New Class of Estimators of Population Mean. DECISION SCIENCES INSTITUTE New Class of Estimators of Population Mean Utilizing Median of Study Variable

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1 New lass of Esiaors of Poulaio Mea DEISION SIENES INSTITUTE New lass of Esiaors of Poulaio Mea Uilizig Media of Sudy Variable S.K. Yadav Dr. RML Avadh Uiversiy Diesh K. Shara Uiversiy of Marylad Easer Shore S.S. Mishra Dr. RML Avadh Uiversiy ABSTRAT This sudy deals wih he esiaio of oulaio ea usig he edia of he sudy variable hrough a ew class of esiaors. The esiaors are esed fro he bias ad ea squared error (MSE) ersecives u o he firs degree of aroxiaio. The oiu value of he cosa or characerizig scalar has bee obaied. The salles value of he MSE of he roosed esiaors is also obaied for his oiu value of he characerizig scalar.the uerical sudy shows ha he roosed esiaors erfor beer ha he coeig esiaors of he oulaio ea uder sile rado salig schee. KEYWORDS: Sile rado salig, Poulaio ea, Raio esiaor, Bias, Efficiecy. INTRODUTION Salig is of crucial iorace wheever he oulaio is huge as i is cosly ad iecosuig o collec iforaio o each ui of oulaio. Naurally, i is esseial o ake he corresodig saisic as a esiaor for he esiaio of oulaio araeer as i has alos all desirable roeries of a good esiaor. I has bee oed i racice ha he corresodig saisic of ay araeer has a sigifica aou of variaio aroud he acual araeer. Oe of our objecives is o fid he esiaor for he araeer which has is disribuio ore ad ore coceraed aroud he rue araeer. The ai of such esiaor is achieved hrough he use of eiher addiioal iforaio o ay araeer of he sudy variable or by he use of he auxiliary variable. Boh he variables have a high degree of correlaio. Raio ye esiaors are used whe sudy ad auxiliary variables are highly osiively correlaed (ochra, 90). Produc ye esiaors are used whe sudy ad auxiliary variables are highly egaively correlaed (Robso, 957). Thus, i his sudy, we search for eve biased esiaor bu havig salig disribuio very close o rue araeer. Alhough he use of auxiliary variable iroves he efficiecy of he esiaor, i is colleced wih he addiioal cos of he survey. Therefore, i is beer if we have iforaio o ay araeer of he sudy variable which ay be uilized for iroved esiaio of he araeer wihou icreasig cos of he survey (Subraai, 06). I his sudy, we have uilized he iforaio o he oulaio edia of he sudy variable which is soe ies easily available i racice. For exale, i he surveys regardig he

2 New lass of Esiaors of Poulaio Mea esiaio of average icoe, average arks, body ass idex, ec., where i is assued ha oulaio ea is ukow, bu oulaio edia is kow sice i does o require he full iforaio o he oulaio values of he sudy variable. LITERATURE REVIEW Over he years, several esiaors of he oulaio ea such as he aive ubiased sale ea esiaor, usual raio esiaor ad odified raio ye esiaors have bee roosed i he lieraure. A suary of he lieraure review of exisig esiaors is reseed i Table. Table : Suary of Lieraure Review of Exisig Esiaors YEAR REFERENES JOURNAL 06 Yadav e al. Ieraioal Joural of Agriculural & Saisical Scieces 06 Subraai Saisics i Trasiio New Series 05 Yadav ad Mishra Saisika 05 Malik ad Sigh Alied Maheaics & ouaio 05 Rashid e al. Alied Maheaics & Iforaio Scieces 0 Oyeka e al. Oe Joural of Saisics 0 Saaullah e al. Alied Maheaics & ouaio 03 Subraai ad Kuaraadiya Saisics i Trasiio New Series 03 Subraai Joural of Moder Alied Saisical Mehods 03 Yadav ad Kadilar Alied Maheaics & ouaio 03 Oyeka e al. Global Joural of Sciece Froier Research Maheaics & Decisio Scieces 03 Yadav ad Kadilar Haceee Joural of Maheaics & Saisics 0 Subraai ad Kuaraadiya Joural of Reliabiliy ad Saisical Sudies 0 Solaki e al. ISRN Probabiliy ad Saisics 0 Sigh ad Solaki Alied Maheaics ad ouaio 0 Tailor e al. ouicaios of he Korea Saisical Sociey 0 Shabbir ad Gua ouicaios i Saisics Theory & Mehods 00 Koyucu ad Kadilar Joural of Alied Saisics 00 Sousa e al. Joural of Saisical Theory ad Pracice PROPOSED ESTIMATORS Moivaed by Subraai (06), we have roosed he followig class of esiaors uilizig oulaio edia of he sudy variable for iroved esiaio of oulaio ea of he sudy variable as follows: M P y ( ) () M Here is a cosa, also kow as characerizig scalar, ad is deeried i such a way ha he roosed esiaor has leas ea squared error.

3 New lass of Esiaors of Poulaio Mea The followig assuios are ade i sudyig he salig roeries of he roosed class of esiaors, M M Bias ( ) y Y ( e0 ) ad M( e ) such ha E ( e 0 ) 0, E( e ) ad M M f f f E( e0 ) y, E( e ), E( e e ) 0. where, M i i U o he aroxiaio of degree oe, he aheaical exressios for bias ad MSE of roosed class of esiaors resecively are, f B( ) B( ) Y[( ) ( ) ( ) ] () M ad f MSE( ) Y [ y ] (3) The oiu value of or equivalely is obaied by arially differeiaig (3) wih resec o which iiizes he ea squared error of he roosed esiaor as, The exressio for he salles MSE for roosed esiaor is obaied by subsiuig he value of i (3) as follows, f MSE i ( ) Y y () EFFIIENY OMPARISONS The roosed esiaors are coared wih he eioed coeig esiaors; ea er ui esiaor, ochra (90) usual raio esiaor, Waso (937) usual regressio esiaor, Bahl ad Tueja (99) ad Subraai (06) esiaors. The resecive codiios uder which i is beer ha above esiaors are give below. V ( 0) MSEi ( ) 0, MSE ( ) MSEi ( ) x 0, MSE( ) MSEi ( ) 0 y, 3

4 New lass of Esiaors of Poulaio Mea x MSE ( 3) MSEi ( ) 0, MSE ( ) MSEi ( ) R R 0, NUMERIAL STUDY For he judge regardig he roeries of he roosed ad he exisig esiaors of oulaio ea, we have cosidered he oulaio give by Subraai (06). The ables give below aed Tables o 5 show he araeers of he oulaio uder cosideraio, biases of differe esiaors uder sudy icludig roosed esiaors, variaces ad ea squared errors of exisig ad roosed esiaors ad erceage relaive efficiecies of he roosed esiaors over oher exisig esiaors of oulaio ea resecively. Table. Various Paraeers of hree aural oulaios Paraeer Pol N 3 5 N 7856 Y M M X R.0999 R.58 5 y 0.50 x Table 3. Bias of he coeig ad roosed esiaors Esiaor Pol

5 New lass of Esiaors of Poulaio Mea Table. Variace ad MSE of he coeig ad roosed esiaors Esiaor Pol Table 5. PRE of he roosed esiaor w.r.. coeig esiaors Esiaor Pol ONLUSION I his sudy, we have roosed a class of esiaors of he oulaio ea uilizig he oulaio edia of sudy variables. We have derived he MSE ad bias of he roosed esiaors esiaig he aroxiaio uo he firs degree. The leas value of he MSE of he roosed esiaors is also achieved o he oiu value of he characerizig scalar. The roosed esiaors are heoreically coared wih he coeig esiaors of he oulaio ea. Fially, i has bee show ha he roosed esiaors are beer ha esiaors eioed above sice hey have he leas MSEs. REFERENES Bahl, S. ad Tueja, R.K. (99). Raio ad roduc ye exoeial esiaor, Iforaio ad Oiizaio Scieces, XII (I), Sigh, H.P. ad Solaki, R.S. (0). Iroved esiaio of oulaio ea i sile rado salig usig iforaio o auxiliary aribue, Alied Maheaics ad ouaio, 8, Subraai, J. (03). Geeralized odified raio esiaor of fiie oulaio ea, Joural of Moder Alied Saisical Mehods, (), 55. Subraai, J. (06). A ew edia based raio esiaor for esiaio of he fiie oulaio ea, Saisics i Trasiio New Series, 7,, -. 5

6 New lass of Esiaors of Poulaio Mea Tailor, R. Parar, R., Ki, Jog-Mi ad Tailor, R. (0). Raio-u-Produc Esiaors of Poulaio Mea usig kow Poulaio Paraeers of Auxiliary Variaes, ouicaios of he Korea Saisical Sociey, 8(), Yadav S.K. ad Kadilar,. (03). Iroved lass of Raio ad Produc Esiaors, Alied Maheaics ad ouaio, 9, Yadav, S.K. ad Mishra, S.S. (05). Develoig Iroved Predicive Esiaor for Fiie Poulaio Mea Usig Auxiliary Iforaio, Saisika, 95,, (A colee lis of refereces is available uo reques) 6