An Efficient Dual to Ratio and Product Estimator of Population Variance in Sample Surveys

Transcription

1 "cece as True Here" Joural of Mahemacs ad ascal cece, Volume 06, cece gpos Publshg A Effce Dual o Rao ad Produc Esmaor of Populao Varace ample urves ubhash Kumar Yadav Deparme of Mahemacs ad ascs A Cere of Ecellece), Dr R M L Avadh Uvers, Fazabad-00, Ida heela Msra Deparme of ad ascs, Uvers of Luckow, Luckow-6007, Ida Ravedra Kumar * Deparme of Mahemacs, Dr Ram Maohar Loha Gov Degree College, Aola, Barell, UP, Ida haledra Verma Deparme of Busess Maageme ad Erepreeurshp, Dr R M L Avadh Uvers, Fazabad-00, UP, Ida haledra Kumar Deparme of Mcrobolog, Dr R M L Avadh Uvers, Fazabad-00, UP, Ida Absrac The prese mauscrp deals wh he esmao of populao varace usg aular varable Here we proposed a effce dual o rao ad produc pe esmaor of populao varace of sud varable ulzg aular formao he form of coeffce of kuross ad he populao mea of he aular varable To he frs order of appromaos, he bas ad he mea squared error of he proposed varace have bee obaed The opmum value of he characerzg scalar has bee obaed Ths opmum value of characerzg scalar mmzes he mea squared error of he proposed esmaor The mmum value of he mea squared error has bee obaed for hs opmum value of he characerzg scalar A comparso of he proposed esmaor has bee made wh he meoed esg esmaors of populao varace Through a emprcal sud, he performaces of dffere esmaors of populao varace are judged b calculag mea squared errors of dffere esmaors I has bee see ha he proposed esmaors has mmum mea squared error amog all meoed esmaors Kewords: Ma varable, aular varable, bas, mea squared error, effcec Correspodg auhor: Ravedra Kumar, Deparme of Mahemacs, Dr Ram Maohar Loha Gov Degree College, Aola, Barell, UP, Ida E-mal: rkpmbl@gmalcom

2 A Effce Dual o Rao ad Produc Esmaor of Populao Varace ample urves 79 Iroduco I s well kow ha for esmag a parameer, he mos suable esmaor s he correspodg sasc o for esmao he populao varace he mos suable esmaor s he sample varace Populao varace s oe of he mpora measures of dsperso Is role our da oda lfe s ver mpora such as we are eresed kowg he esmae of he varao he produco of a parcular crop, blood pressure, emperaure ec The role of aular formao s of ver much mporace as ehaces he effcec of he esmaor for esmag he populao parameer of he ma varable uder sud I s used a boh he sages of desgg ad esmao of surve samplg I he prese mauscrp, we have used a he esmao sage ol I he prese sud, he sud varable s Y ad he aular varable s X ad boh he varables are closel relaed wh each oher Whe he sud varable ad he aular varables have posvel assocao wh each oher ad he le of regresso of he sud varable Y o he aular varable X passes hrough org, he he rao pe esmaors are used for mproved esmao of he parameers of he populao uder sud Whereas he produc pe esmaors are used for mproved esmao of parameers uder cosderao whe he aular varable X ad he sud varable Y are egavel correlao wh each oher The regresso pe esmaors are used for he mproveme esmao, whe he le of regresso does o pass hrough he org Le he populao o be vesgaed s fe ad cosss of dsc ad defable us Le, ),,,, be a radom sample of sze from hs bvarae populao X, Y) usg a mple Radom amplg whou Replaceme RWOR) scheme Le X ad Y be he populao meas of he aular ad he sud varables respecvel, ad le ad are he correspodg sample meas whch are ubased esmaors of populao meas X ad Y respecvel Revew of Leraure of Varace Esmaors As meoed earler, he mos approprae esmaor of populao varace s he correspodg sample varace defed b, s ) 0 The above esmaor s ubased ad up o he frs order of appromao, s varace s:

3 80 A Effce Dual o Rao ad Produc Esmaor of Populao Varace ample urves V ) γ λ ) ) 0 0 where, s ), Y Y,, λ rs µ, µ µ rs r s 0 0 r s µ rs Y Y ) X X ), f γ ad f Isak 983) used he aular formao he form of populao varace ad sample mea of aular varable ad proposed he followg radoal rao esmaor of populao varace as: R s s 3) where s ), X X ), X X The Bas ad Mea quare Error ME) of he esmaor R, up o he frs order of appromaos respecvel are, B R ) γ [ λ ) λ )] ) 0 ME R ) γ [ λ ) + λ ) λ )] 5) 0 0 gh e al 0), based o Bahl ad Tueja 99) epoeal rao pe esmaor for he populao mea, proposed he followg epoeal rao pe esmaor for he populao varace as, s s ep 6) + s The ME of he esmaor, up o he frs order of appromao s gve b,

4 A Effce Dual o Rao ad Produc Esmaor of Populao Varace ample urves 8 λ0 ) ME ) γ λ0 ) + λ ) 7) The produc pe esmaor for he populao varace, based o Bahl ad Tueja 99) produc pe esmaor ca be defed as, s s ep 8) s + The ME of he esmaor, up o he frs order of appromao s gve b, λ0 ) ME ) γ λ0 ) + + λ ) 9) Upadhaa ad gh 00) ulzed he mea of he aular varable ad proposed he followg modfed rao esmaor of populao varace as, X 3 s 0) The bas ad he mea squared error of he esmaor 3 up o he frs order of appromao respecvel are, B ) γ [ C λ C ] ) 3 ME ) γ [ λ ) + C λ C ] ) 3 0 where λ µ 0 µ µ 0 The produc pe esmaor based o he Upadhaa ad gh 00) esmaor 0) ma be proposed as, s X 3)

5 8 A Effce Dual o Rao ad Produc Esmaor of Populao Varace ample urves The bas ad he mea squared error of he esmaor up o he frs order of appromao respecvel are, B ) γ [ C + λ C ] ) ME ) γ [ λ ) + C + λ C ] 5) 0 Asgar e al 0) b usg he populao mea of he aular varable, proposed he followg mproved rao ad produc pe esmaors of populao varace respecvel as, X 5 s ep 6) X + X 6 s ep 7) + X The bas ad mea squared error of above esmaors up o he frs order of appromao respecvel are, B C λ ) γ [ C ] 8) 8 5 ME C ) γ [ λ0 ) + λ C ] 9) 5 B C λ ) γ [ + C ] 0) 8 6 ME C ) γ [ λ0 ) + + λ C ] ) 6 Ma more esmaors of populao varace he leraure have bee proposed b dffere auhors ulzg varous parameers of he aular varable I he prese sud we have also proposed a dual o rao ad produc pe esmaor of populao varace usg coeffce of kuross ad he

6 A Effce Dual o Rao ad Produc Esmaor of Populao Varace ample urves 83 populao mea of he aular varable 3 Proposed Esmaor Movaed b Vshwakarma e al 0) ad Upadhaa ad gh 00), we propose a mproved rao esmaor of populao varace as, * ˆ * + αx α s * 3) X + α where α s a suabl chose characerzg cosa ad s obaed b mmzg he ME of he proposed esmaor * ad X ) ) + g) X g, g / ) * I order o sud he large sample properes of he proposed esmaor *, we defe s e ) ad X + e ) such ha E ) 0 for 0,) ad E e ) γ λ ), + 0 e 0 0 E, E e0 e ) γ λc e ) γ C Epressg * erms of e s 0, ), we have ˆ * ge αge α + e0 ) 3) + α) + α) Epadg he rgh had sde of 3), smplfg ad reag he erms up o he frs order of appromao, we have ˆ * Fα + e0 Fge Fge0e g e 33) α + α) α where, F + α ubracg o boh sdes of 33), we ge ˆ * Fα e0 Fge Fge0e g e 3) α + α)

7 8 A Effce Dual o Rao ad Produc Esmaor of Populao Varace ample urves bas of Takg epecaos o boh sdes of 3) ad pug he values of dffere epecaos, we ge he *, up o he frs order of appromaos as, ˆ * Fα B α ) γ FgλC + g C 35) + α) From 3), we have up o he frs order of appromao as, ˆ * α e0 Fge ) 36) quarg boh sdes of 36), we have ˆ ) e Fge ) + 37) * α 0 [ e0 F g e Fge0e ] Takg epecaos o boh sdes of 37) ad pug he values of dffere epecaos, up o he frs order of appromao, we ge he mea squared error of * as, ME ˆ * α 0 ) γ [ λ ) + F g C Fgλ C ] 38) Whch s mmum for, F λ gc 39) The mmum mea squared error of * for hs opmum value of F s, ˆ * ME ) γ [ λ ) λ ] 30) m α 0 From equao 30) ad ), we have Effcec Comparso ˆ * V ) ME ) γ λ 0 ) 0 m α >

8 A Effce Dual o Rao ad Produc Esmaor of Populao Varace ample urves 85 From above equao, we see ha he proposed esmaor * s beer ha he esmaor 0 as has lesser mea squared error ha he esmaor 0 From equao 30) ad 5), we have ˆ * ME R ) ME ) γ [ λ ) λ ) + λ ] > 0, f, λ ) + λ ] > λ ) ) m α 0 [ 0 Whe he above codo s me ou, he proposed esmaor s beer ha he esmaor R From equao 30) ad 7), we have ˆ * λ0 ) λ0 ) ME ) MEm α ) γ λ ) + λ > 0, f, + λ > λ ) 3) Whe he above codo s fulflled, he proposed esmaor * performs beer ha he esmaor Thus wll be preferable over uder above codo From equao 30) ad 9), we have ˆ * λ0 ) ME ) MEm α ) γ + λ ) + λ > 0, ) Uder he above codo, he proposed esmaor * s beer ha he esmaor as wll have lesser mea squared error as compared o esmaor Thus wll be preferred over From equao 30) ad ), we have ˆ * ME ) ME ) γ [ C λ C + λ ] γ [ C λ ] > 0 5) 3 m α Thus we see ha he proposed esmaor * has lesser mea squared error ha he esmaor 3 Thus wll be more effce ha he esmaor 3 From equao 30) ad 5), we have ˆ * ME ) ME ) γ [ C + λ C + λ ] γ [ C + λ ] > 0 6) m α

9 86 A Effce Dual o Rao ad Produc Esmaor of Populao Varace ample urves Thus we see ha he proposed esmaor * has lesser mea squared error ha he esmaor Thus wll be preferred over he esmaor From equao 30) ad 9), we have ) ˆ * C C ME 5 MEm α ) γ [ λc + λ] γ [ λ] > 0 7) I s clear from above equao ha he proposed esmaor * s havg lesser mea squared error as compared o esmaor Thus wll be more effce ha he esmaor From equao 30) ad ), we have ) ˆ * C C ME 6 MEm α ) γ [ + λc + λ] γ [ + λ] > 0 8) Whch shows ha he proposed esmaor * s beer ha he esmaor 6 as has lesser mea squared error as compared o 6 5 umercal Illusrao To judge he heorecal fdg of he proposed esmaor he form of s performace of he proposed esmaor regardg s mea squared error, a emprcal sud has bee carred ou usg followg wo real populaos The ources ad he parameers for wo populaos are gve below; Populao I- ource: Murh [6] X: oupu Y: umber of workers 5, 5, X , Y , C 0 35, C 0 76, ρ 0 936, λ 667 0, λ , λ 075, λ 3377 Populao II- ource: Gujara [3] X: Average mles per gallo) Y: Top peed mles per hour) 8,, X 568, Y , C 0 8, C 0 8,

10 A Effce Dual o Rao ad Produc Esmaor of Populao Varace ample urves 87 ρ 06935, λ , λ 0 6 8, λ 37, λ 0 Table Perce relave effcec PRE) of dffere esmaors wh respec o 0 Esmaor Pop-I Pop-II * R 5 * * * * * Proposed) *Daa s o applcable 6 Resul ad Cocluso The prese mauscrp deals wh he esmao of populao varace of sud varable usg aular formao We have proposed a dual o rao ad produc pe esmaor of populao varace usg coeffce of kuross ad he mea of he aular varable Up o he frs order of appromao, he bas ad he mea squared error of he proposed esmaor have bee obaed The opmum value of he characerzg scalar whch mmzes he mea squared error of he proposed esmaor s obaed up o he frs order of appromao From he able- of he umercal sud ad gve able- ad he heorecal dscusso seco-, we see ha he proposed esmaor s havg ver less mea squared error as compared o oher meoed esmaors of populao varace Thus s he mos effce esmaor of populao varace amog he class of all meoed esmaor of populao varace Therefore he proposed esmaor should be preferred over all oher esmaors for he esmao of populao varace Refereces [] Amber Asghar, Aamr aaullah ad Muhammad Haf 0): Geeralzed Epoeal Tpe Esmaors for Populao varace urve amplg Revsa Colombaa deesadísca, 37 ), 3

11 88 A Effce Dual o Rao ad Produc Esmaor of Populao Varace ample urves [] Bhal, ad RK Tueja, 99 Rao ad produc pe epoeal esmaor Iformao ad Opmzao ceces : [3] Gujara, D 00), Basc Ecoomercs, ed, The McGraw-Hll Compaes [] Isak, C T 983) Varace esmao usg aular formao, Joural of Amerca ascal Assocao, 78, 7-3 [5] Kha, M ad Aruachalam, A 0) Esmao of Populao Varace usg he Kowledge of Kuross of a Aular Varable uder mple Radom amplg, Ieraoal Joural of Appled cece ad Mahemacs,,, [6] Murh, M 967), amplg Theor ad Mehods, Calcua ascal Publshg oce,kolkaa, Ida [7] gh, R, Chauha, P, awa, & maradache, F 0), Improved epoeal esmaor for populao varace usg wo aular varables, Iala Joural of Pure ad Appled Mahemacs 8, 0 08 [8] Upadhaa, L ad gh, G 00) Cha-pe esmaors usg rasformed aular varable wo-phase samplg AME, Vol 38, 0,, -9 [9] Vshwakarma, GK, Gagele, RK ad gh, R 0) A Effce Vara of Dual o Rao ad Produc Esmaor ample urves, The Phlppe asca, 63,, -9