Comparison of Dual to Ratio-Cum-Product Estimators of Population Mean
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1 Research Joural of Mathematcal ad Statstcal Sceces ISS Vol. 1(), 5-1, ovember (013) Res. J. Mathematcal ad Statstcal Sc. Comparso of Dual to Rato-Cum-Product Estmators of Populato Mea Abstract avee Kumar Boroju 1 ad Kollu Raga Rao 1 Departmet of Statstcs, Osmaa Uverst, Hderabad, IDIA Sr Vekateshwara Veterar Uverst Dar Techolog Program, Kamaredd, zamabad, IDIA Avalable ole at: Receved 17 th October 013, revsed d October 013, accepted 8 th ovember 013 I ths paper, some dual to rato-cum-product estmators of populato mea usg kow parameters of aular varables are cosdered. These estmators are computed ad compared wth respect to bas ad mea squared error usg a smulato stud from the ormal populato. Coeffcet of skewess ad coeffcet of kurtoss are also computed to have a dea about the samplg dstrbuto of dual to rato-cum-product estmators of populato mea. Dual to rato-cumproduct estmators are more effcet tha that of the mea per ut, classcal rato estmator ad lear regresso estmators ad Choudhur ad Sgh (01) estmator s more effcet estmator amog the dual to rato-cum-product estmators of populato mea. Kewords: Rato-cum-product estmator, smulato, stadard error ad relatve bas. Itroducto I ths paper, some dual to rato-cum-product estmators avalable the lterature are revewed ad ther effceces are compared b smulato from the ormal dstrbuto wth kow correlato coeffcet. The estmatos of the populato mea s a persstet ssue samplg practce ad ma efforts have bee made to mprove the precso of the estmates. The lterature o surve samplg descrbes a great varet of techques for usg aular formato b meas of rato, product ad regresso methods 1. Srvekataramaa frst proposed dual to rato estmator. Badopadha proposed dual to product estmator 3. Sgh proposed class of ubased dual to rato estmators 4. Sharma ad Talor suggested a rato-cum-dual to rato estmator as a lear combato of classcal rato estmator ad dual to rato estmator 5. Choudhur ad Sgh proposed a rato-cum-product rato estmator as a lear combato of classcal rato estmator ad the dual to product estmator 6, 7. These estmators are compared wth respect to stadard error, relatve bas, skewess ad kurtoss usg smulato b geeratg radom samples from dfferet bvarate populatos wth kow correlato coeffcets betwee the stud ad aular varables. Dual to Rato-Cum-Product Estmators Let (, ) deote the values of the respose varable ad a aular varable respectvel a fte populato. Let ad deote the sample meas of stud varable ad a aular varable ad assumg that the populato mea ( ) of aular varable s kow advace. The mea per ut estmator of populato mea s gve b T 0 = = = 1 Y The classcal rato estmator of populato mea = 1 Y = s gve b T 1 = T = b The lear regresso estmator s gve b ( ) Ad the product estmator s gve b T = 3 Iteratoal Scece Cogress Assocato 5
2 Research Joural of Mathematcal ad Statstcal Sceces ISS Vol. 1(), 5-1, ovember (013) Res. J. Mathematcal ad Statstcal Sc. Where = = 1, = s = 1, s 1 s ad = - 1 b =, ( ) = Srvekataramaa suggested a dual to rato estmator usg the trasformato =1,, ; where g s = 1. = = 1 + g g =. Ths trasformato elds to ew estmator ( ) estmator of. The dual to rato estmator due to Srvekataramaa s gve b T Ad Badopadha 3 obtaed dual to product estmator as T 5 = 4 = = 1 + g g, or ( ) ad whch s a ubased Sharma ad Talor 5 suggested the followg rato-cum-dual to rato estmator b takg the lear combato of classcal rato estmator ad dual to rato estmator. T6 = + 1 α ( ) K g α whereα =. 1 g Recetl, Choudhur ad Sgh 6,7 proposed a estmator as T + ( α ) α whereα. 7 = 1 ( K + g) = 1 + g ρc S S The populato quattes K =, ρ =, C =, C C S S Y ad S = ( Y )( ) ( 1) = 1 /. S =, S = ( ) /( 1), S = ( Y ) /( 1) Estmates of the bas ad MSE of these estmators ca be obtaed from ther respectve papers. The authors compared the effcec of these estmators wth that of the ether mea per ut estmator or wth the classcal rato estmators. It s ver dffcult to compare these estmators aaltcall wth the remag estmators dscussed ths paper. Comparso of Dual to Rato-Cum-Product Estmators I ths paper, a attempt s made to compare the dual to rato-cum-product estmators usg smulato. Smulato s a alteratve techque to compare the effcec of the estmators wheever the aaltcal comparsos are ot possble. Frst we have geerated the correlated samples (Y, ) of sze 0 from the ormal populato. The correlato samples are geerated usg the algorthm gve b Krsha Redd et.al 8. We have geerated three populatos of sze =0 as follows: Populato-I: Ths populato cossts of the two correlated varables (Y, ) wth equal varaces are geerated usg the ~,4 ad algorthm proposed b Krsha Redd et. al. 8. Ths populato wll have the margal dstrbutos ( ) ( ρ + 1 ρ,4 ) Y ~ where ρ s the populato correlato betwee ad Y. Ths bvarate populato wll have the varace rato = 1. Populato-II: Ths populato cossts of the two correlated varables (Y, ) wth uequal varaces are geerated usg the algorthm proposed b Krsha Redd et. al. 8. The populato cotas the varace of s less tha the varace of Y wth the = 1 = 1 Iteratoal Scece Cogress Assocato 6
3 Research Joural of Mathematcal ad Statstcal Sceces ISS Vol. 1(), 5-1, ovember (013) Res. J. Mathematcal ad Statstcal Sc. margal dstrbutos ~ (,4 ) the varace rato > 1. ( ) ad Y ~ ρ + 1 ρ, 4 ρ + 16 ( 1 ρ ). Ths bvarate populato wll have Populato-III: Ths populato cossts of the two correlated varables (Y, ) wth uequal varaces are geerated usg the algorthm proposed b Krsha Redd et. al. 8. The populato cotas the varace of s greater tha the varace of Y wth the ~, 5 Y ~ ρ + 1 ρ, 5 ρ ρ. Ths bvarate populato wll have margal dstrbutos ( ) the varace rato < 1. ( ) ad ( ) From each of the populato, 0 smple radom samples wthout replacemet of sze =, 30 ad are draw ad computed the above estmators each of the sample. Let T be the value of the estmator T j based o the j th SRSWOR for j=1, 0 ad =0,1, 7. The estmator of populato of mea usg smulato s gve b Y ˆ for = 0,1, The relatve bas of Ŷ s defed as 0 = T j j= 1 Y ˆ Y 0 RB( Y ˆ ) = for = 0,1,..., 7. The stadard error of the estmator s defed as ( ˆ 1 SE Y ) = ( Tj Y ) for = 0, 1,..., 7. The Y 0 j= 1 behavor of the dstrbuto of the estmators are detfed usg the coeffcet of skewess ad coeffcet of Kurtoss. For each estmator the stadard error, relatve bas, skewess ad kurtoss are computed ad preseted agast the sample sze =, 30, ad correlato coeffcet betwee the varables Y ad, r = 0.4, 0.6 ad 0.8. The followg tables preset the results of the smulato stud. Table-1 Comparso of estmators of populato mea from Populato-I wth r=0.4 r Estmator Estmate SE RB Skewess Kurtoss T T T T T T T T T T T T T T T T T T T T T T T T Iteratoal Scece Cogress Assocato 7
4 Research Joural of Mathematcal ad Statstcal Sceces ISS Vol. 1(), 5-1, ovember (013) Res. J. Mathematcal ad Statstcal Sc. Table- Comparso of estmators of populato mea from Populato-I wth r=0.6 r Estmator Estmate SE RB Skewess Kurtoss T T T T T T T T T T T T T T T T T T T T T T T T Table-3 Comparso of estmators of populato mea from Populato-I wth r=0.8 r Estmator Estmate SE RB Skewess Kurtoss T T T T T T T T T T T T T T T T T T T T T T T T Iteratoal Scece Cogress Assocato 8
5 Research Joural of Mathematcal ad Statstcal Sceces ISS Vol. 1(), 5-1, ovember (013) Res. J. Mathematcal ad Statstcal Sc. Table-4 Comparso of estmators of populato mea from Populato-II wth r=0.4 r Estmator Estmate SE RB Skewess Kurtoss T T T T T T T T T T T T T T T T T T T T T T T T Table-5 Comparso of estmators of populato mea from Populato-II wth r=0.6 r Estmator Estmate SE RB Skewess Kurtoss T T T T T T T T T T T T T T T T T T T T T T T T Iteratoal Scece Cogress Assocato 9
6 Research Joural of Mathematcal ad Statstcal Sceces ISS Vol. 1(), 5-1, ovember (013) Res. J. Mathematcal ad Statstcal Sc. Table-6 Comparso of estmators of populato mea from Populato-II wth r=0.8 r Estmator Estmate SE RB Skewess Kurtoss T T T T T T T T T T T T T T T T T T T T T T T T Table-7 Comparso of estmators of populato mea from Populato-III wth r=0.4 r Estmator Estmate SE RB Skewess Kurtoss T T T T T T T T T T T T T T T T T T T T T T T T Iteratoal Scece Cogress Assocato
7 Research Joural of Mathematcal ad Statstcal Sceces ISS Vol. 1(), 5-1, ovember (013) Res. J. Mathematcal ad Statstcal Sc. Table-8 Comparso of estmators of populato mea from Populato-III wth r=0.6 r Estmator Estmate SE RB Skewess Kurtoss T T T T T T T T T T T T T T T T T T T T T T T T Table-9 Comparso of estmators of populato mea from Populato-III wth r=0.8. r Estmator Estmate SE RB Skewess Kurtoss T T T T T T T T T T T T T T T T T T T T T T T T Iteratoal Scece Cogress Assocato 11
8 Research Joural of Mathematcal ad Statstcal Sceces ISS Vol. 1(), 5-1, ovember (013) Res. J. Mathematcal ad Statstcal Sc. Cocluso From the above tables, t s observed that. The estmators cosdered ths paper are almost ubased all the three populatos.. It s also observed that the estmators are asmptotcall ormall dstrbuted, sce the skewess values are ear to zero ad kurtoss values are ear to 3 all the cosdered populatos.. The stadard error of the estmators decreases whe the correlato ad sample sze creases the populatos-i, II ad III. v. From the stadard errors of the estmators t s observed that the estmators proposed b Choudhur ad Sgh 6,7 (T 7 ) ad Sharma ad Talor 5 (T 6 ) are almost equall effcet estmators ad these two estmators showg the stadard error almost equal to the stadard error of the lear regresso estmator (T ). These three estmators (T 7, T 6 ad T ) are more effcet estmators tha that of the mea per ut estmator (T 0 ) ad classcal rato estmator (T 1 ). v. The stadard error of the product estmator (T 3 ) s ver hgh compared to the stadard errors of the remag estmators. v. The dual to rato estmator (T 4 ) ad dual to product estmator (T 5 ) are effcet tha that of the mea per ut estmator (T 0 ) ad rato estmator (T 1 ). v. The dual to rato estmator (T 4 ) s effcet tha that of the dual to product estmator (T 5 ). v. From the above stud, t s clearl observed that the dual to rato-cum-product estmators are more effcet tha that of the mea per ut estmator, classcal rato estmator ad lear regresso estmators. Specall, Choudhur ad Sgh 6,7 ad Sharma ad Talor 5 estmators are more effcet estmators as compared to the other dual to rato-cum-product estmators. Refereces 1. Daa G. ad Perr P.F., Estmato of fte populato mea usg mult-aular formato, METRO - Iteratoal Joural of Statstcs, LV, (007). Srvekataramaa T., A dual to rato estmator sample surves, Bometrca, 67, (1980) 3. Badopadha S. Improved rato ad product estmators, Sakha Seres C, 4(), (1980) 4. Sgh V.P., The Class of Ubased Dual to Rato Estmators, Jour. Id. Soc. Ag. Statstcs, 56(3), 11-1(003) 5. Sharma B. ad Talor R., A ew rato-cum-dual to rato estmator of fte populato mea smple radom samplg, Global Joural of Scece Froter Research, (1), 7-31 (0) 6. Choudhur S. ad Sgh B.K., A Effcet Class of Dual to Product-Cum- Dual to Rato Estmator of Fte Populato Mea Sample Surves, Global Joural of Scece Froter Research, 1(3), 5-33 (01a) 7. Choudhur S. ad Sgh B.K., A Effcet Class of Rato-Cum-Dual to Product Estmator of Fte Populato Mea Sample Surves, Global Joural of Scece Froter Research Mathematcs ad Decso Sceces, 1 (1), 1-11 (01b) 8. Krsha Redd M., Raga Rao K. ad Boroju.K., Comparso of Rato Estmators Usg Mote Carlo Smulato, It. J. Agrcult. Stat. Sc., 6 (), (0) Iteratoal Scece Cogress Assocato 1
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