Some Improved Estimators for Population Variance Using Two Auxiliary Variables in Double Sampling

Transcription

1 Vplav Kumar gh Rajeh gh Deparme of ac Baara Hdu Uver Varaa-00 Ida Flore maradache Uver of ew Meco Gallup UA ome Improved Emaor for Populao Varace Ug Two Aular Varable Double amplg Publhed : Rajeh gh Flore maradache (Edor) O IMPROVEMET I ETIMATIG POPULATIO PARAMETER() UIG AUXILIARY IFORMATIO Educaoal Publhg (Columbu) & Joural of Maer Regular (Bejg) UA - Cha 03 IB: pp. -

2 Abrac I h arcle we have propoed a effce geeraled cla of emaor ug wo aular varable for emag ukow populao varace of ud varable.we have alo eeded our problem o he cae of wo phae amplg. I uppor of heorecal reul we have cluded a emprcal ud.. Iroduco Ue of aular formao mprove he preco of he emae of parameer.ou of ma rao ad produc mehod of emao are good eample h coe. We ca ue rao mehod of emao whe correlao coeffce bewee aular ad ud varae pove (hgh) o he oher had we ue produc mehod of emao whe correlao coeffce bewee aular ad ud varae hghl egave. Varao are pree everwhere our da-o-da lfe. A agrculur eed a adequae uderadg of he varao clmac facor epecall from place o place (or me o me) o be able o pla o whe how ad where o pla h crop. The problem of emao of fe populao varace of he ud varable wa dcued b Iak (983) gh ad gh ( ) gh e al. (008) Grover (00) ad gh e al. (0).

3 Le ad are aular varae havg value ) ad he ud varae havg ( value ( ) repecvel. Le V (...) he populao havg u uch ha povel correlaed ad egavel correlaed wh. To emae we aume ha ad are kow where ( Y) ad ( Z). ( X) Aume ha large o ha he fe populao correco erm are gored. A ample of e draw from he populao V ug mple radom ample whou replaceme. Uual ubaed emaor of populao varace where ( ). ( ) Up o he fr order of appromao varace of gve b var( ) 00 (.) where μ pqr pqr p / q / r / μ μ μ ad μ p q r pqr ( Y) ( X) ( Z). Eg Emaor Le ( e ) ( e ) ad ( e ) 0 ( ) ( ) ( ) where ( ) ad. Alo le E(e ) E(e ) 0 ; p q r beg he o-egave eger.

4 E(e0 ) E(e 0 00 e) E(e ) 0 00 E(ee ) ad E(e ) E(e e ) 0 Iak (983) uggeed rao emaor for emag a- ; where ubaed emaor of (.) Up o he fr order of appromao mea quare error of gve b ME( [ ] 0 (.3) ) gh e al. (00) propoed he epoeal rao-pe emaor a- ep (.) Ad epoeal produc pe emaor 3 a- 3 (.) Followg Kadlar ad Cg (00) gh e al. (0) propoed a mproved emaor for emag populao varace a- k ep ( k ) (.) where k a coa. Up o he fr order of appromao mea quare error of 3 ad are repecvel gve b 00 ME ( ) 00 0 (.)

5 00 ME ( 3) 00 0 (.8) k ( k ) ME ( ) 00 k ( k ) k 0 ( k ) 0 0 (.9) ( / ) where k ( Improved Emaor Ug gh ad olak (0) we propoe ome mproved emaor for emag populao varace a- p c D (.0) (c d) q (a b) (.) a b p q c D (a b) k ( k ) (.) (c d) a b where a b c d are uabl choe coa ad k a real coa o be deermed o a o mme ME. Epreg ad erm of [ pe e pe ] 0e e we have (.3) 0 d where. (c d) [ q e e q e ] 0 0e (.) b where. (a b) [ e p k (e e ) (k )q ] 0 e (.)

6 The mea quared error of emaor are obaed b ubracg quarg boh de ad ha akg epecao- from each emaor ad ME( [ p ] 0 (.) ) p Dffereag (.) wh repec o we ge he opmum value of a- p 0 (op) 00 ME(. [ q ] 0 (.) ) Dffereag (.) wh repec o we ge he opmum value of a q 0 (op) 00. [ A k B ( k )C k ( k )E ( k )F] ME( ) (.8) Dffereag (.8) wh repec o k we ge he opmum value k of a k (op) C D F E. B C E where A 00 B p 00 C q 00 D p 0 E F q. pq 0 0. Emaor I Two Phae amplg I cera praccal uao whe o kow a pror he echque of wo phae amplg or double amplg ued. Allowg RWOR deg each phae he wo phae amplg cheme a follow: The fr phae ample ( V) of a fed e draw o meaure ol ad order o formulae he a good emae of ad repecvel. 8

7 Gve ol. he ecod phae ample ( ) of a fed e draw o meaure Eg Emaor gh e al. (00) propoed ome emaor o emae wo phae amplg a: ep (.) 3 ep (.) k ep ( k )ep (.3) ME of he emaor ad 3 are repecvel gve b 00 ME ( ) 00 0 (.) 00 ME ( 3) 00 0 (.) [ A k B ( k ) C k D ( k )E] ME( ) (.) Ad k (op) C E D (B C) Where 00 A B D 0 E 00 0 C 00 Propoed emaor wo phae amplg The emaor propoed eco 3 wll ake he followg form wo phae amplg; c d (.) (c d) 9

8 (a b) (.8) a b c d (a b) k ( k ) (.9) (c d) a b Le ( e ) ( e ) 0 ( e ) ( Where ( ad Alo ) ( ) E(e ) E(e ) 0 e ) ( E(e ) E(e ) E(e0e) 0 E(e0e ) E(e e ) 00 E(ee ) Wrg emaor ad [ e p (e e )] ) 0 0 ( ( e ) ) E(ee) E(ee ) erm of 00 0 e we have repecvel 0 (.0) [ e q ] 0 e (.) [ p k ((e e ) e (k )q ] 0 e (.) olvg (.0)(.) ad (.)we ge he ME of he emaor repecvel a- ad 00 ME ( ) p 00 p 0 (.3) Dffereae (.3) w.r.. we ge he opmum value of a- 0

9 (op) p ME ( ) q 00 q 0 (.) Dffereae (.) wh repec o we ge he opmum value of a (op) ME(. q 0 00 [ A Bk (k ) C k D (k ) ] ) E (.) Dffereae (.) wh repec o k we ge he opmum value of k a k (op) C D E B C Where A 00 B p 00 C p q 00 D p 0 E q 0. Emprcal ud I uppor of heorecal reul a emprcal ud carred ou. The daa ake from Murh(9): c ρ c 0.3 c 0.90 ρ X.88 Y 99. Z 08.88

10 I Table. perce relave effcec of varou emaor of repec o wre wh Table.: PRE of he emaor wh repec o Emaor PRE

11 I Table. perce relave effcec of varou emaor of repec o wo phae amplg: wre wh Table.: PRE of he emaor wo phae amplg wh repec o Emaor PRE Cocluo I Table. ad. perce relave effcece of varou emaor are wre wh repec o. From Table. ad. we oberve ha he propoed emaor 3

12 uder opmum codo perform beer ha uual emaor Iak (983) emaor ad gh e al. (00 ) emaor.. Referece Bahl. ad Tueja R.K. (99): Rao ad Produc pe epoeal emaor. Iformao ad Opmao cece Vol. XII I 9-3. Grover L. K. (00): A correco oe o mproveme varace emao ug aular formao. Commu. a. Theo. Meh. 39:3. Iak C. T. (983): Varace emao ug aular formao. Joural of Amerca acal Aocao. KadlarC. ad CgH. (00) : Improveme emag he populao mea mple radom amplg. Appled Mahemac Leer 9 (00) 9. Murh M.. (9).: amplg Theor ad Mehod acal Publhg oc. Calcua Ida. gh H. P. ad gh R. (00): Improved Rao-pe Emaor for Varace Ug Aular Iformao. J.I..A..(3)-8. gh H. P. ad gh R. (00): A cla of cha rao-pe emaor for he coeffce of varao of fe populao wo phae amplg. Algarh Joural of ac -9. gh H. P. ad gh R. (003): Emao of varace hrough regreo approach wo phae amplg. Algarh Joural of ac gh H. P ad olak R.. (0): A ew procedure for varace emao mple radom amplg ug aular formao. acal paper DOI 0.00/ gh R. Chauha P. awa. ad maradache F. (0): Improved epoeal emaor for populao varace ug wo aular varable. Iala Jour. Of Pure ad Appled. Mah gh R. Kumar M. gh A.K. ad maradache F. (0): A faml of emaor of populao varace ug formao o aular arbue. ude amplg echque ad me ere aal. Zp publhg UA. gh R. Chauha P. awa. ad maradache F. (008): Almo ubaed rao ad produc pe emaor of fe populao varace ug he kowledge of kuro of a aular varable ample urve. Ocogo Mahemacal Joural Vol. o