Journal of Reliability and Statistical Studies; ISSN (Print): , (Online): Vol. 11, Issue 1 (2018): 51-66

Transcription

1 Joral of Reliabilit ad Statistical Stdies; ISS (Prit: , (Olie: Vol., Isse (08: 5-66 ESTIMATIO OF FIITE POPULATIO MEA USIG KOW OEFFIIET OF VARIATIO I THE SIMULTAEOUS PRESEE OF O - RESPOSE AD MEASUREMET ERRORS UDER DOUBLE SAMPLIG SHEME Dharmedra Kmar adav, Moika Devi * ad 3 Sbhash Kmar adav Deptt. of Statistics, Ramaja ollege, Uiversit of Delhi, ew Delhi, Idia * Sri Kara aredra Agricltre Uiversit, Jober, Jaipr, Idia 3 Deptt. of Statistics, Babasaheb Bhimrao Ambedkar Uiversit, Lckow, Idia E Mail: dkmar.adava@gmail.com; * mscagstats@gmail.com; 3 drskstats@gmail.com * orrespodig Athor Received Febrar 9, 08 Modified Ma, 08 Accepted Je, 08 Abstract The preset article cosiders the problem of fiite poplatio mea estimatio, whe the orespose ad measremet errors are preset simltaeosl tilizig kow iformatio o coefficiet of variatio of std variable. We have developed a estimator of poplatio mea which is improved ad efficiet, sig Hase ad Hrwitz (946 techiqe. Asmptotic expressios of the bias ad variace of sggested estimator have bee fod correct p to approximatio of degree oe.the optimm vale of characterizig scalar for which variace of proposed estimator is miimm has also bee calclated. We have made a theoretical efficiec compariso of proposed estimator with sal Hase Hrwitz estimator. To ampl corroborate the theoretical fidigs, a simlatio std has also bee carried ot sig R software. Ke Words: Estimatio, oefficiet of Variatio, o- respose, Measremet Errors, Doble Samplig, Simlatio.. Itrodctio I samplig theor it is a ver commo assmptio that the observatios take over the its der cosideratio are measred correctl o characteristics der std. Bt i practice, it has bee see that this assmptio is hardl satisfied i varios real world applicatios ad data get cotamiated with ma errors amel frame errors, o- respose, ad measremet errors. The errors of measremet also referred to as respose errors sall occr at the time of data collectio de to discrepac betwee observed vale ad tre vale of the characteristic der std. These measremet errors make the reslt ivalid ad the estimates ths obtaied are ot reliable. If measremet shows ver small variatios i errors of measremet so that it ca be eglected, the statistical ifereces draw, based o observed data will still be valid. O the other had, whe the measremet errors are ot so small that ca be eglected, the ifereces draw o the parameters der cosideratio ma be some how accrate ad valid bt the ma ofte approaches to expected, desirable ad fortate coseqeces [Srivastava ad Shalabh (00].

2 5 Joral of Reliabilit ad Statistical Stdies, Je 08, Vol. ( ochra (968, Shalabh (997, Sd ad Srivastva (000 ad Sigh ad Karpe (008, 009, 00 are some mai athors who discssed abot varios measremet errors. Misra ad adav (05 proposed regressio tpe estimator sig kow coefficiet of variatio der measremet errors. Misra et al (06, 07 have discssed the impact of committig errors i measremet for estimatig poplatio mea ad poplatio variace i presece of measremet errors i the data obtaied from srve samplig. osider a fiite poplatio of its. Let be the std variable. Sppose that we have selected a sample of its b sig the techiqe of simple radom samplig withot replacemet (SRSWOR o the std variable. Frther it is assmed that, i,,, is the observed samplig it havig measremet error i place of the tre vale i of the std variable. Let Where, i is the measremet error associated with the ith it of the observatio which is radom i atre with zero mea ad fixed variace. o-respose reslts whe respodets do ot provide the desired iformatio to the iterviewers. o respose rate is represeted b the proportio of o-respodets i the sample. It is worth otable that oe shold take ito cosideratio the maximm respose rate as o-respodets ma be of differet characteristics to respodets. o-respose ma lead to the biased ad iefficiet reslts of the srve. I srves samplig, orespose is a ver commo pheomeo ad it is more likel i mail srves rather tha the persoal iterviews. Mch research work is devoted to cosider the problem of o-respose i srve samplig. Foradori (96 stdied the sb-samplig of the o-respodets techiqe for estimatig the poplatio total of the std variable i two stages sig eqal probabilit samplig. Tripathi ad Khare (997 geeralized the approach of the problem of the sb samplig of o-respodets p to the case of exted of mltivariate. Okafor (00, 005 frther lead this method i elemet samplig ad two-stage samplig respectivel for differet two sccessive occasios. Sigh ad Kmar (0, Kmar (04, Sigh ad aqvi (05 ad Khare et al (05 are some remarkable developmets i the cotext of o-respose. Devi et al. (06 sggested the estimatio of poplatio mea for two stage samplig der the mechaism of radom respose i case of o-respose. Hase ad Hrwitz (946 first took ito accot the case of o-respose i mail srves, which are commol sed for data collectio i advaced cotries de to their low cost. The Hase ad Hrwitz (946 techiqe cosists i selectig sb-sample of iitial o-respodets of sbseqet data collectio with a more expesive method. The techiqe is, geerall, applicable to mail srves. The problem of o-respose is ver ofte i mail srves. The approach cosists i takig a radom sb-sample of the persos wh o have ot bee reached ad make a major effort to iterview everoe i the sb-sample. It was show that biased estimatio is possible despite the o-observatio

3 Estimatio of fiite poplatio mea sig kow coefficiet of of certai its i the iitial sample. The smmer of the proposed techiqe is give below:. The qestioaire is mailed to all the selected respodets ito the sample.. Whe the time of respose is over, the o-respodets are idetified ad sbsample of the o- respodets is selected. 3. Data is collected from the o-respodets fod i the sbsample throgh codctig the iterview. 4. The data so obtaied from two parts of the srve is combied for estimatig the poplatio parameters. It is importat to metio that the approach sed b Hase ad Hrwitz (946 was based o a determiistic respose mechaism. Ths, the poplatio der cosideratio was spposed to be divided i two differet parts, i.e. respodig part ad o-respodig part. Uits belogig to the respodig part respod with probabilit oe while those i the other part respod with probabilit zero. Let the poplatio der cosideratio of size is divided i two differet classes, i.e. the persos respodig at the first attempt belog to the respose class ad the persos ot respodig at first attempt will be termed as the orespose classes havig sbpoplatio sizes ad respectivel, sch that +. Let be the vale of the i-th respose variable. The poplatio mea of the std variable ca be expressed as, Where ad sch that + W + W are the proportios of its i the respose ad o-respose classes ad ad are the poplatio meas i these classes. Ths i i ad i i Let be the size of the simple radom sample draw from the poplatio havig its b simple radom samplig withot replacemet (SRSWOR techiqe. Let ot of deotes the mber of its respodig i the sample ad its deote the orespodig its. Let be the size of sbsample from the o-respodets to be iterviewed so that. The biased estimators of ad are give b, Let Let w where ˆ ˆ, be the mea of observatios i the sb-sample ad defie, + h h h i h i i i ad (.

4 54 Joral of Reliabilit ad Statistical Stdies, Je 08, Vol. ( The estimator w is a biased estimator of with variace V f ( S + S w where, poplatio mea sqare for the o-respose class. For improved estimatio of the poplatio mea of the std variable, the importat cocepts amel measremet error ad o-respose are merged together. I the preset work we have cosidered that sb sampled part of o-respodig class cotais measremet errors i.e. where ih ih ih ih with std variable i sbsample of o-respose class ad vales ad tre vales respectivel. are measremet errors associated ih ad ih are observed. Proposed Estimator for Estimatig Poplatio Mea Misra et al (07 sggested the give below regressio tpe estimator of poplatio mea of the std variable i presece of measremet errors as, Motivated b the work of Misra et al (07, we have proposed the followig Modified Hase Hrwitz estimator of fiite poplatio mea i the simltaeos presece of both o- respose as well as measremet error. ˆ h HM + h + k h (.3

5 Estimatio of fiite poplatio mea sig kow coefficiet of where i i ad h h i h i Here we have cosidered that sb sampled part h of o-respodig class cotais measremet errors. We have sed the followig approximatios for calclatio of Bias ad Variace of proposed estimator. h Where ( + e h o ( e ˆ + ( E( e E e o, ( A, ( E e A 0 E e o + + +, where θ 4 4 ( ( ( ( θ, E( e e β β, + ( 3 o 4 3,, r E i ( ( The proposed estimator der the aforesaid approximatio ca be expressed as, r ˆ + + k h HM h h HM + ( ( + e o + k + e o ( + e

6 56 Joral of Reliabilit ad Statistical Stdies, Je 08, Vol. ( e e e k HM + + o+ ( + o ( + e e e k HM + + o + ( + o ( + e k e e HM + { + eo + k( eo e + e 8 } HM + + o + ( + o ( + Takig expectatios o both sides, where, ad represets the followig expectatios- E 3 represets the expectatio of total possible samples havig their size h take from a samples of size m, E represets expectatio of all possible samples of size draw from, Estads for the expectatio of total possible samples havig their sizes take from. ( HM o + ( o + { 8 } { ( 8 } E E E E e k e e e E HM EE + + E 3 eo + k E3 eo E3 e + E3 e ( ( ( ( ( { 8 } ( HM + + 3( E E E k E e 4 HM 8 4 ( E E E k

7 Estimatio of fiite poplatio mea sig kow coefficiet of ( HM E E E E k ( HM E E E k ( HM E k ( ( HM HM Bias E ( k HM Bias ( ( 3 3, opt k A θ θ + After pttig the vale of K opt thebias is ( ( ( HM opt A Bias θ θ

8 58 Joral of Reliabilit ad Statistical Stdies, Je 08, Vol. ( Bias Bias Bias ( ( 3 θ HM A opt A 3( 8 + θ ( ( 3 θ θ A ( + θ 3 θ θ HM opt 4 A 4 8 ( HM opt ( 3 θ 4 4 A 4 ( + θ 3 θ A 3. Variace of Proposed Estimator The variace of proposed estimator ca be partitioed ito two parts. We will fid the variace for both parts separatel ad the combied both variace together. V( HM E( V( HM /, + V( E( HM /, The variace of first part ca be obtaied as k ˆ h HM + h + h ( + e HM + ( ( + e o + k + e o k + + e + + e + e ( ( HM o o

9 Estimatio of fiite poplatio mea sig kow coefficiet of k + + e + + e + e ( ( HM o o + + e + k + e + e HM + { + eo + k( eo e + e 8 } HM + { eo + k( eo e + e 8 } ( ( HM o o { ( 8 } HM e o + k eo e + e { 8 } ( HM eo + k( eo e + e ( ( { 4 } HM o + o + o + o o e k e e e e ke ke e { 4 0 } HM o ( ( + ( + ( ( + ( E k E e k E e k k E e e Pttig the vales of E( eoe, E( eo, E( e A 3( HM HM 4 θ E( V ( ( + k + k k( k + 3( HM HM 4 θ ( ( ( + + ( + E V k k A k k ow takig expectatio ( ( ( ( 3 ( 4 E V H M + k + k A k k + θ

10 60 Joral of Reliabilit ad Statistical Stdies, Je 08, Vol. ( ow secod term of variace ˆ HM + h + k h h ( e + HM + ( ( + e o + k + e o e e e k HM + + o+ ( + o ( + e e e k HM + + o + ( + o ( e + k + e + e HM + { + eo + k( eo e + e 8 } HM + { eo + k( eo e + e 8 } ( ( HM o o E stads for the expectatio of total possible samples havig their sizes h take from the sample havig size. { ( } ( HM + ( ( ( o + o + 8 ( + E ( e E E e k E e E e E e 8 4 HM 8 4 ( E owthe variace ca be writte as

11 Estimatio of fiite poplatio mea sig kow coefficiet of HM 8 4 ( ( ( V E V V V ( E( HM V ( + V A 8 Igorig A de to higher order terms: V E S S 8 ( ( HM + where S ( i i Total Variace MSE( HM S + S 8 3( + ( ( + k + k A 4 k k + θ The optimm vale of k for which MSE of proposed estimator miimm is give as 3( θ kopt, A ( 3 + θ After pttig the optimm vale of the characterizig scalar k, the miimm mea sqared error of proposed estimator is calclated as,

12 6 Joral of Reliabilit ad Statistical Stdies, Je 08, Vol. ( 8 MSE( HM S S opt + where ( ( 3 BD 6AB A D B D 4 A E 3 A θ A, B, D, E ( B A + (3. 4. ostrait Problem We have assmed that poplatio coefficiet of variatio is kow to s from the pilot srves ad we sig this iformatio i or estimatio procedre. The MSE of proposed estimator is give as MSE ( HM S + S 8 3( ( ( 4 θ + + k + k A k k + sh The poplatio coefficiet of variatio k is kow. h So or problem is to miimize V( HM, provided that poplatio coefficiet of variatio is kow. ow the optimizatio problem is sh φ V ( HM λ k h whereλ is a Lagrage mltiplier. d ( ( ( 3 3 φ s h + k + ka k λ dk θ h Eqatig to zero

13 Estimatio of fiite poplatio mea sig kow coefficiet of ( 3( s k + ka λ θ θ k h h s h 3( h θ 3( + ka θ λ Usig the vale of k i costrait s h 3( s h λ h θ 3( h θ + ka 3( 3( λs s h h + h + ka θ θ h Pttig the vale of λ i k h s k 3( 3( h + ka θ s θ h 3( θ 3( + ka θ

14 64 Joral of Reliabilit ad Statistical Stdies, Je 08, Vol. ( k s h h 5. Theoretical Efficiec ompariso We have made a theoretical efficiec compariso betwee the proposed estimator ad sal Hase Hrvitz estimator. Variace of sal Hase Hrvitz estimator is give as ` f V( w S + S MSE of proposed estimator is give b MSE( HM S S opt + 8 ( 3 BD 6AB A D B D 4 A E The sggested estimator will perform better tha the sal Hase- Hrvitz estimator if, which gives optimalit coditio 6. A simlatio Std We demostrate the performaces of both estimators throgh simlatio std b geeratig a sample from ormal distribtio sig R software. The cosidered poplatio is ver mch relevat i varios socio-ecoomic sitatios. Descriptio of data is give as follows 5000, 4000, 000, 500, 300, 00, h 50. (5000,, 4 Variace (MSE of Hase Hrwitz estimator ad proposed estimator for the sitatio, whe the o-respose as well as the measremet errors both are preset simltaeosl. Hase Hrwitz estimator V ( HH Proposed estimator MSE( HM opt

15 Estimatio of fiite poplatio mea sig kow coefficiet of oclsio From the simlatio std, it is evidet that the proposed estimator performs better tha the sal Hase Hrvitz estimator i the simltaeos presece of orespose ad measremet errors as it is more efficiet tha it. Therefore proposed estimator is recommeded to srve practitioers to estimate poplatio mea i the case whe srve data is cotamiated with o-respose as well as measremet errors. Ackowledgmet The athors are ver mch thakfl to the aomos referees ad editor of JRSS for their sefl sggestios which improved the earlier draft. Refereces. ochra, W.G. (963. Samplig Techiqes, Secod Editio, Wile Easter Private Limited, ew Delhi.. ochra, W.G.(968. Errors of measremet i statistics, Techometrics, 0, p Devi, M., Sisodia,B.V.S. ad Azfer, M.(06. Estimatio of poplatio mea i two-stage samplig der a radom respose mechaism i the presece of orespose, Iter. Jor. Agri. Statistics, (, p Foradori, G. T. (96. Some o-respose samplig theor for two stage desigs, Istitte of Statistics, orth arolia State ollege. 5. Hase, M.H. ad Hrwitz, W..(946. The problem of o-respose i sample srves, Joral of America Statistical Associatio, 4, p Khare, B. B., Jha, P. S. ad Kmar, K. (05. Estimatio of poplatio proportio sig axiliar character i the presece of orespose, Joral of Reliabilit ad Statistical Stdies, 8(, p Kmar,S. (04. Variace estimatio i presece of radom o-respose, Joral of Reliabilit ad Statistical Stdies, 7(, p Misra, S. ad adav, D. K. (05. Estimatig poplatio mea sig kow coefficiet of variatio der measremet errors, Statistics ad Iformatics i Agricltral Research, Excel Idia Pblisher, ew Delhi, p Misra, S., Dipika ad adav, D. K.(06. Some improved estimators for estimatig poplatio variace i the presece of measremet errors, Joral of Statistics Applicatios & Probabilit, atral Sciece Pblishig, USA, (5, p Misra, S., adav, D.K. ad Dipik. (06. A efficiet estimator for estimatig poplatio variace i presece of measremet errors, Iteratioal Joral of Mathematics ad its Applicatios, 4(, p Misra, S., adav D.K. ad Dipika (07. Estimatio of poplatio mea sig axiliar iformatio i presece of measremet errors, Iteratioal Joral of Egieerig Scieces & Research Techolog, 6(6, p Okafor, F.. (00. Treatmet of o-respose i sccessive samplig, Statistica, 6(, p

16 66 Joral of Reliabilit ad Statistical Stdies, Je 08, Vol. ( 3. Okafor, F.. (005. Sb-samplig the o-respodets i two-stage samplig over two sccessive occasios, Joral of the Idia Statistical Associatio, 43(, p Shalabh ( 997. Ratio method of estimatio i the presece of measremet errors, Jor. Id. Soc. Ag. Statistics, (, p Sigh, H. P. ad Karpe,. (008. Ratio-prodct estimator for poplatio mea i presece of measremet errors, Joral of Applied Statistical Scieces, 6, p Sigh, H.P. ad Karpe,. ( 009. A geeral procedre for estimatig the geeral parameter sig axiliar iformatio i presece of measremet errors, ommicatio of the Korea Statistical Societ, 6(5, p Sigh, H. P. ad Karpe,. (00. Estimatio of mea, ratio ad prodct sig axiliar iformatio i the presece of measremet errors i sample srves, Joral of Statistical Theor ad Practice, 4(, p Sigh, H.P. ad Kmar,S. (0. Sbsamplig the o-respodets i clster samplig o samplig o two sccessive occasios, Statistics i Trasitio- ew Series, (, p Sigh, R.K. ad aqvi,. (05. Estimatio of poplatio mea sig axiliar iformatio i o-respose, Joral of Reliabilit ad Statistical Stdies, 8(, p Srivastava, A.K. ad Shalabh (00. Effect of measremet errors o the regressio method of estimatio i srve samplig, Joral of Statistical Research, 35, p Sd,. ad Srivastava, S. K. (000. Estimatio of poplatio mea i repeat srves i the presece of measremet errors, Joral of Idia Societ of Agricltral Statistics, 53, p Skhatme, P.V, Skhatme, B.V.,Skhatme, S. ad Ashok,. ( 984. Samplig Theor of Srves with Applicatios, 3 rd Ed., Iowa State Uiversit Press, Ams, Iowa(USA ad Idia Societ of Agricltral Statistics, ew Delhi. 3. Tripathi, T. P. ad Khare, B.B. (997. Estimatio of mea vector i presece of orespose, ommicatios i Statistics-Theor ad Methods, 6(9, p