An Improved Suggestion in Stratified Random Sampling Using Two Auxiliary Variables

Transcription

1 Raje ing Dearmen of Maemaic, RM Univeriy Deli CR, onea- 309, India acin Malik Dearmen of aiic, Banara Hindu Univeriy aranai-005, India Florenin marandace Dearmen of Maemaic, Univeriy of ew Mexico Gallu, M 8730, UA An Imroved uggeion in raified Random amling Uing Two Auxiliary ariable Publied in: acin Malik, eeraj Kumar, Florenin marandace (Edior) UE OF AMPIG TECHIQUE & IETOR COTRO WITH CAPACIT COTRAIT Pon Ediion, Bruel, Belgium, 06 IB

2 Abrac In i aer, we ugge an eimaor uing wo auxiliary variable in raified random amling following Malik and ing []. Te rooe eimaor a an imrovemen over mean er uni eimaor a well a ome oer conidered eimaor. Exreion for bia and ME of e eimaor are derived u o fir degree of aroximaion. Moreover, ee eoreical finding are uored by a numerical examle wi original daa. Key word: udy variable, auxiliary variable, raified random amling, bia and mean quared error.. Inroducion Te roblem of eimaing e oulaion mean in e reence of an auxiliary variable a been widely dicued in finie oulaion amling lieraure. Ou of many raio, roduc and regreion meod of eimaion are good examle in i conex. Diana [] uggeed a cla of eimaor of e oulaion mean uing one auxiliary variable in e raified random amling and examined e ME of e eimaor u o e k order of aroximaion. Kadilar and Cingi [3], ing e al. [7], ing and iwakarma [8],Koyuncu and Kadilar [] rooed eimaor in raified random amling. ing [9] and Perri [6] uggeed ome raio cum roduc eimaor in imle random amling. Bal and Tueja [] and ing e al. [] uggeed ome exonenial raio ye eimaor. In i caer, we ugge ome exonenialye eimaor uing e auxiliary informaion in e raified random amling. 9

3 Ue of amling Tecnique & Invenory Conrol wi Caaciy Conrain Conider a finie oulaion of ize and i divided ino raa uc a were i e ize of raum (=,,...,). We elec a amle of ize n from eac raum by imle random amle wiou relacemen amling uc a n n, were n i e raum amle ize. A imle random amle of ize n i drawn wiou relacemen from e raum uc a e (yi, xi, zi) denoe e oberved value of y, x, and z on e i uni of e raum, were i=,, 3... To obain e bia and ME, we wrie y w y e 0, x w x X e, z w z Z e uc a, e Ee Ee 0 E r were, w r E r y x X z Z n n y w y, y yi, n i r X Z i i w, w and ( y ) 00 (.) imilar exreion for X and Z can alo be defined. 0

4 An Imroved uggeion in raified Random amling Uing Two Auxiliary ariable W fy W fx And Ee 0 00, Ee, 00 W fz e, e e, E 00 Z W f yz E 0 0 e e, were, Z X W f yx E 0 0 X W fxz 0 and E e e, XZ y y, x X i x i z Z, x X y z yz And, f i z Z y, x X z Z n i -. Eimaor in lieraure In order o ave an eimae of e udy variable y, auming e knowledge of e oulaion roorion P, aik and Gua [5] and ing e al. [] reecively rooed following eimaor yx xz i i y X x (.) y X x ex X x (.)

5 Ue of amling Tecnique & Invenory Conrol wi Caaciy Conrain Te ME exreion of ee eimaor are given a ME (.3) ME (.) Wen e informaion on e wo auxiliary variable i known, ing [0] rooed ome raio cum roduc eimaor in imle random amling o eimae e oulaion mean of e udy variable y. Moivaed by ing [0] and ing e al. [7], ing and kumar rooe ome eimaor in raified amling a X x Z z 3 yex ex X x Z z x X z Z yex ex x X z Z X x z Z 5 yex ex X x z Z x X Z z 6 yex ex x X Z z (.5) (.6) (.7) (.8) Te ME equaion of ee eimaor can be wrien a ME( 3) ME( ) ME( 5) (.9) (.0) (.)

6 An Imroved uggeion in raified Random amling Uing Two Auxiliary ariable ME( 6) (.) Wen ere are wo auxiliary variable, e regreion eimaor of will be X x b Z z 7 y b (.3) yx yz Were b and b. Here yx x z x and z are e amle variance of x and z reecively, and yz are e amle covariance beween y and x and beween z reecively. Te ME exreion of i eimaor i: ME W f ρ ρ ρ ρ (.) 7 y yx yz yx yzρ xz 3. Te rooed eimaor Following Malik and ing [], we rooe an eimaor uing informaion on wo auxiliary aribue a X x yex X x m Z z ex Z z m b X x b Z z (3.) Exreing equaion (3.) in erm of e, we ave e e m e e e0 ex ex b Xe b Ze m e me me me mmee me me0e me0e 0 bex bez (3.) quaring bo ide of (3.) and neglecing e erm aving ower greaer an wo, we ave 3

7 Ue of amling Tecnique & Invenory Conrol wi Caaciy Conrain e 0 me me b e X b e Z (3.3) Taking execaion of bo e ide of (3.3), we ave e mean quared error of fir degree of aroximaion a ME( ) 00 P P P3 u o e (3.) Were, m P P B P 3 B m B 0 B mm0 m B B 0 m B m B m 0 0 m B 0 m B 00 (3.5) Were, B W f ρ W f yx y x x and B W f ρ W f yz y z z Te oimum value of m and mwill be B m B m B B 0 B0000 B B000 B (3.6) Puing oimum value of m and m from (3.6), we obained min ME of rooed eimaor.. Efficiency comarion In i ecion, e condiion for wic e rooed eimaor i beer an y,,, 3,, 5, 6, and 7. Te variance i given by

8 An Imroved uggeion in raified Random amling Uing Two Auxiliary ariable ( y ) 00 (.) To comare e efficiency of e rooed eimaor wi e exiing eimaor, from (.) and (.3), (.), (.9), (.0), (.), (.) and (.), we ave 3 ( y ) - ME( ) P P P 0 (.) ) - ME( ) - ME( ME( ) P P P 0 ) ME( P P P3 0 (.3) (.) ME( 3 ) - ME( ) - ME( 5 ) - ) - ME( 6 ) ME( ) ME( ) ME( ) ME( P P P 0-3 P P P 0-3 P P P 0-3 P P P 0 (.5) (.6) (.7) (.8) Uing (.) - (.8), we conclude a e rooed eimaor ouerform an e eimaor conidered in lieraure. 5. Emirical udy In i ecion, we ue e daa e in Koyuncu and Kadilar []. Te oulaion aiic are given in Table 3... In i daa e, e udy variable () i e number of eacer, e fir auxiliary variable (X) i e number of uden, and e econd auxiliary variable (Z) i e number of clae in bo rimary and econdary cool. Table 5.: Daa aiic of Poulaion =7 =7 3=03 =70 5=05 6=0 5

9 Ue of amling Tecnique & Invenory Conrol wi Caaciy Conrain n=3 n= n3=9 n=38 n5= n6=39 = = 6 = = =7.73 = = =.66 = = = = = =88.93 = =309. = =9.79 = = = = = = = = = = = = =0.99 = = = = = = = = = = 98.8 = = 3.36 = 98.8 = 7.0 = 33.7 = = = = = 0539 = = 5968 = = = 805 = 057 = = = = = = =

10 An Imroved uggeion in raified Random amling Uing Two Auxiliary ariable We ave comued e re relaive efficiency (PRE) of differen eimaor of wi reec o y and comlied in able 5.: Table 5.: Percen Relaive Efficiencie (PRE) of eimaor.o. Eimaor PRE y Concluion In i aer, we rooed a new eimaor for eimaing unknown oulaion mean of udy variable uing informaion on wo auxiliary variable. Exreion for bia and ME of e eimaor are derived u o fir degree of aroximaion. Te rooed eimaor i comared wi uual mean eimaor and oer conidered eimaor. A numerical udy i carried ou o 7

11 Ue of amling Tecnique & Invenory Conrol wi Caaciy Conrain uor e eoreical reul. In e able 5., e rooed eimaor erform beer an e uual amle mean and oer conidered eimaor. Reference [] Bal,. and Tueja, R.K. (99): Raio and roduc ye exonenial eimaor. Infrm.andOim. ci., XII, I, [] Diana, G. (993). A cla of eimaor of e oulaion mean in raified random amling. aiica53(): [3] Kadilar,C. and Cingi,H. (003): Raio Eimaor in raiified Random amling. Biomerical Journal 5 (003), 8-5. [] Koyuncu,. and Kadilar, C. (009) : Family of eimaor of oulaion mean uing wo auxiliary variable in raified random amling. Comm. In a. Teory and Me., 38:, [5] aik,.d and Gua, P.C., 996: A noe on eimaion of mean wi known oulaion roorion of an auxiliary caracer. Jour. Ind. oc. Agri. a., 8(), [6] Perri, P. F. (007). Imroved raio-cum-roduc ye eimaor.ai. Tran. 8:5 69. [7] ing, R., Kumar, M., ing,r.d.,and Caudary, M.K.(008): Exonenial Raio Tye Eimaor in raified Random amling. Preened in Inernaional ymoium On Oimiaion and aiic ( I..O.).-008 eld a A.M.U., Aligar, India, during 9-3 Dec 008. [8] ing, H., P. and iwakarma, G. K. (008): A family of eimaor of oulaion mean uing auxiliary informaion in raified amling. Communicaion in aiic Teory and Meod, 37(7), [9] ing, M. P. (965): On e eimaion of raio and roduc of oulaion arameer. ankya 7 B, [0] ing, M.P. (967): Raio-cum-roduc meod of eimaion. Merika, 3-7. [] ing, R.,Cauan, P., awan,. and marandace,f. (007): Auxiliary informaion and a riory value in conrucion of imroved eimaor. Renaiance Hig re. [] Malik and ing (03): An imroved eimaor uing wo auxiliary aribue. Alied Maemaic and Comuaion, 9(03),