Use of Non-Conventional Measures of Dispersion for Improved Estimation of Population Mean

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1 Amerca Joural of Operaoal esearch 06 6(: DOI: 0.59/.aor Use of o-coveoal Measures of Dsperso for Improve Esmao of Populao Mea ubhash Kumar aav.. Mshra * Alok Kumar hukla hak Kumar am agar gh 4 Deparme of Mahemacs a ascs (A Cere of Ecellece Dr. ML Avah Uvers Fazaba U.P. Ia Deparme of ascs D.A.V College Kapur U.P. Ia Cere for Ecoomc ues a Plag chool of ocal ceces Jawaharlal ehru Uvers ew Delh Ia 4 Kamla ehru Isue of Maageme & Egeerg Dr APJ Abul Kalam Techcal Uvers Ia Absrac I hs paper we have propose some mprove mofe rao pe esmaors for populao mea of su varable usg aular formao he form of o-coveoal measures of sperso G s mea fferece Dowo s meho a probabl weghe momes gve b Ab (06 wh lear combao of populao coeffce of kewess a Kuross of aular varable. The large sample properes of he propose esmaors have bee sue up o he frs orer of appromao ha s he bases a he mea square errors. A comparso has bee mae wh he esg esmaors of populao mea. The coos uer whch he propose esmaors perform beer ha he oher esg esmaors have bee meoe. A emprcal su s also carre ou o usf he heorecal fgs. The heorecal as well as he emprcal fgs show he mproveme of he propose esmaors over oher esg esmaors for he esmao of populao mea. Kewors u varable Aular varable Bas Mea quare Error Effcec. Irouco Wheever he populao s large s ver me akg a cosl o ge he formao o each of he populao u. amplg s a ver goo alerave o overcome hs problem. I s ver aural o esmae a parameer of he populao b s correspog sasc. Thus o esmae he populao mea he mos approprae esmaor s he sample mea. The sample mea s ubase bu seems o have large amou of varao. ow our am s o seek a esmaor whch ma be base bu shoul have lesser mea square error as compare o sample mea. The use of aular formao supple b he aular varable fulflls our am o f more a more effce esmaors. The aular varable s he varable abou whch he epermeer has full formao a s collece wh he ma varable uer su whou creasg he cos of he surve. The aular varable ma be posvel or egavel correlae wh he su varable uer coserao. Whe he aular varable has posve relaoshp wh he ma varable uer su a he le of regresso of o passes hrough org rao pe esmaors are use o esmae populao parameers uer * Correspog auhor: sa_00@ahoo.co. (.. Mshra Publshe ole a hp://oural.sapub.org/aor Coprgh 06 cefc & Acaemc Publshg. All ghs eserve coserao. Whe has egave relaoshp wh he ma varable uer su prouc pe esmaors are use oherwse regresso pe esmaors are use for he esmao of parameers uer vesgao. I he prese su we have cosere he posve correlao case ol so ol rao pe esmaors are sue alog wh he propose rao pe class of esmaors. Le he fe populao of eres coss of sc a efable us a le (... be a bvarae sample of sze ake from ( usg a WO scheme. Le a respecvel be he populao meas of he aular a he su varables a le a be he correspog sample meas whch are ubase esmaors of a respecvel. Le ρ be he correlao coeffce bewee a.. oaos Followg oaos have bee use hs mauscrp - ze of he populao - ze of he sample - u varable - Aular varable - Populao meas - ample meas

2 70 ubhash Kumar aav e al.: Use of o-coveoal Measures of Dsperso for Improve Esmao of Populao Mea - Populao aar Devaos - Populao Covarace bewee a C C - Coeffces of Varao M - Mea of he aular varable ρ - Correlao coeffce bewee a s b - egresso coeffce of o s ( β - Coeffce of kewess of ( ( aular varable 4 ( ( ( β ( ( ( 4 - ( ( Coeffce of Kuross of aular varable Q Q QD - Quarle Devao 4 G - G s Mea Dfferece π D - Dowo s Parameers ( π ( - Probabl Weghe Momes - Bas of he esmaor B(. V (. - Varace of he esmaor ME(. - Mea square error of he esmaor ME( PE( e e p *00 - Perceage relave ME( effcec of he esmaor p p over e. evew of Esg Esmaors As meoe above he mos approprae esmaor for populao mea s he sample mea ha s mea per u esmaor gve b o I s a ubase esmaor of populao mea a s varace up o he frs orer of appromao s V ( 0 f ( Cochra (940 keepg m ha he mea per u esmaor has suffcel large varace use he aular varable o esmae he populao mea of he ma varable a propose he raoal rao esmaor as I s a base esmaor of populao mea a s bas a mea square error up o he frs orer of appromao respecvel are f B( r [ ρ ] f ME( r [ ρ ] where ( Kalar a Cg (004 suggese some mprove rao pe esmaors of populao mea as ( C ( C ( ( β β ( C ( β C 4 ( C β 5 β The bases a he mea square errors of above esmaors up o he frs orer of appromaos respecvel are Where 4 f B( (...5 f ME ( [ ( ρ ] (...5 ( β C C 5 β C C β Kalar a Cg (006 suggese some more mprove rao pe esmaors of populao mea as

3 Amerca Joural of Operaoal esearch 06 6(: ( ( ρ 6 ρ ( C ρ 7 ρ ( ρ C ( ρ C 8 ( ( β ρ ρ 9 ( ρ ( ρ β 0 β The bases a he mea square errors of above esmaors up o he frs orer of appromaos respecvel are Where f B( ( f ME ( [ ( ρ ] ( (4 6 ρ β 9 ρ C 7 C ρ 0 ρ ρ β 8 ρ ρ a a Ta (00 suggese wo mofe rao pe esmaors of populao mea usg formao o aular varable he form of coeffce of skewess a kuross respecvel as ( β ( β ( ( β β β The bases a he mea square errors of above esmaors up o he frs orer of appromaos respecvel are f B( ( f ME ( [ ( ρ ] ( (5 Where β β β β C ubrama a Kumarpaa (0a 0b 0c suggese some mofe mprove rao pe esmaors of populao mea usg mea wh coeffce of skewess kuross a coeffce of varao as ( M ( M M ( C M 4 ( M ( β M 5 ( M ( β M 6 The bases a he mea square errors of above esmaors up o he frs orer of appromaos respecvel are Where 5 f B( ( f ME ( [ ( ρ ] ( (6 M β M C 4 C M 6 β M Jeela e al. (0 suggese he followg mofe rao pe esmaor of populao mea usg coeffce of skewess a quarle fferece of aular varable as ( 7 ( β QD QD The bas a he mea square error of above esmaor up o he frs orer of appromao s gve b f B( 7 7 f ME ( 7 [ 7 ( ρ ] (7 Where 7 β QD Ab e al. (06 rouce some mofe mprove esmaors of populao mea usg some o-coveoal parameers of aular varable as

4 7 ubhash Kumar aav e al.: Use of o-coveoal Measures of Dsperso for Improve Esmao of Populao Mea ( ( G 8 G ( ρ ( ρ G 9 G ( C G 0 G ( ( D D ( ρ ( ρ D D ( C D D ( ( 4 ( ρ ( ρ 5 ( C 6 The bases a he mea square errors of above esmaors up o he frs orer of appromaos respecvel are where f B( ( f ME ( [ ( ρ ] ( (8 G 8 D 4 ρ ρ G 9 ρ ρ D ρ ρ 5 4. Propose Esmaors C C G 0 C C D C 6. C Movae b Ab e al. (06 a ubrama (0 we have propose he followg mprove mofe rao pe esmaors of populao mea usg specfc parameer of aular varable alog wh he o-coveoal parameers of aular varable as b ( p ( δ G ( δ G b ( p ( δ D ( δ D b ( p ( δ ( δ Where δ QD. The bases a he mea square errors of propose esmaors up o he frs orer of appromaos respecvel ca be obae as f B ( p p ( f ME( p [ ( ρ ] p ( (9 Where δ p δ G δ p δ D p 5. Effcec Comparso δ δ I hs seco we have compare heorecall he propose esmaor wh he esg esmaors of populao mea a have gve he coos uer whch he propose esmaor performs beer ha he oher esmaors of populao mea uer smple raom samplg whou replaceme scheme. 5(a. Comparso wh mea per u esmaor From equao (9 a equao ( we see ha he propose esmaors are beer ha he mea per u esmaor f ME( V ( 0 or p f [ p ρ ] 0 or ρ p or p ρ ± ( (0

5 Amerca Joural of Operaoal esearch 06 6(: (b. Comparso wh usual rao esmaor The propose esmaors are beer ha he usual rao esmaor f ME( ME( 0 or r f [( p ρ ρ ] 0 or p ρ ρ ( ( ( 5(c. Comparso wh Kalar a Cg (004 esmaors From equao (9 a equao ( he propose esmaors are beer ha he Kalar a Cg (004 esmaors f ME( ME( 0 or f [ p ] 0 or ± ( (...5 ( 5(. Comparso wh Kalar a Cg (006 esmaors From equao (9 a equao (4 s see ha he propose esmaors are beer ha he Kalar a Cg (006 esmaors uer he coos f ME( ME( 0 or f [ p ] 0 or ± ( ( ( 5(e. Comparso wh a a Ta (00 esmaors The propose esmaors are beer ha a a Ta (00 esmaors f ME( ME( 0 or f [ p ] 0 or ± ( ( (4 5(f. Comparso wh ubrama a Kumarpaa (0a 0b 0c esmaors From equao (9 a equao (6 s see ha he propose esmaors are beer ha ubrama a Kumarpaa (0a 0b 0c esmaors uer he coos f ME( p ME( 0 or f [ p ] 0 or ± ( ( (5 5(g. Comparso wh Jeela e al. (0 esmaor The propose esmaors are beer ha Jeela e al. (0 esmaor f ME( ME( 7 0 or p f [ p 7 ] 0 or ± ( (6 p 7 5(h. Comparso wh Ab e al. (06 esmaors From equao (9 a equao (6 s see ha he propose esmaors are beer ha Ab e al. (06 esmaors uer he coos f ME( p ME( 0 or f [ p ] 0 or ± ( ( (7 6. umercal Illusrao I hs seco we have use he aa of Kalar a Cg (004. The have cosere he aa of ol Aegea ego of Turke uer smple raom samplg scheme. We have apple our propose a oher rao pe esmaors for he aa o amou of apple prouco (as su varable a umber of apple rees (as aular varable 06 vllages of Aegea ego 999. Usg hese aa we have compue ME of propose esmaors alog wh he oher rao pe esmaors meoe hs mauscrp a hese esmaors are compare wh each oher wh respec o her ME values. We observe he sascs abou he populao. I s worh oable ha he correlao bewee he su a he aular varable s 86% a we have ake a sample of sze 40. I woul be mpora o meo ha he sample sze has o effec o he effcec comparsos of he esmaors. The populao parameers for he above aa are ρ C C.0 β. β M QD 56.5 G D

6 74 ubhash Kumar aav e al.: Use of o-coveoal Measures of Dsperso for Improve Esmao of Populao Mea Table. Cosas Bases a ME of Propose a oher esmaors Esmaor Cosa Bas Mea quare error 0 l r p p p The values of he relae cosas bases a he mea square error of he esg a propose mprove rao esmaors are gve Table-. 7. esuls a Coclusos I hs mauscrp we have propose a mprove esmaor of populao mea usg a specal parameer of aular varable alog wh some o-raoal measures of sperso of aular varable. The epressos for he bas a mea square error have bee obae up o he frs orer of appromao. A heorecal as well as umercal comparso of propose esmaors has bee mae wh oher esg esmaors of populao mea. From he above Table- ca be observe ha he relae cosas bases a mea square errors of he propose mprove rao pe esmaors are smaller ha he usual rao esmaor a he oher esg rao esmaors leraure. Thus he propose esmaors perform beer ha he usual rao esmaor a he oher esg mofe rao esmaors erms of ME whch caes ha he propose esmaors are more effce. Therefore s recommee ha he propose esmaors ma be use for he effce esmao of populao mea. I ca be furher oe ha amog he propose esmaors he esmaor p performs beer ha he oher wo propose esmaors. EFEECE [] Ab M. Abbas. herwa.a.k. a azr H.Z. (06. Improve ao Esmaors for he Populao Mea Usg o-coveoal Measures of Dsperso Paksa Joural of ascs a Operaos esearch II [] Cochra W. G. (940. amplg Techques Thr Eo Wle Easer Lme. [] Jeela M. I. Maqbool. a Mr. A. (0. Mofe rao esmaors of populao mea usg lear combao of coeffce of skewess a quarle evao. Ieraoal Joural of Moer Mahemacal ceces ( [4] Kalar C. a Cg H. (004. ao esmaors smple raom samplg. Apple Mahemacs a Compuao [5] Kalar C. a Cg H. (006. A Improveme esmag he populao mea b usg he correlao coeffce Haceepe Joural of Mahemacs a ascs 5 ( [6] ubrama J. (0. Geeralze Mofe ao Esmaor for Esmao of Fe Populao Mea Joural of Moer Apple ascal Mehos -55. [7] ubrama J. a Kumarapaa G. (0a. Esmao of Populao Mea usg Co-effce of varao a Mea of a Aular Varable. Ieraoal Joural of Probabl a ascs ( 8. [8] ubrama J. a Kumarapaa G. (0b. Esmao of populao mea usg kow mea a co-effce of skewess. Amerca Joural of Mahemacs a ascs ( 0-07.

7 Amerca Joural of Operaoal esearch 06 6(: [9] ubrama J. a Kumarapaa G. (0c. Mofe rao esmaors usg kow mea a co-effce of kuross. Amerca Joural of Mahemacs a ascs ( [0] a Z. a Ta B. (00: ao meho o he mea esmao usg coeffce of skewess of aular varable ICICA 00 Par II CCI

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

An Efficient Dual to Ratio and Product Estimator of Population Variance in Sample Surveys "cece as True Here" Joural of Mahemacs ad ascal cece, Volume 06, 78-88 cece gpos Publshg A Effce Dual o Rao ad Produc Esmaor of Populao Varace ample urves ubhash Kumar Yadav Deparme of Mahemacs ad ascs