +e +e +e k +θ e + -k +θ e (7) we epad he erms f () ad cllec he erms up he frs rder f apprma, we have he fllw resuls: α Therem :The bas B(.) ad mea squared errr (m.s.e.) M(.) f he prpsed sequece f esmars a, ( =,,,4), he erms f rder ( - ) are ve b B a = E( - ) a IJRITCC Ma 7, hp:// 88

4 Ieraal Jural Rece ad Iva Treds Cmpu ad Cmmuca I: ad = α- αθ f C k + - k +α - k θ f ρ C C + f C - ρ C C 3 a M = where f = -, f = -, 3 4 fc +f3c - f3ρcc +α - k θ fc +α- kθfρcc f =( - ) E e = f C, E e = f C, E e = f C, E e = f C, E e = f C 3 4 Ee e = f ρ C C, Ee e = f ρ C C, E e e = f C E e e = f ρ C C, E e e = f C, E e e = f ρ C C, E(e e )= f ρ CC 3 3 E(e e )= f ρ C C, E e e = f ρ C C, E e e = f ρ C C ad C= λ -, C = λ -, C λ -, ρ = λ - / λ - λ -, ρ = λ - / λ - λ ρ = λ - / λ - λ -, κ =ρ C /C, 4 4 κ =ρ C /C, κ =ρ C /C, κ =ρ C /C, λ =μ / μ μ μ, p/ q/ r/ pqr pqr p q pqr r eave eer. = μ = / -Y -X -Z, p,q,r be (9) (8) Crllar : I s bvus ha bases ad m.s.e. s f,,..,4.) (8)ad (9) respecvel. a, (=,,,4) ca be baed b subsu he values f θ, ( = 4. Mmum M..E f a, (=,,..., 4) The pmal cd, ha s, he cd uder whch a, (=,,...,4) has mmum M..E, s baed as α = (p) ρc k- θc Hece he mmum M..E f he class f esmars a, (=,,...,4) s deed b M( a), ( =,,...,4). 4 M( a) = f- fρ C +f3c - f3ρcc () IJRITCC Ma 7, hp:// 88

5 Ieraal Jural Rece ad Iva Treds Cmpu ad Cmmuca I: Remark: The values f α (p) (=,,,4) ca be baed b pu he values f s ve b θ (=,,...,4) resvel. 5. Effcec Cmparss f a, (=,,...,4) I hs sec, he cds fr whch he prpsed esmars uder pmum cds are beer ha r, m.s.e. s f hese esmars up he rder ( - ) are derved as, (). The where M( ) = f C + f C - f ρ C C M ( )= 4 f C +f C -κ +f θc θ -κ (3) r m. M( )= f - ρ + f - ρ C 4 m. ME = f - ρ + f - ρ C () 4. ρ. s he mulple crrela ceffce bewee Y X ad Z. (4) (5) 5.. Cmpars f a ( =,,,4) wh r We have see frm () ad ha M ( a ) M ( r ) f - fρc <. whch s alwas rue as f >. (6) 5.. Cmpars f a ( =,,,4) wh r I s bvus frm () ad (3) ha M ( a ) M ( r ) f C ρ <. (7) C We bserve ha f s he same characersc as bu ake a prevus ccas, bh C ad Cwll apprmael be equal. I hs case a s superr r prvded ρ <. (8) 5.3. Cmpars f a, ( =,,...,4) wh Frm () ad (4) we fd ha M a m. M ( ) f - < ρ < ad f ρ - C < - f C - ρ C (9) 3 IJRITCC Ma 7, hp:// 883

6 Ieraal Jural Rece ad Iva Treds Cmpu ad Cmmuca I: Here ρ s he crrela ceffce bewee -Y ad -X. We kw ha crrela ceffce les bewee - ad, whch we have see here. s cds s alwas sasfed f d als sasfed, he we ca sa, ( =,,...,4) a s alwas mre effce ha. 5.4 Cmpars f a, ( =,,...,4) wh () Frm () ad (5) culd be ccluded ha M a M ( () ) f We have see ha f, f C - ρ C C + C + f C ρ - <. () 3. C - ρcc + C C - CC + C C - ρcc + C C-C 3 3 ρ < -< ρ <.. f C - ρ C C + C as f s alwas psve. fc ρ - < - f C - ρ C C + C ur esmars ad. 3 mre effce ha (). We kw ha., ( =,,...,4) ρ ad ur esed cds les bewee he pssble raes f ρ. ecep whe ρ.. Bu a ha cds he predced value f Y, he mulple lear reress equa f Y X ad Z ma be sad be perfec predc frmula. 6. umercal Illusras We csder he daa used b Aders (958) demsrae wha we have dscussed earler. 5 famles have bee bserved fr he fllw hree varables. : Head leh f secd s : Head leh f frs s : Head breadh f frs s X=85.7, Y=83.84, Z=5., C =.356, C =.49478, C =.8573,. ρ =.55688, ρ =.56879, ρ =.438, ρ =.63964, csder = ad = 7. We have cmpued m.s.e. s f he dffere prpsed varace esmars ad perce relave effcec (PRE) f dffere esmars f wh respec Isak (983) varace esmar are cmpled Table. Table a are Esmars r () a, ( =,,...,4) ME/ Varace PRE IJRITCC Ma 7, hp:// 884

7 Ieraal Jural Rece ad Iva Treds Cmpu ad Cmmuca I: Frm Table we bserve ha uder pmum cd ur prpsed esmars a, (=,,...,4) are beer ha her esmar such as, r, smlar aure.. Therefre ur suesed esmars hs wrk are mre jusfable cmpare wh he prevus wrk f Ccluss The esmars csdered hs wrk are mre reasable ad frm he abve resuls, we ca cclude ha he esmars whch are prpsed hs paper are mre effce ha her esmars csder hs paper. Therefre, he use f hese esmars ma be recmmeded b he surve sascas. Refereces [] ADERO, T.W. (958): A Irduc Mulvarae ascal Aalss, Jh Wle & s, Ic., ew Yrk. [] CHAD, L. (975): me ra pe esmars based w r mre aular varables, Upublshed Ph.D. hess, Iwa ae Uvers, Ames, Iwa (UA). [3] IGH, G.. : O he use f rasfrmed aular varable he esma f ppula mea w phase sampl, ascs I Tras.Vl.5. 3,pp [4] IAKI, C.T. (983): Varace esma us aular frma, Jur. Amer. as. Assc., 78, (38), 7-3. [5] CHADRA, P., h, H.P. & Mahur,. (9) : A cha-pe esmar fr ppula varace us w aular varables w-phase sampl, ascs I Tras.-ew seres.vl..,pp [6] Upadha, L.. ad h. H.P (6) : Alms ubased ra ad prduc-pe esmars f fe ppula varace w phase sample surves, ascs I Tras, Vl 7,. 5 pp IJRITCC Ma 7, hp:// 885