Application of Calibration Approach for Regression Coefficient Estimation under Two-stage Sampling Design

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1 Authors: Pradp Basak, Kaustav Adtya, Hukum Chadra ad U.C. Sud Applcato of Calbrato Approach for Regresso Coeffcet Estmato uder Two-stage Samplg Desg Pradp Basak, Kaustav Adtya, Hukum Chadra ad U.C. Sud CAR-da Agrcultural Statstcs Research sttute, ew Delh Emal: ad 1. troducto ow a days, survey data are complex ad multvarate ature whch volves clusterg, stratfcato, uequal probablty of selecto, mult-stages ad mult-phases. The tradtoal method of estmato of regresso coeffcet s ordary lest squares (OLS) estmato whch s based o the assumpto that sample observatos are draw depedetly. Ths assumpto of depedece holds oly f the sample observatos are draw usg smple radom samplg wth replacemet (SRSWR) but for other samplg desgs t does ot hold. Oe such complex desg s two-stage samplg desg whch s wdely used large scale surveys. twostage samplg, sample s selected two stages. the frst stage, clusters are selected ad the secod stage, a specfed umber of elemets are vestgated from the selected clusters. The clusters whch form the uts of samplg at the frst stage are kow as prmary stage uts (PSU) ad the elemets wth the clusters are kow as secod stage uts (SSU). As for example, case of crop, surveyg felds ca be take as frst stage uts ad plots wth the felds ca be take as secod stage uts. Ksh ad Frakel (1974) suggested use of samplg desg weghts the estmato procedure as a alteratve to the OLS. Estmato of regresso coeffcet based o maxmum lkelhood estmato was suggested by Holt, Smth ad Wter (1980). the presece of auxlary formato, calbrato approach was suggested by Devlle ad Särdal, 199for the mprovemet of the estmator of populato parameters. Work o calbrato approach based estmato of populato parameters lke mea, total, proporto, covarace has already bee doe uder u-stage or mult-stage desgs, see for example Adtya et al. (016), Plkusas ad Pumputs (007, 010).Thus, uder the avalablty of auxlary formato the two-stage desg, the theory of calbrato approach s used here for the mprovemet of the estmator of populato regresso coeffcet.. Methodology Let U=(1,,,k,,) be a fte populato of sze comprsg of clusters as U1, U,..., U,..., U wth sze of the clusters 1,,..., 3,..., respectvely. These clusters are othg but prmary stage uts (psus) ad uts wth the clusters are secod Verso 1.0 Sce we update ths documet frequetly, we request you to dowload a fresh copy each tme Page 1 of 5

2 Authors: Pradp Basak, Kaustav Adtya, Hukum Chadra ad U.C. Sud stage uts (ssus). At the frst stage, a sample of clusters s of sze populato of clustersu ad at the secod stage, a sample of uts s draw from the s draw from s of sze by usg ay probablty samplg scheme. Let, the th selected cluster, U of sze ad be the frst ad secod order cluso probablty at the frst stage ad at the secod stage j t s k / ad kl / respectvely. Let, a 1/, ak / 1/ k / ad a a a. / k k Let, y ad x be the varables uder study. Here, y s depedet varable ad x s explaatory varable. Let us assume, auxlary varable z s assocated wth depedet varable y ad formato o auxlary varable z s avalable at psu level. Let, the sample observatos correspodg to the jth ut of th cluster are deoted by yk, xk ad z k. ow, the populato total of varables y, x ad z are gve by respectvely, where, ad t z Z z k 1 k 1 1 respectvely. We have assumed that, t y t y k y 1 k 1 1 y x t x t ad x k x 1 k 1 1 t t Z s the th cluster total of y, x ad z Z s kow for all psu s. Populato regresso coeffcet B uder two-stage samplg desg s gve by B xk X yk Y 1 k1 x X k 1 k1 k k. 1 k 1 1 k where X x ad Y y The usual π-estmator of ths populato regresso coeffcet, B s gve by Bˆ 1 k1 ˆ / ˆ / a x t y t k k x k y 1 k1 a x tˆ / k k x (1) where,, tˆ x a ˆ tx, tˆ x ak / xk, tˆ 1 k 1 y a ˆ ty, tˆ y ak / yk. 1 Thus, usg calbrato approach the estmator of populato total of varable y s obtaed as tˆ c y w ˆ ty.fally, the estmator of populato regresso coeffcet uder two-stage desg 1 k 1 Verso 1.0 Sce we update ths documet frequetly, we request you to dowload a fresh copy each tme Page of 5

3 Authors: Pradp Basak, Kaustav Adtya, Hukum Chadra ad U.C. Sud s obtaed as Bˆ c ˆ / ˆc / w a x t y t k / k x k y 1 k1 k / k x 1 k1 w a x tˆ / () 3. Emprcal Evaluato A populato of 84 mucpaltes of Swede cotag formato o several varables was used for emprcal evaluato. The populato was grouped to 50 clusters each cotag 5 to 9 mucpaltes. At the frst stage, some clusters were selected from the 50 clusters usg smple radom samplg wthout replacemet ad at the secod stage some mucpaltes were selected from each selected clusters usg same sample desg. From the selected mucpaltes observatos were recorded o the varables 1985 Mucpal taxato (RMT85, measured mllos of kroor), total umber of seats the mucpal coucl (S8) ad umber of mucpa employees 1984 (ME84). The objectve was to study the patter of relatoshp betwee varables RMT85 ad S8 usg ME84 as the auxlary varable. From ths populato, three dfferet combatos of sample: ) 0, 4, 80, ) 0,, 40, ad s s ) 10,, s 0, were draw. the emprcal evaluato, two estmators of fte populato regresso coeffcet were cosdered for comparso purpose: ) π-estmator, ˆB gve by (1) (deoted as Est-π), ) Calbrated estmator, B ˆ c gve by () (deoted as Est-CAL). The performace of the estmators were evaluated by the crtera of percetage absolute relatve bas (ARB) ad percetage relatve root mea square error (RRMSE) µ µ 1 M B B µ ARB( B) 100 ad µ 1 M B B RRMSE( B) M M B B where B deotes the estmated value of populato regresso coeffcet at smulato ru, ˆ wth true value B. The result of the emprcal study dcates that the calbrated estmator (EST-CAL1) has a lower ARB as compared to the π-estmator (EST-π). Smlarly, terms of RRMSE the estmator EST- CAL1 has a advatage as compared to the exstg π-estmator. The results are dsplayed through a graphcal represetato Fgure 1 ad Fgure. Verso 1.0 Sce we update ths documet frequetly, we request you to dowload a fresh copy each tme Page 3 of 5

4 ARB, % RRMSE, % Authors: Pradp Basak, Kaustav Adtya, Hukum Chadra ad U.C. Sud Est- π Est-CAL Est- π Est-CAL =0, =4,=80 =0, =,=40 =10, =,=0 Fgure 1 Fgure =0, =4,=80 =0, =,=40 =10, =,=0 4. Cocludg Remarks Ths study dscusses about the calbrated estmator of populato regresso coeffcet the presece of psu level auxlary formato. The calbrated estmator foud to be satsfactory as compared to the exstg OLS estmator whch volates the depedece of observatos assumpto uder two-stage samplg desg. Refereces (f ay) 1. Adtya, K., Sud, U. C., Chadra, H. ad Bswas, A. (016).Calbrato based regresso type estmator of the populato total uder two stage samplg desg. Joural of da Socety of Agrcultural Statstcs, 70(1), Devlle, J. C. ad Sardal, C. E. (199).Calbrato estmators survey samplg. Joural of the Amerca Statstcal Assocato, 87, Holt, D., Smth, T. M. F. ad Wter, P. D. (1980).Regresso aalyss of data from complex surveys. Joural of the Royal Statstcal Socety. Seres A (Geeral),143, Ksh, L. ad Frakel, M.R. (1974).ferece from complex samples. Joural of the Royal Statstcal Socety, B36, Plkusas, A. ad Pumputs, D. (007). Calbrated estmators of the populato covarace. Acta Applcadae Mathematcae, 97, Plkusas, A. ad Pumputs, D. (010). Estmato of fte populato covarace usg calbrato. olear Aalyss: Modellg ad Cotrol, 15(3), Särdal, C. E., Swesso, B. ad Wretma, J. (199).Model Asssted Survey Samplg. Sprger Verlag, ew York. Verso 1.0 Sce we update ths documet frequetly, we request you to dowload a fresh copy each tme Page 4 of 5

5 Authors: Pradp Basak, Kaustav Adtya, Hukum Chadra ad U.C. Sud Terms - Do ot remove or chage ths secto ( t should be emaled back to us as s) Verso 1.0 Sce we update ths documet frequetly, we request you to dowload a fresh copy each tme Page 5 of 5

Comparison of Dual to Ratio-Cum-Product Estimators of Population Mean

Comparison of Dual to Ratio-Cum-Product Estimators of Population Mean Research Joural of Mathematcal ad Statstcal Sceces ISS 30 6047 Vol. 1(), 5-1, ovember (013) Res. J. Mathematcal ad Statstcal Sc. Comparso of Dual to Rato-Cum-Product Estmators of Populato Mea Abstract