EFFICIENT ESTIMATOR IN SUCCESSIVE SAMPLING USING POST-STRATIFICATION
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1 EFFICIET ETIMATOR I UCCEIVE AMPLIG UIG POT-TRATIFICATIO M. Trved* ad D. hula ** ABTRACT It s ofte see that a populato havg large umber of elemets remas uchaged several occasos but the value of uts chages. The sample surves are also ot lmted to oe-tme qures. I ths paper, a estmator has bee troduced uder successve surves. Ths estmator s ubased ad effcet over Post-stratfcato estmator. I ths paper, mmum varace of the optmum estmator has bee derved ad comparatve stud s corporated. Ke ords : Post-stratfcato, uccessve Occasos, Optmum, Estmator *Departmet of Appled Mathematcs,Brla Isttute of Techolog, Mesra, Rach (JH **Departmet of tatstcs, Lucow Uverst, Lucow(UP E-mal : mash_trved976@ahoo.com ( M. Trved Correspodg Author
2 . ITRODUCTIO: Toda, t s ofte see that sample surves are ot lmted to oe tme qures. If the value of stud character of a fte populato s subject to chage over tme, a surve carred out o a sgle occaso wll provde formato about the characterstcs of the surveed populato for the gve occaso ol ad ca ot gve a formato o the ature or the rate of chage of the characterstc over all occasos or more recet occaso. mth, T.M.F. (99, Holt, D. ad mth, T.M.F. (979, Jagers, P. Ode, A. ad Trulsso, L. (985 ad Jagers P. (986 have doe a lot of spred wor the area of post-stratfcato sample surve. Further the theor of Post-stratfcato sample surve was exteded b Cochra,.G. (977, Gupta,.C. ad Kapoor, V.K. (977, Ravdra gh ad uatme, B.V. (969 ad (973, gh, D. ad Chaudhar, F.. (984, uatme, P.V. uatme, B.V., uatme,. ad Aso, C. (984. Data regardg chagg propertes of the populatos of ctes or coutes, such as uemplomet statstcs, are collected regularl o a sample bass, to estmate the chages from oe occaso to the ext or to estmate the average over a certa perod. A mportat aspect of cotuous surves s the structure of the sample o each occaso. To meet these requremets, successve samplg provdes a strog tool for geeratg the relable estmates at dfferet occasos. Theor of successve samplg appears to have started wth the wor of Jesse (94. He was poeer to utlze the etre formato collected the prevous occasos. Further the theor of successve samplg was exteded b Patterso (950, Rao ad Graham (964, Gupta(979, Das(98, Chaturved ad Trpath ad ma others. Feg ad Zou (997 used the auxlar formato o both the occasos for estmatg the curret mea successve samplg. There are several tpes of procedures to adopt for estmatg the populato parameters: ( the same sample ma be used o each occaso ( a ew sample ma be tae o each occaso, ( a part of the sample ma be retaed whle the remader of the sample ma be draw afresh. ome codtos to cosder are that ( for estmatg chage from oe occaso to the ext, t ma be best to reta the sample o each occaso, ( for estmatg the mea o each occaso, t ma be best to draw a fresh sample o each occaso, ad ( f t s desre to estmate the mea o each occaso ad also the chage from oe occaso to the ext, t ma be best to reta part of the sample ad draw the remader of the sample afresh.
3 . OTATIO : Let a populato be of sze,that s sampled over two occasos. Assume that the sze of the populato remas uchaged so, but values of uts chage over occasos. Assume that we have two parts of the sample secod occaso. Frst part of the sample cossts of retaed uts of frst occasos ad the secod part has the. - The populato mea o the -th occaso,,,. uts draw afresh secod occaso where - The populato mea square error for the th occaso,,,. Y -The sample mea based o uts observed o the frst occaso. -The sample mea based o uts observed o the secod occaso ad commo wth the frst occaso. -The sample mea based o x -The sample mea based o o the frst occaso. - the proporto uts draw afresh o the secod occaso. uts commo to both the occasos ad observed - the sample sze of the sample draw afresh o the secod occaso. - sample uts observed o the frst occaso. - uts observed o the secod occaso ad commo wth the frst occaso. 3. ETIMATIO TRATEGY : (I ( ( e assumed the populato of sze remas uchaged over both occasos. e have uts costtutes the sample o the frst occaso of whch are retaed o the secod occaso whle are draw afresh o the secod occaso from (- uts. ow we have post stratfed the uts to strata, whch are draw afresh secod occaso. hle the uts, whch retaed from frst occaso rema uchaged. 3
4 4. THE ETIMATOR : For estmatg Y based o successve samplg usg post-stratfcato scheme, we propose a estmator ( l, such that.. (4. where, s a costat. The motvato of tag ths costat s tae from Agrawal, M.C. ad Pada, K.B. (993 ad (995 ad the form of equato (4. s mportat for varous applcatos, as to be metoed secto 5. Ths appears to belog to the class of estmators ow as composte estmators. ee, secto 5 Her, trudler, ad Che (007, ad also FCM (993. s the mea based o post stratfed uts s the mea based o the matched sample, whch s termed as l l ( x β here β the regresso coeffcet of the varate of the secod occaso o the varate of the frst occaso s assumed to be ow. I addto,, ad x are defed before. The the proposed estmator ca be wrtte as ( [ β( x ] 4.. (4. THEREM 4. The estmator s ubased for Y Proof: Clearl s a ubased estmator of Y wth varace gve b V (. To estmate the mea Y o the secod occaso we have two estmators, oe based o the sample draw afresh o the secod occaso ad the after post stratfed to -strata s. I addto, the other based o the sample commo to both the occaso. The estmator based o s a ubased estmator of Y such that E [ ] E ( Y.. (4.. ow, the estmator based o the matched sample s also ubased estmator for Y such that E [ l ] E [ β { x} ] ( β { E( E( x } Y β.. (4.. Y Y Y { }
5 5 Hece, the estmator s ubased estmator for, such as, E ( [ ] ( ( l pass E E ( Y Y Y.. (4..3 From equato o. (4.. & (4... THEOREM 4. The varace of the estmator s gve b V ( ( ( ( ( ( ( ρ Proof: e have the estmator ( l s The V ( ( ( ( ( ( l l, Cov Var Var.. (4.. The estmator s a stratfed mea the V ( V ( ( (.. (4.. I addto, the estmator l s a regressed estmator, whch has the varace. V( ( [ ] l x V β V ( ( [ ] x V V ( β ρ ρ ( ρ. (4..3 here β ρ After that oe ca easl get Cov ( /, l.. (4..4 ow b puttg the values from (4.., (4..3 ad (4..4 equato (4.. oe gets
6 V ( ( ( ( ( ( ρ ( δ / 5 OPTIMUM CHOICE For gettg the optmum value of, we have to dfferetate Var ( expresso wth respect to ad the equate t to zero. The we ca easl get Var Opt ( l Cov (, l Var ( l Var ( Cov(, l B puttg ths value varace expresso oe ca get V( V ( l [ Cov(, l ] V ( opt V( l V( Cov (, l. (5.. (5. The derved equato (5. s the expresso of the optmum varace. It has recetl bee oted that uder a dfferet cotext, Grager ad ewbold (986, usg Bates ad Grager (969, arrved at a varace estmate of ths same form. I that case t was used for the combato of forecasts for tme seres. However, t ma be used for a combato of estmators of the form ( COMPARIO: e see the varace expresso V ( [ Var( ] ( [ Var( l ] ( Cov(, l s [ Var ] ( [ Var ( l ] ( Cov (, l Ad V ( ( here V ( s s the varace of geeral sstematc samplg scheme. Obvousl, oe ca sa easl Var ( Var ( Because Var ( Var ( ad Var ( here Var( are the varaces of sample mea of post stratfed sample ad sample mea of smple radom sample scheme. 6
7 REFERECE. Agrawal, M.C. ad Pada, K.B. (993: A effcet estmator Post stratfcato, METRO, Vol. 5, 3-4, Agrawal, M.C. ad Pada, K.B. (995: A effcet estmator Post stratfcato, METRO, LIII, 3-4, Bates, J.M., ad Grager, C..J. [969], The combato of forecasts, Oper. Res. Q. 0, Cochra,.G. (977: amplg techques, Joh el ad os Ic. ew Yor. 5. Das, A. K. (98 : Estmato of Populato rato o two occasos.jour. Id.oc. Ag. tatst., 34, Feg,. ad Zou, G. (997 : ample Rotato Method wth auxlar Varable. Commu. tatst. Theo-Meth.,6, 6, FCM (993: Idrect Estmators Federal Programs. (U Federal Commttee o urve Methodolog, org Paper #, 8. Gupta,.C. ad Kapoor, V.K. (977: Fudametals of Appled tatstcs: ultachad ad sos, ew Delh. 9. Gupta, P.C. (979 : amplg o Two uccessve Occassos. Jour. tatst. Res.,3, Grager, C..J. ad Paul ewbold (986: The Combato of Forecasts, Forecastg Ecoomc Tme eres, d edto, Academc Press, pp Her, K., trudler, M., ad Che,. (007: A Emprcal Evaluato of Varous Drect, thetc, ad Tradtoal Composte mall Area Estmators, Amerca tatstcal Assocato, urve Research Methods ecto, proceedgs from August 007 Jot tatstcal Meetgs, page 3. (O CD.. Holt, D. ad mth, T.M.F. (979: Post-stratfcato, J.R. tat. oc., A, 4, Jagers, P. Ode, A. ad Trulsso, L. (985: Post-stratfcato ad rato estmator, It. tat. Rev., 53, Jagers P. (986 : Post-stratfcato agast bas samplg, It. tat. Rev., 54, Jesse, R. J. (94: tatstcal vestgato of a sample surve for obtag form facts. Iowa Agr. Expt. ta. Res. Bull., Patterso, H.D. (950 : amplg o successve occasos wth partal replacemet of uts. Jour. Roal tatst. Assoc., cr. B,, Rao, J..K. AD Graham, J. E. (964 : Rotato Desg for samplg o repeated occasos. Jour. Amer. tatst. Assoc., 59, Ravdra gh ad uatme, B.V. (969: Optmum stratfcato. A. Ist. tat. Math.,, Ravdra gh ad uatme, B.V. (973: Optmum stratfcato wth rato ad regresso methods of estmato. A. Ist. tat. Math, 5, mth, T.M.F. (99: Post stratfcato, the statstca, 40, gh, D. ad Chaudhar, F.. (984, Theor ad aalss of sample surve desg: ew Age Iteratoal (P Lo (formall le Easter Ltd. ew Delh.. uatme, P.V. uatme, B.V., uatme,. ad Aso, C. (984: amplg Theor of urve wth Applcatos, Iowa tate Uverst Press, Ida ocet of Agrcultural tatstcs, ew Delh. 7
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