A FAMILY OF ESTIMATORS FOR ESTIMATING POPULATION MEAN IN STRATIFIED SAMPLING UNDER NON-RESPONSE

Transcription

1 Florentn marache A FAMILY OF ETIMATOR FOR ETIMATIG POPULATIO MEA I TRATIFIED AMPLIG UDER O-REPOE MAOJ K. CHAUDHARY, RAJEH IGH, RAKEH K. HUKLA, MUKEH KUMAR, FLORETI MARADACHE Abstract Khoshnevsan et al. (007) proposed a general famly of estmators for populaton mean usng nown value of some populaton parameters n smple rom samplng. The objectve of s paper s to propose a famly of combned-type estmators n stratfed rom samplng adaptng e famly of estmators proposed by Khoshnevsan et al. (007) under non-response. The propertes of proposed famly have been dscussed. We have also obtaned e expressons for optmum sample szes of e strata n respect to cost of e survey. Results are also supported by numercal analyss. 1. Introducton There are several auors who have suggested estmators usng some nown populaton parameters of an auxlary varable. Upadhyaya ngh (1999) ngh et al. (007) have suggested e class of estmators n smple rom samplng. Kadlar Cng (003) adapted Upadhyaya ngh (1999) estmator n stratfed rom samplng. ngh et al. (008) suggested class of estmators usng power transformaton based on e estmators developed by Kadlar Cng (003). Kadlar Cng (005), habbr Gupta (005, 06) ngh Vshwaarma (008) have suggested new rato estmators n stratfed samplng to mprove e effcency of e estmators. Khoshnevsan et al. (007) have proposed a famly of estmators for populaton mean usng nown values of some populaton parameters n smple rom samplng (R), gven by g ax b t y (ax b) (1 )(ax b) 3

2 Florentn marache where a 0 b are eer real numbers or functons of nown parameters of auxlary varable X. Koyuncu Kadlar (008, 09) have proposed famly of combned-type estmators for estmatng populaton mean n stratfed rom samplng by adaptng e estmator of Khoshnevsan et al. (007). These auors assumed at ere s complete response from all e sample unts. It s fact n most of e surveys at nformaton s usually not obtaned from all e sample unts even after callbacs. The meod of sub-samplng e non-respondents proposed by Hansen Hurwtz (1946) can be appled n order to adjust e non-response n a mal survey. In e next sectons, we have tred to propose a famly of combned-type estmators consderng e above famly of estmators n stratfed rom samplng under non-response. We have dscussed e propertes of proposed famly of estmators. We have also derved e expressons for optmum sample szes of e strata n respect to cost of e survey.. amplng trateges Estmaton Procedure Let us consder a populaton consstng of unts dvded nto strata. Let e sze of stratum s, ( 1,,..., ). We decde to select a sample of sze n from e entre populaton n such a way at n unts are selected from e unts n e stratum. Thus, we have n n 1. Let Y X be e study auxlary characterstcs respectvely w respectve populaton mean Y X. It s consdered at e non-response s detected on study varable Y only auxlary varable X s free from non-response. Let by y be e unbased estmator of populaton mean Y for e stratum, gven n 1 yn1 n yu y (.1) n where y n1 yu are e means based on n 1 unts of response group u unts of sub-sample of non-response group respectvely n e sample for e stratum. x be e unbased estmator of populaton mean X, based on n sample unts n e stratum. Usng Hansen-Hurwtz technque, an unbased estmator of populaton mean Y s gven by st 1 y p y (.) e varance of e estmator s gven by e followng expresson 1 1 ( 1) V(yst ) p y Wp y 1 n 1 n (.3) 4

3 Florentn marache where y y are respectvely e mean-square errors of entre group non-response group of study varable n e populaton for e stratum. n, p W on-response rate of e stratum n e u populaton..1 Proposed Estmators Motvated by Khoshnevsan et al. (007), we propose a famly of combned-type estmators of populaton mean Y, gven by g ax b yst (.1.1) (axst b) (1 )(ax b) where x st p x (unbased for X ) 1 X p X. 1 Obvously, s based. The bas ME can be obtaned on usng large sample approxmatons: y st Y 1 e ; x st X 0 1 e 1 such at Ee Ee V yst 1 1 E e0 p f Y W Y Y Y 1 n V x st 1 E e1 p f X X X E Cov yst, xst p fyx YX YX 1 e e where f n, n X varable n e populaton for e between Y X n e be e mean-square error of entre group of auxlary stratum. stratum s e correlaton coeffcent 5

4 Florentn marache Expressng T C n terms of e 0,1, we can wrte (.1.1) as Y 1 e 1 e g 0 1 (.1.) ax where. ax b uppose e1 1 1 s expable. Expng e rght h sde of (.1.) up to e frst order of approxmaton, we obtan < 1 so at g g g 1 T Y Y e ge e C 0 1 e1 ge0e1 Tang expectaton of bo sdes n (.1.3), we get e bas of e estmator B 1 g g 1 fp Y 1 R X gryx (.1.3) as (.1.4) quarng bo sdes of (.1.3) en tang expectaton, we get e ME of e estmatort C, up to e frst order approxmaton, as 1 ME fp Y g R X gryx p W Y 1 1 n (.1.5) Optmum choce of On mnmzng MET C ME fp 1 opt gr f p Thus opt w.r.t., we get e optmum value of as g R fp X gr fp YX YX X s e value of at whch ME T C would attan ts mnmum. (.1.6) 3. Optmum n w respect to Cost of e urvey Let C0 be e cost per unt of selectng n unts, C 1 be e cost per unt n enumeratng n 1 unts C be e cost per unt of enumeratng u unts. Then e total cost for e stratum s gven by C C0n C1n 1 Cu 1,,..., 6

5 Florentn marache ow, we consder e average cost per stratum W EC n C0 C1W1 C Thus e total cost over all e strata s gven by E C 0 C 1 n C0 C1W1 C 1 Let us consder e functon ME T C C0 W (3.1) (3.) where s Lagrangan multpler. Dfferentatng e equaton (3.) w respect to n separately equatng to zero, we get e followng normal equatons. p g R gr 1W p Y X Y X n n n Y W C0 C1W1 C 0 (3.3) p W Y W n C 0 n (3.4) From e equatons (3.3) (3.4) respectvely, we have p n Y py n C g R X gryx W C0 C1W1 C 1 W Y (3.5) (3.6) Puttng e value of e opt C B YA from equaton (3.6) nto e equaton (3.5), we get (3.7) Where A C0 C1W1 B Y g R X gryx W Y 7

6 Florentn marache ubsttutng opt from equaton (3.7) nto equaton (3.5), n can be expressed as p n B A C BWY A C AWY B (3.8) The n terms of total cost C0 can be obtaned by puttng e values of opt n from equatons (3.7) (3.8) respectvely nto equaton (3.1) 1 p C0 1 AB ow we can express n opt p 1 Thus nopt W. AB C WY n n terms of total cost C0 C0 4. umercal Analyss C WY p B A C BWY A C AWY B (3.9) (3.10) can be obtaned by equaton (3.10) by puttng dfferent values of For numercal analyss we have used data consdered by Koyuncu Kadlar (008). The data concernng e number of teachers as study varable e number of students as auxlary varable n bo prmary secondary school for 93 dstrcts at 6 regons (as 1: Marmara, : Agean, 3: Medterranean, 4: Central Anatola, 5: Blac ea, 6: East oueast Anatola) n Turey n 007 (ource: Mnstry of Educaton Republc of Turey). Detals are gven below: Table o.4.1: tratum means, Mean quare Errors Correlaton Coeffcents Y tratum o. n Y X Y X XY Y

7 Florentn marache Table o.4.: % Relatve effcency (R.E.) of 5. Concluson W w.r. to y st at opt, a 1, b 1 R. E. T C We have proposed a famly of estmators n stratfed samplng usng an auxlary varable n e presence of non-response on study varable. We have also derved e expressons for optmum sample szes n respect to cost of e survey. Table 4. reveals at e proposed estmator T has greater precson an e usual estmator References y st under non-response. 1. Hansen, M. H., Hurwtz, W.. (1946): The problem of non-response n sample surveys. Journal of Amercan tatstcal Assocaton, 41, Kadlar, C., Cng, H. (005): A new estmator n stratfed rom samplng. Communcaton n tatstcs Theory Meods, 34, Kadlar, C., Cng, H. (003): Rato estmator n stratfed samplng. Bometrcal Journal, 45, Khoshnevsan, M., ngh, R., Chauhan, P., awan,., marache, F. (007): A general famly of estmators for estmatng populaton mean usng nown value of some populaton parameter(s). Far East Journal of Theoretcal tatstcs,, C 9

8 Florentn marache 5. Koyuncu,., Kadlar, C. (008): Rato product estmators n stratfed rom samplng. Journal of tatstcal Plannng Inference, 139, 8, Koyuncu,., Kadlar, C. (009): Famly of estmators of populaton mean usng two auxlary varables n stratfed rom samplng. Communcaton n tatstcs Theory Meods, 38:14, habbr, J. Gupta,. (005): Improved rato estmators n stratfed samplng. Amercan Journal of Maematcal Management cences, 5, habbr, J. Gupta,. (006): A new estmator of populaton mean n stratfed samplng. Communcaton n tatstcs Theory Meods, 35, ngh, H., P., Talor, R. ngh. Km, J. M. (008): A modfed estmator of populaton mean usng power transformaton. tatstcal papers, Vol-49, o.1, ngh, H., P., Vshwaarma, G. K. (008): A famly of estmators of populaton mean usng auxlary nformaton n stratfed samplng. Communcaton n tatstcs Theory Meods, 37(7), ngh, R., Cauhan, P., awan,. marache, F. (007): Auxlary nformaton a pror values n constructon of mproved estmators. Renassance Hgh press. 1. Upadhyaya, L.., ngh, H.P. (1999): Use of transformed auxlary varable n estmatng e fnte populaton mean. Bometrcal Journal. 41, Publshed n "Pastan Journal of tatstcs Operatonal Research", Vol. V, o. 1, pp ,