Multivariate Ratio Estimator of the Population Total under Stratified Random Sampling

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1 Open Journal of Statstcs, 0,, ttp://dx.do.org/0.436/ojs Publsed Onlne July 0 (ttp:// Multvarate Rato Estmator of te Populaton Total under Stratfed Random Samplng Oscar O. Ngesa, George O. Orwa, Romanus O. Oteno, Henry M. Murray Mnstry of State for Plannng, Natonal Development and V030, Narob, Kenya Department of Statstcs and Actuaral Scence, Jomo Kenyatta Unversty of Agrculture and Tecnology, Narob, Kenya Emal: oscanges@yaoo.com Receved Marc 7, 0; revsed Aprl 0, 0; accepted Aprl 30, 0 ABSTRACT Olkn [] proposed a rato estmator consderng p auxlary varables under smple random samplng. As s expected, Smple Random Samplng comes wt relatvely low levels of precson especally wt regard to te fact tat ts varance s greatest amongst all te samplng scemes. e extend ts to stratfed random samplng and we consder a case were te strata ave varyng wegts. e ave proposed a Multvarate Rato Estmator for te populaton mean n te presence of two auxlary varables under Stratfed Random Samplng wt L strata. Based on an emprcal study wt smulatons n R statstcal software, te proposed estmator was found to ave a smaller bas as compared to Olkn s estmator. Keywords: Rato Estmator; Stratfcaton; Auxlary Varables; Lagrange s Multpler. Introducton Auxlary varables ave been used to ncrease precson of estmators especally n regresson and rato estmators []. Ts s partcularly so n cases of complex surveys, more so n stuatons were some nformaton on te survey varable mgt be mssng [3]. Tese classcal metods of estmaton are based on drect estmators,.e., tose wc use te response varable, y and nformaton provded by an auxlary varable, x, gly correlated wt te man varable [4].. Revew of Multvarate Rato Estmators Olkn [] proposed a multvarate generalzaton of te rato estmator. Olkn proposed an estmator for te populaton total, denoted by Y MR, and defned as y y y YMR X X p X p (.) x x x wc n oter context can also be wrtten as; Y Y Y + Y (.) MR R R p R p were y YR X s te component of te populaton x t total rato estmate afflated to te auxlary varable are te wegts wc maxmze te precson of Y, subject to a lnear constrant MR p. Ts estmate of populaton total also wll be accurate f p te regresson lne of Y on X, X,, X p s a stragt lne gong troug te orgn. Te populaton totals for te auxlary varables X must be explctly known. 3. Te Proposed Estmator Consder a populaton wc as been dvded nto L strata, wt te strata beng dsjont, te sample elements from eac stratum are sampled and wen te measurement y s done, measurement for te unt n te t t stratum, two auxlary varables, say, x and x t are also measured for tat unt. Let Y MRE denote te proposed multvarable estmator under te stratfed random samplng sceme for te populaton total. Y MRE s terefore defned as; L Y MRE YMR (.3) were te ndvdual components are defned as follows: Y MR YR YR for te st stratum. Y MR YR YR for te nd stratum. Y Y Y for L t te stratum. MRL L R L R L L Ts can furter be represented n a sngle equaton as follows; Y Y Y (.4) were MR R R,,, L are te varous strata. Copyrgt 0 ScRes.

2 O. O. NGESA ET AL Varance of te Proposed Estmator To compute te values of te wegts, te general Equaton (.4) s used and ts wll cater for eac stratum by just cangng te value of n respectve strata. Subtractng Y to te rgt and sde and left and sde of equaton (.4) yelds Y Y Y YR Y (.5) MR R But t s known tat te sum of te wegts n eac stratum s, so. Ts mples tat Y = Y Y Y Y Y Y (.6) Replacng Equaton (.6) to te rgt and sde of Equaton (.5), yelds MR R R Y Y Y Y Y MR R R Y Collectng te lke terms wt respect to wegts yelds Y Y Y Y Y Y (.7) MR R R Squarng eac sde and takng Expectaton on eter sde, assumng neglgble bas, Equaton (.7) leads to MR R, R R V YR V Y V Y Cov Y Y Equaton (.8) can be wrtten n notaton as follows, V Y V V V MR were V Varance Y V R Varance YR and V Covarance Y, Y R R, (.8) (.9) e ten proceed to fnd te values of te wegts and tat mnmze te varance V Y MR subject to te lnear constrant. To aceve ts, we form a functon wc as te varance and te lnear constrant mentoned above. V Y MR (.0) wt beng te Lagrange s Multpler. From Equaton (.9), V Y V V V MR replacng ts nto Equaton (.0) yelds; V V V To mnmze ts functon wt respect to te wegts and, we dfferentate partally te functon wt respect to tese wegts eac at a tme. V V (.) V V V V (.) For optmzaton, we equate te partal dervatve Equatons (.) and (.), eac to zero. Tese yelds; V V V V V V (.3) (.4) It follows tat Equatons (.3) and (.4) are equal, ten Te s common and can be cancelled out. e proceed to collect lke terms wt respect to te wegts and ts yeld V V V V (.5) It s known tat, ence. From ts Equaton (.5) wll reduce to and V V V V V V V V V V Ten t follows, by makng te subject of te formula, V V V V V V Openng te brackets n te denomnator yelds V V V V V To get te value of wegt, we use te lnear constrant V V V V V wc may be wrtten as, V V V V V V V V V V V V V V V V (.6) (.7) Equatons (.6) and (.7) gve te wegts tat mnmze te varance V Y for stratum. MR Copyrgt 0 ScRes.

3 30 O. O. NGESA ET AL. Tese wegts can now be substtuted n te proposed model to get te populaton total. 5. Emprcal Study An emprcal study was carred out to estmate te populaton total of a smulated populaton and compare te performance of te proposed model to tat of Olkn []. In ts secton we smulated a populaton (y, x and x ), wc as 0 strata,,,0 n wc eac stratum dffers from oters. Ts dfference was aceved by usng dfferent error terms e wle generatng te y usng y ax bx e. Te coeffcents a and b are randomly generated from a unform dstrbuton wle y, x and x are randomly gene-rated from normal dstrbuton wt dfferent parameters. A sample of sze 300 was selected randomly from te smulated populaton ndex-wse, tat s f ndex s selected ten te sample elements wll ave y, x and x. Ts was repeated for all te ten strata, te selected sample was used n te proposed model to estmate te po- pulaton total. Te ten strata were agan joned togeter to form one uge stratum, ndex-wse sample of sze 000, was selected and ten usng Olkn s model, te populaton total was estmated. Te procedure above was repeated for 000 samples and te populaton totals usng eac model was recorded. 8. Smulaton Results 6. Descrpton of te Study Populaton Te populaton total estmates of te two metods were compared to tat of te true populaton (smulated) total. Te True populaton total s 8,35,645. Table summarzes te statstcs correspondng to eac estmator. Fgures and sow te plotted values of te populaton total estmates of proposed model and Olkn s model, respectvely, repeated for 000 smulatons eac. 7. Computatonal Procedure In order to sow te dfference n varablty between te two metods, te two plots above are now combned nto one grap usng a common scale n te Fgure Conclusons Table. Summary statstcs for eac metod. From te summary table above, t can be seen tat te proposed estmator gves a total wt a very small bas as compared to te Olkn s. Also, te proposed model can be seen to ave a small Root Mean Square Error (RMSE) Mn. Medan 3rd Qrt Max Mean Bas RMSE Proposed Metod,8, ,83, ,85,3,83, Olkn s Metod,746, ,8, ,903,358,8, Fgure. Plot of te populaton totals wt proposed model for te 000 samples. Copyrgt 0 ScRes.

4 O. O. NGESA ET AL. 303 Fgure. Plot of te populaton totals wtout stratfcaton for te 000 samples. Fgure 3. Fgures and plotted on a common scale. as compared to Olkn s estmator. Te combned grap also sows tat te populaton total estmate s more varable n Olkn s as compared to te proposed model. Te lmtng condton to allow te use of ts estmator s te requrement of exstence of lnear relatonsp Copyrgt 0 ScRes.

5 304 O. O. NGESA ET AL. troug te orgn between te varable of nterest, y, and te auxlary varables. REFERENCES [] I. Olkn, Multvarate Rato Estmaton for Fnte Populatons, Bometrka, Vol. 45, No. -, 956, pp []. G. Cocran, Samplng Tecnques, 3rd Edton, ley, New York, 977. [3] L. Y. Deng and R. S. Ckura, On te Rato and Regresson Estmaton n Fnte Populaton Samplng, Amercan Statstcan, Vol. 44, No. 4, 990, pp [4] P. V. Sukatme and B. V. Sukatme, Samplng Teores of Survey wt Applcatons, Iowa State Unversty Press, Ames, 970. Copyrgt 0 ScRes.