A Note on Ratio Estimators in two Stage Sampling

Transcription

1 Iteratoal Joural of Scetfc ad Research Publcatos, Volume, Issue, December 0 ISS 0- A ote o Rato Estmators two Stage Samplg Stashu Shekhar Mshra Lecturer Statstcs, Trdet Academy of Creatve Techology (TACT), Bhubaeswar, Odsha, Ida. Abstract- I ths paper a umber of rato estmators two stage samplg are cosdered ad ther effceces are compared wth a estmator wthout use auxlary formato. umercal llustrato s provded to compare the effceces. I I. ITRODUCTIO large scale sample surveys t s usual practce to adopt mult stage samplg to estmate the populato mea or total of the study varable y. The ma purpose of usg mult stage samplg place of urestrcted ustage samplg s to reduce the cost of survey operatos eve f estmates derved from mult stage samplg are lkely to be less effcet the those of the urestrcted u-stage samplg. Sometmes, all auxlary varable x compared wth y may be avalable ether at prmary stage or of secodary stage or at both the stages to case the effcecy of estmate of the fte populato parametrc fuctos such as populato meas or total. Thus, the followg we cosder dfferet rato estmators of populato mea of the study varable y two stage samplg usg kowledge o a sgle auxlary varable x. II. OTATIOS ow, cosder a fte populato U parttoed to frst stage uts (fsu) deoted by U, U, U. Let M be the umber M M M M U of secod stage uts (,,...,). Defe ad y j x. Let ad j deote values of the study varable y ad the auxlary varable x respectvely for the jth ssu of U = (j =,,, M, =,,.,). Defe, Y M M j y j ad M M j x j ( =,,.., ) The populato mea of y, Y u Y ad the populato mea of x, u, where u M M. Further, defe R Y ad R Y ( =,,.., ) S by uy Y S u Sy uy Yu M Sy yj Y,,,...,. M j S x,,,...,. M x j M j

2 Iteratoal Joural of Scetfc ad Research Publcatos, Volume, Issue, December 0 ISS 0- M Sxy xj yj Y, M j =,,,. Defe, m y y y m m j j js j m x x x m m j j js j y uy uy s x ux ux s sby uy y s ux x sy uy y ux x m sy yj y m j s x x m j m x j m s y y x x S S b, S S xy j j m j y xy by SySx S S by C, Cby Y S S x y C x, C y, Y ( =,,.,) ( =,,., ) III. RATIO ESTIMATORS Cosder the followg rato type estmators uder two-stage samplg scheme. T u y, 0 uy T. ux wthout usg auxlary formato o x.

3 Iteratoal Joural of Scetfc ad Research Publcatos, Volume, Issue, December 0 ISS 0- y x T u uy T, u y u x T u uy T u u x Smth (99), Murthy (97) These estmators belog to the class of estmators cosdered by Pada (998). j T requres advace kowledge o T ad T requre advace kowledge o kowledge o,,..., ad. (,,...,) ; T ad T requre advace : IV. BIASES AD MEA SQUARE ERRORS OF ESTIMATORS As kow, T 0 s a ubased estmator of Y. T, T, T, T, ad T are based estmates of Y ad upto S S y S Sx xy ) Y u Y m M Y S S x xy ) uy m M Y S S y ) Y Y. S S y S S x xy ) Y COV u Y, u Y m M Y S S S x xy ) Y u m M Y. 0 To 0 ) the mea square errors (MSE) of T, T, T, T ad T are gve by S u S m M by y 0, the bases are

4 Iteratoal Joural of Scetfc ad Research Publcatos, Volume, Issue, December 0 ISS 0- ) Sby R S RSxy u Sy R Sx RSxy m M ) S by u Sy RSx R Sxy m M ) Sby R S RSxy u Sy m M ) Sby R S RSyx u (Sy R Sx RS xy) m M ) S by u (Sy R Sx RS xy) m M V. COMPARISO OF EFFICIECIES The suffcet codtos uder whch T, T, T, T, T would be more effcet tha T 0 (wthout usg auxlary varable ) are gve Table. Table R Cx C for all (=,,...) R b Cy Cby T: T: C T: C C b Cby Cx C for all. ( =,,..., ) T: b Cy Cby R C T: x for all. ( =, R Cy x y,...,) VI. ESTIMATIO OF VARIACES We have Est(S y ) sy u sxy m M Est(S ) s u s m M Est(S ) s u s m M x by by y

5 Iteratoal Joural of Scetfc ad Research Publcatos, Volume, Issue, December 0 ISS 0- Est u S u s Est u Sx u sx m M m M Est u Sy u sy m M m M Thus, Est ) Est MSE (T ) xy xy m M m M s ˆ ˆ by R s Rsy u s Rˆ s Rs ˆ m M y x xy s u s Rˆ s Rˆ s m M by y x xy Est ) s R s Rs u s m M Est ) ˆ ˆ by y y s R s Rs u d m M ˆ d s R s Rˆ s where Est MSE (T ) ˆ ˆ by y y x xy s u s Rˆ s Rs ˆ m M by y x xy VII. UMERICAL ILLUSTRATIOS For umercal llustrato of effceces of dfferet estmators, we cosder data from 97 cesus of Ida, descrbed below. The populato cossts of 0 blocks (ssu) dvded to = wards (fsu) of Berhampur Cty of Odsha. The umber of block (M ) wards are,,,,,, 0,,,,,,,,. The two varables.e. umber of educated females, female populato are used as y, x, respectvely. For comparso of Mea square error (MSE) of T 0, T, T, T, T, T, we cosder 0% samplg fracto at both the stages. Let the frst stage sample sze be = ad the szes of the secod stage sample.e. m ( =,,, ) are assumed to be,,,,,,,,,,,,, ad respectvely.

6 Iteratoal Joural of Scetfc ad Research Publcatos, Volume, Issue, December 0 ISS 0- Table : Statstcal Computatos M m u R (u Y Y) (u ) S y (u )(u Y Y) S x Sxy Calculated Results: = = Y 99. R = 0. S S 9. S.97 Remarks : For the gve llustrato, t s observed that by y Table : Comparso of Mea Square Errors (MSES) Estmator MSE Effcecy T 0 T T T T T MSE T MSE T MSE T MSE T MSE T MSE T 0 VIII. COCLUSIO The rato estmators cosdered a two stage samplg set up have bee compared wth the usual ubased estmator wthout use of auxlary formato, as regards ther effceces. The umercal llustrato shows that T ad T have early equal effcecy ad are more effcet tha other compettve estmators. REFERECES [] Murthy, M.. (97). Samplg Theory ad Methods. Statstcal Publshg Socety, Calcutta, Ida. [] Pada, P. (998). Some Strateges two-stage samplg usg auxlary formato. Upublshed Ph.D. dssertato, Utkal Uversty, Odsha, Ida. [] Sukhatme, P.V., ad Sukhatme, B. V. (970) Samplg theory of Surveys wth Applcatos. Id. Soc. Agr. Stat., ew Delh AUTHORS Frst Author Stashu Shekhar Mshra, MSC, M.phl(Statstcs), CIC, LLB, Assstat Professor, Dept. MCA TACT, F-, Chadaka, Idustral Estate, Bhubaeswar-, Odsha, Ida, Mal-ID-stashumshra0@gmal.com