IMPROVED RATIO AND PRODUCT TYPE ESTIMATORS OF FINITE POPULATION MEAN IN SIMPLE RANDOM SAMPLING

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1 REVISTA IVESTIGAIO OPERAIOAL VOL. 6, O., 7-76, 6 IMPROVED RATIO AD PRODUT TPE ESTIMATORS OF FIITE POPULATIO MEA I SIMPLE RADOM SAMPLIG Gajndra K. Vshwaarma, Ravndra Sngh, P.. Gupa, Sarla Par Dparmn of Appld Mahmacs, Indan School of Mns, Dhanbad-6, Jharhand, Inda vshwag@rdffmal.com Dparmn of Mahmacs and Sascs, Banashal Unvrs, Japur-, Rajashan, Inda ravndrasngh@gmal.com ABSTRAT In hs papr, mprovd rao and produc p smaors hav bn dvlopd for smang h fn populaon man of h sud varabl usng aular nformaon n smpl random samplng SRS. Th prssons for h bas and man squar rror MSE of h proposd smaors ar oband undr frs ordr of appromaon. Thorcal and mprcal suds hav bn don o dmonsra h ffcncs of h proposd smaors ovr ohr wll nown smaors. KEWORDS: Sud varabl, Aular varabl, bas, MSE, Effcnc. MS: 6D5 RESUME En s rabajo smadors dl po razón produco mjorados han sdo dsarrrollados para smar la mda d una poblacón fna d la varabl bajo sudo usando nformacón aular n l musro smpl alaoro msa. Las prsons dl ssgo dl rror cuadráco mdo EM d los smadors propusos son obndos bajo apromacon d prmr ordn. Esudos órcos mpírcos han sdo dsarrollados para dmosar las fcncas d los smadors propusos rspco a oros smadors bn conocdos.. ITRODUTIO Th us of aular nformaon has bcom ndspnsabl for mprovng h prcson of h smaors of populaon paramrs such as h man and varanc of a varabl undr sud. A gra var of chnqus such as h rao, produc and rgrsson mhods of smaon ar commonl nown n hs rgard. Aular nformaon can b usd hr a h dsgn sag or a h smaon sag or a boh h sags. Kpng hs fac n vw, larg numbr of smaors hav bn suggsd n samplng lraur. Som noworh conrbuons n hs drcon hav bn mad b ochran9, Robson957, Murh96, Sngh 967, Saha 979, Bahl and Tuja 99, Sngh and Espjo, Sngh and Talor 5, Kadlar and ng 5, Sngh and Vshwaarma 7,, Shabbr al., and man ohrs. L U dno a fn populaon conssng of uns U, U,..., U }. Also, l, dno h 7 { sud varabl and h aular varabl ang valus,,,,...,, rspcvl, on h h un

2 U of h populaon U. On h assumpon ha h populaon man of s nown, h sma of populaon man of s oband b slcng a sampl of sz n n < from h populaon U usng smpl random samplng whou rplacmn SRSWOR schm. Th convnonal rao and produc smaors of ar gvn b R P whr and ar h sampl mans of and, rspcvl. Bahl and Tuja 99 suggsd h followng ponnal p rao and produc smaors for h populaon man : p p. PROPOSED ESTIMATORS W dfn h followng mprovd rao and produc p smaors for h populaon man n SRSWOR: 5 6 whr and ar ral consans o b drmnd such ha h MSEs of and ar mnmzd. Furhr, s obsrvd ha h smaors and rducs o a s of smaors {,, } b assgnng suabl valus o h consans and as follows: Usual unbasd smaor: for Bahl and Tuja 99 rao smaor: for Bahl and Tuja 99 produc smaor: for To oban h bas and MSE of h smaors and, w consdr Thn, w hav, E E E, E E f n S S S whr,, f,,,, n S S S, S, 7 7

3 7. S ow, prssng 5 and 6 n rms of,, and ranng h rms of s upo h scond dgr, w oban 9 Tang h pcaons n, 9, and usng rsuls n 7, w oban h bas of h smaors and o h rms of ordr n O as B B whr,. Agan, from and 9, b nglcng h rms of s havng dgr grar han on, w hav Squarng boh sds of and, ang h pcaon, and usng rsuls n 7, w oban h MSE of h smaors and o h rms of ordr n O as MSE MSE 5.. Opmal Valus of and Th opmal valus of and, for whch h MSE of h smaors and ar mnmzd, ar oband b usng h followng condons: MSE 6 MSE 7 On solvng 6 and 7, w hav

4 whr and 9 dno h rspcv opmal valus of and. Also, usng hs opmal valus of and n and 5, rspcvl, w oban h mnmum aanabl MSE of h smaors and as MSE MSE mn mn Rmar. Th mnmum aanabl MSEs n corrsponds o h MSEs of asmpoc opmum smaors AOEs opmal valus,.., and, whch ar oband on rplacng and, n 5 and 6 b hr rspcv and. So, w hav and MSE MSE To h frs dgr of appromaon, h MSE of h varous smaors lsd abov ar: MSE MSE R P V MSE MSE 5. EFFIIE OMPARISOS For mang ffcnc comparsons of h smaors and wh h sng smaors, w hav from, 5, and o 5, MSE < V f > 6 MSE < V f > 7 MSE < MSE f R < 7

5 v MSE < MSE f P < 9 v MSE < MSE f < v MSE < MSE f <. EMPIRIAL STUD To amn h mrs of h proposd smaors and ovr ohr sng smaors, w hav consdrd hr naural populaon daa ss as follows: Populaon I - [Sourc: Johnson 97] : Prcnag of hvs affcd b dsas : Man Januar mpraur Z : Da of flowrng of a parcular summr spcs numbr of das from Januar, n, 5,, Z,.,.9,.7, Z Z.,.7, Z. Populaon II - [Sourc: Sngh 969] : umbr of fmals mplod : umbr of fmals n srvc Z : umbr of ducad fmals 6, n, 7.6, 5., Z 79.,.777,.7, Z Z.,.56,.577, Z.6 Populaon III - [Sourc: Sl and Torr 96] : Log of laf burn n sc : Poassum prcnag Z : hlorn prcnag, n 6,.66,.657, Z.77,.79,.996, Z.7,.,.95, Z.79 Z Th prcnag rlav ffcncs PREs ar oband for varous suggsd smaors of wh rspc o h usual unbasd smaor and h fndngs ar prsnd n Tabl. 7

6 Tabl : Prcnag Rlav Effcncs PREs of varous smaors wh rspc o Esmaors Aular varabls usd Populaon I Populaon II Populaon III -... R P Z Z Z OLUSIOS From Tabl, s obsrvd ha: For all h populaon daa ss, h PRE of h proposd rao smaor s mor han ha of h usual unbasd smaor, h rao smaor R and h Bahl and Tuja 99 rao smaor. For all h populaon daa ss, h PRE of h proposd produc smaor s mor han ha of h usual unbasd smaor, h produc smaor P and h Bahl and Tuja 99 produc smaor. So, h proposd smaors and ouprforms h ohr sng smaors of h samplng lraur, and hnc can b prfrrd for praccal applcaons. Acnowldgmn: Th auhors ar hanful o h dor and h larnd rfrs for hr valuabl commns and suggsons owards h mprovmn of h papr. REEIVED SEPTEMBER, REVISED MA, 5 REFEREES [] BAHL, S. and TUTEJA, R.K. 99: Rao and produc p ponnal smaor. Informaon and Opmzaon Scncs,, [] OHRA, W.G. 9: Th smaon of h lds of h cral prmns b samplng for h rao of gran o oal produc. Th Journal of Agrculural Scnc,, [] JOHSTO, J. 97: Economrc mhods nd dn.. Mc Graw Hll Boo o., Too. [] KADILAR,. and IGI, H. 5: A nw smaor usng wo aular varabls. Appld Mahmacs and ompuaon, 6, 9-9. [5] MURTH, M.. 96: Produc mhod of smaon. Th Indan Journal of Sascs, Srs A, 6, [6] ROBSO, D. S. 957: Applcaon of mulvara polas o h hor of unbasd rao-p smaon. Journal of Amrcan Sascal Assocaon, 5, 5-5. [7] SAHAI, A. 979: An ffcn varan of h produc and rao smaors. Sasca rlandca,,, 7-5. [] SHABBIR, J., HAQ, A. and GUPTA, S. : A nw dffrnc-cum-ponnal p smaor of fn populaon man n smpl random samplng. Rvsa olombana d Esadsca, 7, [9] SIGH, H.P. and ESPEJO, M.R.: On lnar rgrsson and rao-produc smaon of a fn populaon man. Journal of h Roal Sascal Soc, 5,

7 [] SIGH, H. P. and TAILOR, R.5: Esmaon of fn populaon man usng nown corrlaon coffcn bwn aular characrs. Sasca, LV,, 7-. [] SIGH, H.P. and VISHWAKARMA, G.K. 7: Modfd ponnal rao and produc smaors for fn populaon man n doubl samplng. Ausran Journal of Sascs, 6,, 7-5, [] SIGH, H.P. and VISHWAKARMA, G.K. : Som smaors of fn populaon man usng aular nformaon n sampl survs. Journal of Appld Sascal Scncs, 6,, -. [] SIGH, M.P. 967: Rao-cum-produc mhod of smaon. Mra,, -7. [] SIGH, M.P. 969: omparson of som rao-cum-produc smaors. Sanha, B,, [5] STEEL, R.G.D. and TORRIE, J.H. 96: Prncpls and Procdurs of Sascs. Mc Graw Hll Boo o. 76