Joural of Scietific Research Vol. 6 8 : 3-34 Baaras Hidu Uiversity Varaasi ISS : 447-9483 Geeralized ad trasformed two phase samplig Ratio ad Product ype stimators for Populatio Mea Usig uiliary haracter i the presece of Uit orespose o study ad auiliary haracter B. B. Khare Utarsh ad Supriya Khare Departmet of Statistics Istitute of Sciece BHU Varaasi-5 Idia. mail- bbhare56@yahoo.com mail- utarshparmar@gmail.com mail- suprih@gmail.com bstract I this paper we have proposed geeralized ad trasformed two phase samplig ratio ad product type estimators for populatio mea usig auiliary character i the presece of uit orespose o the study ad auiliary character. he properties of the proposed estimators have bee studied. comparative study has bee made with the relevat estimators ad a empirical study has bee give i the support of the problem. Keywords: Mea square error populatio mea auiliary character uit orespose o study ad auiliary character.. Itroductio he use of a auiliary character to icrease the efficiecy of the estimator for populatio parameters is widely used i researches related to the field of socioecoomic agriculture ad biomedical scieces. he research wor o the estimatio of populatio parameters usig auiliary characters by several authors have bee reviewed by ripathi et al. 994 Khare 3 ad Khare et.al 3. For a eample i a forest surveys the average amout of timber of a tree ca be estimated by usig the diameter of the tree as a auiliary character. I this cotet several estimators for populatio mea have bee proposed by Kiregyera 984 Srivastava et al.989 Sahoo et al.993 Kadilar ad igi 5 ad Khare ad Kumar Further the research wor i the presece of orespose o study character has bee reviewed by Khare et al. 4. hile coductig a sample survey it may ot be possible to collect iformatio o all the uits selected i the sample due to orespose. I this case Hase ad Hurwitz 946 have suggested a method of sub-samplig from o-respodets ad the estimator for populatio mea based o available iformatio o respodig uits ad sub sample uits draw from the o-respodig uits i the sample has bee proposed. Further several problems o estimatig the populatio mea i the presece of orespose o the study character through sample surveys i the presece of o-
4 B. B. KHR UKRSH D SUPRI KHR respose have bee cosidered by Rao 986 9 Khare ad Srivastava 993 95 97 Sigh et al. 8 9a b ad Khare ad Kumar. I this paper we have proposed geeralized ad trasformed two phase samplig ratio ad product type estimators for populatio mea usig auiliary character i the presece of uit orespose o the study ad auiliary character. he properties of the proposed estimators have bee studied. comparative study has bee made with the relevat estimators ad a empirical study has bee give i the support of the problem.. Proposed stimators: j th Let y ad deote the study character ad auiliary character havig values j ad j j... with their populatio meas ad. he populatio of size is supposed to be divided ito respodig ad o- respodig uits such that. ccordig to the Hase Hurwitz 946 a sample of size is draw from the populatio of size by usig simple radom samplig without replacemet SRSOR scheme it has bee observed that oly uits are respodig ad uits are ot respodig i the sample of size for the study character y. Further by maig some etra effort a sub-sample of size r r / > is draw from o-respodig uits by usig SRSOR samplig scheme ad related iformatio o available uits are collected by persoal iterview for study character y. I case whe there is o orespose o the auiliary character ad the populatio mea of the auiliary character is ot ow we draw the first phase sample of size < from the populatio of size by usig simple radom samplig without replacemet SRSOR scheme ad estimate the populatio mea by the first phase sample mea based o uits. Further a secod phase sample of size < is draw from first phase sample of size by usig SRSOR samplig scheme ad observed that oly uits are respodig ad uits are ot respodig for the study character y. gai we draw the sub-sample of size r r / > from o-respodig uits by usig SRSOR samplig scheme ad collect the related iformatio o r uits usig persoal iterview method by maig etra effort.
GRL D RSFORMD O PHS SMPLIG RIO D... 5 Usig Hase ad Hurwitz 946 techique of sub samplig from orespodets the estimator for based o robservatios o the study character y is give as follows: y y y. where y ady are the meas of character y based o the estimator is give by f V y S y S y ad r uits. he variace of. where f S y ad S y are the populatio mea squares of study character y for the etire populatio ad for the o-respodig part of the populatio. r Similarly the sample mea of values of based o values o character is give as follows:.3 where deote the meas of auiliary character based o adr uits ad he variace of the estimator is give by f V S S.4 where ad S S are the populatio mea squares of study character for the etire populatio ad for the o-respodig part of the populatio. I case whe there is o orespose o the auiliary character several researchers have proposed two phase samplig estimator for populatio mea of study character. But if there is also uit orespose o the auiliary character alog with the study character i this case samplig desig ad the estimatio procedure will be differet. o estimate we first draw a sample of size for populatio of size by usig SRSOR method of samplig ad observed that uits respod ad uits
6 B. B. KHR UKRSH D SUPRI KHR do ot respod. he from uits we draw a subsample of size r r / > ad observe iformatio o character. ow we estimate by by usig ad r uits o character usig Hase Hurwitz 946 techique. he estimator for is give as follows.5 where ad are the sample meas of based o ad r uits. Further we draw a sample of size from ń ad observe uits respod ad uits do ot respod. gai we draw a subsample of size r r / > from orespodig uits ad the estimate for ad based o r uits. he covetioal ad alterative ratio product ad regressio type estimators for populatio mea uder this scheme are give by Khare ad Srivastava 993 995 which are give as follows: y y t t.6 Further Khare ad Kumar 9 have proposed trasformed two phase samplig ratio ad product type estimators for populatio mea i presece of orespose which are give as follows: y D y D t 3 t4 D D.7 I the preset cotet we have proposed geeralized ad trasformed two phase samplig ratio ad product type estimators for populatio mea usig auiliary character i the presece of uit orespose o the study ad auiliary character which are give as follows: y R y P y.8 3. Bias ad mea square error of the proposed estimator: I order to derive the epressio for Bias ad Mea square error ad bias of the proposed estimator R ad P we have:
GRL D RSFORMD O PHS SMPLIG RIO D... 7 Let y such that l ad l <. l y y y y y 3. 3. } { } { R 3.3 where > & < } { } { P
8 B. B. KHR UKRSH D SUPRI KHR 3.4 where > & < ow the Bias ad MS of the estimators P R ad are give as follows: }] { [ Bias y y Bias R y y Bias P y y 3.5 3.6 3.7
GRL D RSFORMD O PHS SMPLIG RIO D... 9 } { MS y y y y R MS y y y y P MS y y y y ow differetiatig 3.8 with respect to ad equatig to zero we have 3.8 3.9 3.
3 B. B. KHR UKRSH D SUPRI KHR. y y opt fter puttig the value of opt. from 3. i 3.8 the miimum value of MS is give as follows: MS ] [ mi y y y y MS gai differetiatig 3.9 with respect to D ad equatig to zero we obtai the optimum value of which is give by. y y opt fter puttig the value of opt. from 3.3 i 3.9 the miimum value of MS R is give as follows: ] [ mi y y y y R MS Similarly differetiatig 3. w.r.t. D ad equatig to zero we obtai the optimum value of which is give as follows: 3. 3. 3.3 3.4
GRL D RSFORMD O PHS SMPLIG RIO D... 3 opt. y y 3.5 fter puttig the value of opt. from 3.5 i 3. the miimum value of MS P is give by: y y MS P mi [ y y ] 3.6 o study the precisio of the estimator R ad P a empirical study is cosidered. 4. mpirical Study: 9 Village/ow/ ward wise populatio of urba area uder police- statio- Baria ahasil- hampua Orissa has bee tae uder cosideratio from District esus Haboo 98 Orissa published by Govt. of Idia Siha. he last 5% villages i. e. 7 villages have bee cosidered as o-respose group of the populatio. he values of the parameters of the populatio uder study are as follows: 45.38 59.83 y. 7667. 764. y 6899. 549 ρ y.95 ρ y. 875. where average umber of literate persos i the village y by usig average umber of o-worers i the village as auiliary character.
3 B. B. KHR UKRSH D SUPRI KHR able: Relative efficiecy i % of the estimators with respect to differet values of 9 5 3. y for the stimators / /4 /3 / y 548.677 465.6998 38.769 t 3 6.9357 338.83 65.877 8.759 7.9459.5856 t 4 6.9357 338.83 65.877 8.759 7.9459.5856 83.7947 98.547 8.9773 54.54 8.467.48 R 83.7947 98.547 8.9773 54.54 8.467.48 P 83.7947 98.547 8.9773 54.54 8.467.48 Figures i paretheses give the MS. From the able we observed that the MS of the proposed estimators R ad P are equal. It is also observed that the value of MS. of the proposed ad relevat estimators decreases as the value of decreases. 5. oclusio: I the case of uit orespose o the study ad auiliary character geeralized two phase ad trasformed two phase ratio ad product type estimators R ad P are foud to be more efficiet tha the relevat estimatorst 3 t4 ad y. So it is suggested to use R ad P i case of two phase samplig with uit orespose o the study ad auiliary character. 6. Refereces:. Hase M.H. ad Hurwitz.. 946: he problem of o-respose i sample surveys J. mer. Statis. ssoc. 4: 57-59.. Kadilar. ad igi H. 5: ew estimator usig two auiliary variables. pplied Mathematics ad omputatio 6 9-98.
GRL D RSFORMD O PHS SMPLIG RIO D... 33 3. Khare B.B. 3: Use of auiliary iformatio i sample survey upto - review I Mathematics ad Statistics i gieerig Biotechology ad Sciece dited by D.K. Se ad P.K. Mishra. M/S etre for Bio-Mathematical Studies Idia 76-87. 4. Khare B.B. ad Kumar S. 9: rasformed two phase samplig ratio ad product type estimators for populatio mea i the presece of o respose. ligarh J. Stat. Vol. 9 9-6. 5. Khare B.B. ad Kumar S. : stimatio of the populatio mea usig ow coefficiet of variatio of the study character i presece of orespose ommu. Statist. -heory Meth. 4 44-58. 6. Khare B.B. ad Kumar S. : hai regressio type estimators usig additioal auiliary variables i two phase samplig i presece of orespose at. cad. Sci. letr. Idia 33& 369-375. 7. Khare B.B. ad Srivastava S. 995: Study of covetioal ad alterative two phase samplig ratio product ad regressio estimators i presece of orespose. Proc. at. cad. Sci. Idia 65 II 95-3. 8. Khare B.B. ad Srivastava S. 997: rasformed ratio type estimators for the populatio mea i the presece of orespose. ommu. Statist. - heory Meth US. 6 7 779-79. 9. Khare B.B. ad Srivastava S. : Geeralized stimator for populatio mea i presece of orespose It. J. Math. & Stat. Sc. 9 75-87.. Khare B.B. ad Srivastava S. 993: stimatio of populatio mea usig auiliary character i presece of orespose at. cad. Sci. lett. Idia 63-4.. Khare B.B. Srivastava U. ad Kumar K. 3: Geeralized chai ratio i regressio estimator for populatio mea usig two auiliary characters i sample surveys. J. Sci. Res. BHU Vol-5747-53.. Khare B.B.: stimatio of populatio parameters usig the techique of subsamplig from o-respodets i sample surveys - Review. Proc. at. cad. Sci. Sec 843 337-343 4. 3. Kiregyera B. 984: Regressio type estimators usig two auiliary variables ad the model of double samplig from fiite populatio Metria 3 5-6. 4. Rao P. S. R. S. 986: Ratio estimatio with sub samplig the o respodets Survey Methodology 7-3. 5. Rao P. S. R. S. 99: Regressio estimators with sub samplig of o respodets I-Data Quality otrol heory ad Pragmatics ds. Guar. Liepis ad V.R.R. Uppuluri Marcel Deer ew or 9-8.
34 B. B. KHR UKRSH D SUPRI KHR 6. Sahoo J. Sahoo L.. ad Mohaty S. 993: regressio approach to estimatio i two-phase samplig usig two auiliary variables urret Sciece 65 73-75. 7. Sigh H.P. ad Kumar S. : Improved estimatio of the populatio mea uder two phase samplig with sub-samplig of o-respodets Jour. of Stat. Plaig ad Iferece 4 536-55. 8. Sigh H.P. ad Kumar S. 8: regressio approach to the estimatio of the fiite populatio mea i the presece of o-respose ustr. & ew Zeala. Jour. of Statist. 54 395-48. 9. Sigh H.P. ad Kumar S. 9b: geeral procedure of estimatig the populatio mea i the presece of o-respose uder double samplig usig auiliary iformatio SOR-Statistics ad Operatio Research 33 7-83.. Sigh H.P. ad Kumar S. 9a: geeral class of estimators of the populatio mea i survey samplig usig auiliary iformatio with sub samplig the o-respodets he Korea J. ppl. Statis. 387-4.. Srivastava S. Rai Srivastava S.R. ad Khare B.B. 989: hai ratio-type estimators for ratio of two populatio meas usig auiliary characters ommu. Statist.-heory Meth. 8 397-396.. ripathi. P. Das. K. ad Khare B.B. 994: Use of auiliary iformatio i sample surveys- Review ligarh Joural of Statist 4: 79-34.