Efficient Estimators for Population Variance using Auxiliary Information

Transcription

1 Global Joural of Mahemacal cece: Theor ad Praccal. IN 97-3 Volume 3, Number (), pp Ieraoal Reearch Publcao Houe hp:// Effce Emaor for Populao Varace ug Aular Iformao ubhah Kumar Yadav Dep. of Mah & a, Dr. RML Avadh Uver, Fazabad, U.P., Ida E-mal: Abrac I h paper a faml of effce emaor, emag populao varace of he varable uder ud ug aular formao ha bee propoed. The epreo for ba ad mea quared error (ME) have bee obaed upoo ( ). A comparo ha bee made wh he geeral faml of emaor for populao varace of R. gh e.al. (7), whch coa ome well kow emaor of populao varace a a parcular member uch a Iak (983), Upadhaa ad gh (999) ad Kadlar ad Cg () ec. A mproveme ha bee how over above faml of emaor hrough a emprcal ud. Keword: Aular formao, varace emaor, ba, mea quared error, effcec. Mahemac ubjec Clafcao : Prmar D5 ecodar F Iroduco ad Noao The ue of aular formao ma creae he preco of he emaor. Whe he varable uder ud Y hghl correlaed wh he aular varable X, rao ad produc pe emaor are ued for mproved emao of populao parameer. Ma auhor have propoed rao pe emaor for he emao of populao varace ug dffere kow parameer of he aular varable. Iak (983) wa he fr who ued aular formao o emae he varace of he varable uder ud. He ha how ha h emaor beer ha he uual emaor, whch doe o ulze he aular formao he ee of havg leer mea quared error. I he ere of mproveme Upadhaa ad gh (999) ha gve a

2 37 ubhah Kumar Yadav emaor ug he kow populao coeffce of kuro of aular varable ad he howed ha h emaor beer ha ad. Kadlar ad Cg () propoed a emaor whch ulze he kow populao coeffce of varao ad howed ha h emaor beer ha he all above emaor. I he pree ud, we ugge a ew faml of emaor for emag populao varace of he varable uder ud. Maeral ad Mehod Le he populao co of N u ad a ample of ze draw from h populao ug mple radom amplg whou replaceme. Le Y ad X be he value for he h u (,,..., N ) of he populao for he ud varable ad aular varable repecvel. Furher, le ad be he ample mea of he ud ad aular varable repecvel. I order o ud he large ample propere of he propoed faml of emaor, we defe ( + e ) ad ( + e ) wh Ee ( ), (,) cae of mple radom amplg whou replaceme, gorg fe populao correco erm, he followg epecao could be obaed eher drecl or b he mehod due o Kedall ad uar (977) a Ee ( ) ( ) λ, Ee ( ) ( ) λ ad Eee ( ) ( ) λ μr Where λ r r/ / μ μ ad μ ( Y) ( X), (r, ),,, 3, N r r N uggeed faml of emaor R. gh e.al. (7) uggeed a faml of emaor for populao varace a ( a b) (3.) α( a b) + ( α)( a b) where ( a ), b are eher real umber or he fuco of he kow parameer of he aular varable uch a coeffce of varao C ad coeffce of kuro β ( ) λ. The ME of h faml of emaor gve b λ α λ α λ (3.) ME( ) ( ) v( ) + v ( )

3 Effce Emaor for Populao Varace 37 where a v ( a b) C ( λ ) The mmum ME for α op where C v ( λ ) ME m [ λ λ () ( ) ( ) (3.3) The rao pe emaor, gve able are -faml wh he ME a ME( ) ( λ ) αv( λ ) + α v ( λ ) (3.),,., v Wh v, v, v, v 3, ( C) ( λ) C ad v. ( C λ ) ( + λ ) 5 v λ, ( λ C ) Table : ome member of -faml of emaor. Emaor Value of α a b Iak (983) emaor [ C C Kadlar & Cg () emaor 3 [ β( ) β( ) [ ( ) C β( ) C β 5 [ C β( ) C β ( ) C β ( ) β ( ) C C β ( )

4 37 ubhah Kumar Yadav [ ( ) + β + β( ) Upadhaa & gh (999) emaor - β ( ) Movaed b Nurel Koucu ad Cem Kadlar (9), we propoe a ew faml of emaor for populao varace a ξ ( a b) k [ α( a b) + ( α)( a b) (3.5) where k uabl choe coa o be deermed. Now epreg he emaor ξ erm of (,), (.5) ca be wre a e ξ ( + e )( + αve ) (3.) k Epadg he rgh had de of (.) o he fr order of appromao ad ubracg from boh he de, we ge ξ k ( + e )( αve + α v e ) Now ξ k (3.7) ( + e αve + α v e αvee ) Takg epecao o boh de of (.7), we ge he ba of he emaor ξ a k B( ξ ) [ α v ( λ ) αv( λ ) + ( k ) (3.8) quarg boh de o equao (.7), gve ( ξ ) k ( + e αve + α v e αve e ) + k ( + e αve + α v e αve e ) Now akg epecao boh de, we ge ME of ξ upo O ( ) a ME( ξ ) [ { k ( λ ) + (3k k) α v ( λ ) αv(k k)( λ )} + ( k ) (3.9) A The mmum of ME(ξ ) obaed for opmum value of k whch k op. B Where A [ α v ( λ ) αv( λ ) +

5 Effce Emaor for Populao Varace 373 Ad B [( λ ) + 3α v ( λ ) αv( λ ) + Thu he mmum ME of he faml ξ of emaor A MEm ( ξ ) [ (3.) B The rao pe emaor, gve able are ξ - faml wh he ME a ME( ξ ) [ { k ( λ ) + (3k k) α v ( λ ) αv (k k)( λ )} + ( k ),., Table : ome member of ξ - faml of emaor. Emaor ξ ξ k k [ C C ξ3 k [ ( β β ( ) ξ k [ ( ) β C β ( ) C ξ5 k [ C ( β C β ( ) ξ k [ ( + β + β ( ) Value of α A b C β ( ) β ( ) C C β ( ) - β ( ) Ma more emaor ca be formed ju b pug dffere value of he parameer a ad b of aular varable. Effcec comparo To he fr order of appromao, afer gorg fe populao correco, he varace ad ME of emaor ad repecvel are gve a

6 37 ubhah Kumar Yadav V ( ) [ λ ME ) [( λ ) ( λ ) + ( λ ( ) ad V ( ) ME( ) [( λ ) ( λ ) > ( λ ) > ( λ ) Whe h codo aeded, more effce ha. mlarl -faml more effce ha f, ME( ) < ME( ),,3,,5,. f, ha ( λ ) < ( + v )( λ ) Now uggeed emaor ξ are more effce ha, (,., ) emaor f, ME ( ξ ) < ME( ),,3,,5,. m ha A [ < [( λ ) + v ( λ ) v ( λ B ) f, ad he mmum ME of ξ - faml le ha he mmum ME of faml MEm ( ξ ) < MEm ( ),,3,,5,. ha A [ < [( λ ) ( λ B ) Emprcal ud We ued he daa R. gh e.al (7), gve able3, whch wa earl ued b Kadlar ad Cg () for he comparo of effcece of he ξ - faml o faml of emaor of R. gh e.al (7).

7 Effce Emaor for Populao Varace 375 Table 3: Daa ac. N,, C. 8, C., Y 5. 37, X , 9. 89, λ 5. 7, λ 8. 3, λ The ME of ξ ad, (,., ) ad perceage relave effcec (PRE) each ξ o correpodg gve he followg able. Reul From he able, we ee ha he propoed ξ - faml of emaor beer ha he faml of emaor of R. gh e.al (7) he ee of havg leer mea quared error. Table alo how ha propoed faml much beer ha he emaor o ulzg aular formao. Table : ME ad Effcece comparo. faml ξ - faml PRE of Emaor ME Emaor ME emaor ξ over (.9) (.93) 3 (.999) (.9) 5 3 (.99 5 (.985 ) ) PRE of over PRE of ξ over ξ (.859) ξ (.8557) ξ (.85859) ξ (.859) ξ (.8577) ξ (.8573) Cocluo From he reul of he emprcal ud ad heorecal dcuo, cocluded ha he propoed ξ - faml of emaor for emag populao varace uder opmum codo perform much beer ha he uual emaor ad alo beer ha - faml of emaor propoed b R. gh e.al. (7), whch coa ome well kow emaor of Iak (983), Upadhaa ad gh (999) ad Kadlar ad Cg () ec.

8 37 ubhah Kumar Yadav Referece [ Iak, C.T., 983, Varace emao ug aular formao, Jour. Amer. a. Aoc., 78, pp [ Kadlar, C. ad Cg, H.,, Rao emaor for he populao varace mple ad rafed radom amplg, Appled Mahemac ad Compuao, 73, pp [3 Kedall, M. ad uar, A., 977, The advaced heor of ac, Vol I, Charle Grff & Co., Lodo. [ Koucu, N ad Kadlar, C., 9, Effce emaor for he populao mea, Haceepe Joural of Mahemac ad ac, 38 (), pp [5 R. gh e.al, 7, Aular formao ad a pror value coruco of mproved emaor, Reaace Hgh Pre, U..A., pp [ rvaava,.k. ad Jhajj, H.., 98, A cla of emaor ug aular formao for emag fe populao varace akha, C,, pp [7 Upadhaa, L.N. ad gh, H.P., 999, A emaor for populao varace ha ulze he kuro of a aular varable ample urve Vkram Mahemacal Joural, 9, pp. -7.