An Unbiased Class of Ratio Type Estimator for Population Mean Using an Attribute and a Variable

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1 Advace Comutatoal Scece ad Techology ISS Volume, umber 7) Reearch Ida Publcato htt:// A Ubaed Cla of Rato Tye Etmator for Poulato Mea Ug a Attrbute ad a Varable Shah Bhuha, Pravee Kumar Mra ad Sach adav Deartmet of Mathematc ad Stattc, Dr. Shakutala Mra atoal Rehabltato Uverty, Ida. Deartmet of Stattc, Luckow Uverty, Luckow U.P.), Ida. Abtract I th aer, we have codered a cla of exoetal tye rato etmator ug the auxlary formato both the form attrbute ad varable. We rooe a mroved cla of etmator ug Jack-Kfe techque. Further, t ha bee how that the rooed Jack-Kfe etmator ubaed ad ha leer mmum mea quare error uder the otmum value of characterg arameter a comared to ome commoly ued etmator avalable the lterature. A emrcal tudy cluded a a llutrato. Keyword Ubaed Cla of Etmator, Jack-Kfe techque, Ubaede ad Mea Square Error.. ITRODUCTIO I may tuato auxlary formato ued to mrove the reco of a etmator. Th auxlary formato may be the form of a varable or a attrbute or both. For examle - Heght dfferet for male ad female whch how that ex a helful attrbute whle dealg wth heght ad alo heght related wth the weght whch a varable; amout of mlk roduced by cow deed o the breed a well a o the det; yeld of wheat cro deed o the varety of wheat ad maure a well etc. I uch tuato, we ca take the advatage of the avalable formato o varable ad attrbute to creae the effcecy of the etmator. Let y be the tudy varable, x be the auxlary varable ad be the auxlary attrbute. Alo, let coder a fte oulato of ze ad we deote by

2 4 Shah Bhuha ad Pravee Kumar Mra, X X ad P be the oulato mea of y, x ad reectvely; S ), SX X X ) ad S ) P be the oulato varace. Further, o the ba of a mle radom amle of ze draw wthout relacemet from the oulato of ze, we deote by X x ad X x X ) y, be the amle mea of y, x ad reectvely;, y) ad ) be the amle varace. Followg ak ad Guta 996) ad Abu Dayyeh 3), we rooe a cla of etmator aumg that the auxlary oulato mea ad auxlary oulato roorto are kow x y X P.) It may be oted that the amle mea, rato etmator, Srvatava tye geeralzed rato etmator 967), ak ad Guta 996) rato etmator, ak ad Guta Srvatava tye geeralzed rato etmator are the member of the rooed cla of etmator.. BIAS AD MSE OF THE PROPOSED ESTIMATOR I order to obta the ba ad mea quare error MSE), let u deote by y e ), x X e ), P e ) E e ) E e ) E e ), E e ) fc E e ) fcx E e ) fcp E ee) fx C CX,, E e e f C C ) XP X P Subttutg the value from.).), we get E e e ) fpc CP e e e ee ee e e e e Thu, the ba ad mea quare error of the rooed cla gve by Ba ) f CX CP XCX C PCPC XPCX CP =.).) f A ay.3)

3 A Ubaed Cla Of Rato Tye Etmator For Poulato Mea 4 = f B ay MSE ) f C C C C C C C C C X P X X P P XP X P where / f The otmzg value of ad mmzg the MSE are.4) ) C X P PX ot XP ) CX ) C P X XP ot XP ) CP The mmum MSE wth the cla of rooed etmator gve by.5).6) m. XP MSE ) f R ) C M ay).7) where R the multle correlato coeffcet of y o x ad.. XP 3. THE PROPOSED JACK KIFE ESTIMATOR Let a radom amle of ze = m draw from the fte oulato of ze by SRSWOR ad lt t to two radom ub-amle of ze m each. Defe the followg etmator x ) ) ) y X P x ) ) ) y X P x 3 y X P 3.) ) ) ) ) where y, y, x, x ad ) ), are the reectve amle mea of the tudy varable, auxlary varable ad auxlary attrbute baed o lt amle ad each of ze m ad yxad, are the reectve mea of the tudy varable, auxlary varable ad auxlary attrbute baed o the etre amle. Proceedg mlarly, a.3), we have Ba ) f ma Ba Let u defe ba gve by ) f ma Ba ) f A K ay) 3 3.) a a alteratve etmator of oulato mea, o that t Ba ) f m CX CP XCX C PCPC XPCX CP K ay) 3.3)

4 4 Shah Bhuha ad Pravee Kumar Mra Let u ow rooe the followg Jack-Kfe etmator 3) R R K f m) R K f m m ) where 3.4) The t ca be ealy verfed that the rooed Jack-Kfe etmator a ubaed etmator of oulato mea uto the frt order of aroxmato.ow, mea quare error of the rooed etmator defed a MSE ) E By aalogy from.4), we have E 3 ) MSE 3) f B E 3 ) R E ) RE 3 ) ) R) 3.5) 3.6) Coder E ' ) E E E E 4 3.7) From.4) E MSE f B ) ) m ;, 3.8) Takg y e ), x X e ), P e );, ) ) ) ) ) ) E e ) E e ) E e ) ;, ) ) ) ug.) we ca wrte ) ) ) ) ) ) ) ) ) e e e e e e e e e wth 3.9) ) ) e e 3.) To the frt order of aroxmato, we have E ) ) E e e e e e e ) ) ) ) ) ) E e ) ) ) ) ) ) ) ) ) ) e ) E e e ) E e e ) E e e ) E e e ) ) ) ) ) ) ) ) ) E e e ) E e e ) E e e ) E e e )

5 A Ubaed Cla Of Rato Tye Etmator For Poulato Mea 43 ow, ug the reult gve Sukhatme ad Sukhatme 997) E e e ) C ) ) E e e C C, ) ) E e e ) CX ) ) ) ) ) X X E e e E e e ) ) ) ) ) XPCX CP, E e e ) CP ) PC CP ) ) ) ) E e e ) XPCX CP, E e e ) ) ) X C CX E e e ) PC CP 3.) we get E ) ) B 3.) Puttg the value from 3.8) ad 3.) 3.7), we get E ) MSE ) B B f B 4 m ow, coder 3 3 E E E 3 ) ) E 3 ) 3.3) 3.4) To the frt order of term ) ) ) E 3 ) ) E e e e e e e ;, E e ) ) ) ) ) e ) E ee ) E e e ) E e e ) E e e ) ) ) ) ) E ee ) E ee ) E e e ) E e e ) Subttutg the followg reult gve by Sukhatme ad Sukhatme 997) E e e ) f C ) E e e ) f C ) X E e e ) f C ) P E e e ) f C C ) X X E e e ) f C C ) ) X X E e e ) f C C XP X P E e e ) f C C ) ) P P E e e ) f C C XP X P E e e ) f C C ) P P we get E f B 3 ) ) ;, 3.5)

6 44 Shah Bhuha ad Pravee Kumar Mra Puttg the value from 3.5) 3.4), we get E 3 ) ) f B Subttutg the value from 3.6), 3.3), ad 3.6) 3.5), we get ) MSE f R R B f B R) 3.6) 3.7) whch ame a that of.4) but th rooed jack kfe etmator better tha the rooed etmator gve ecto ad the ee of ubaede. Alo, the otmum value of the arameter are alo ame a gve by.4) ad.5); thereby gvg the ame mmum MSE a that of.6). 4. COMPARATIVE STUD Let u coder the followg etmator for oulato mea:. Samle mea SRSWOR) y v. or MSE ) M R. XPC y. Rato etmator ug auxlary varable X x v. or 4.) X X. XP X MSE ) M f C C ) R ) C 4.) y 3. ak ad Guta 996) Rato etmator ug auxlary attrbute 3 P v. or 3 P P. XP P MSE ) M f C C ) R ) C 4.3) 4. Bahl ad Tuteja 99)Exoetal Rato etmator ug auxlary varable 4 yex X x or X x v. CX MSE 4 ) M f X C ) R. XP X ) C 5. Ug Bahl ad Tuteja 99) Exoetal Rato etmator ug auxlary varable, 4.4)

7 A Ubaed Cla Of Rato Tye Etmator For Poulato Mea 45 Sawa ) rooed the followg etmator ug auxlary attrbute 5 yex P or P v. CP MSE 5 ) M f PC ) R. XP P ) C 4.5) 5. EMPIRICAL STUD Coder the data from the data ource: Advace Data from Vtal ad Health Stattc, umber 347, October 7, 4 CDC) dealg the heght of the eole of dfferet age grou of the Uted State. X P C C C 36,, 4.8, 39.63,.5,.3673, X.3649,.896,.94679,.49,.539, R P X P XP. XP Table 5.: PRE of varou etmator wth reect to amle mea Etmator PRE or COCLUSIO The comaratve tudy of the rooed Jack-Kfe etmator etablhe t uerorty the ee of ubaed ad mmum mea quare of error over amle mea, rato etmator ad exoetal rato etmator ug auxlary varable ad auxlary attrbute uder the otmum codto.

8 46 Shah Bhuha ad Pravee Kumar Mra REFERECES [] Abu Dayyeh W. A., Ahmed M.S., Ahmed R.A. ad Muttlak H. A. 3) :Some etmator of a fte oulato mea ug auxlary formato, Aled Mathematc ad Comutato, 39, [] Bahl S. ad Tuteja R. K. 99): Rato ad Product tye exoetal etmator, Iformato ad Otmzato cece, Vol.XII, I, [3] Bhuha, S. 3). Imroved Samlg Stratege Fte Poulato. Scholar Pre, Germay. [4] Bhuha S. ). Some Effcet Samlg Stratege baed o Rato Tye Etmator, Electroc Joural of Aled Stattcal Aaly, 5), [5] Bhuha S., Guta R. ad Padey S. K. 5). Some log-tye clae of etmator ug auxlary formato, Iteratoal Joural of Agrcultural ad Stattcal Scece, ), [6] Bhuha S. ad Katara, S. ). O Clae of Ubaed Samlg Stratege, Joural of Relablty ad Stattcal Stude, 3), 93-. [7] Bhuha, S. ad Kumar S. 6). Recet advace Aled Stattc ad t alcato. LAP Publhg. [8] Bhuha S., Padey A. ad Sgh R.K. 9) Imroved Clae of Regreo Tye Etmator ; Iteratoal Joural of Agrcultural ad Stattcal Scece ISS: ), 5), [9] Bhuha S. ad Padey A. ). Modfed Samlg Stratege ug Correlato Coeffcet for Etmatg Poulato Mea, Joural of Stattcal Reearch of Ira, 7), - 3. [] Bhuha S., Sgh, R. K. ad Katara, S. 9). Imroved Etmato uder Mdzuo Lahr Se-tye Samlg Scheme, Joural of Relablty ad Stattcal Stude, ), [] Bhuha S., Maalda R.. ad Guta P. K. ). Imroved Samlg Stratege baed o Modfed Rato Etmator, Iteratoal Joural of Agrcultural ad Stattcal Scece, 7), [] Gray H. L. ad Schucay W. R. 97): The Geeralzed Jack-kfe Stattc, Marcel Dekker, ew ork. [3] ak V. D. ad Guta P. C. 996): A ote o etmato of mea wth kow oulato roorto of a auxlary character. Jour. Id. Soc. Agr. Stat., 48), [4] Queoulle M. H. 956): ote o ba etmato, Bometrka, 43, [5] S.K.Srvatava 967): A etmator ug auxlary formato amle urvey, Calcutta Stattcal Aocato Bullet, 6, -3. [6] Sukhatme P. V. ad Sukhatme B. V. 97): Samlg theory of urvey wth alcato. Iowa State Uverty Pre, Ame, U.S.A.