Department of Statistics, Banaras Hindu University Varanasi , India 2 Chair of Department of Mathematics, University of New Mexico, Gallup, USA

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1 A Famly of eda Based Estmators Smple Radom Samplg Hemat K.Verma, Rajesh Sgh ad Floret Smaradache Departmet of Statstcs, Baaras Hdu Uversty Varaas-5, Ida har of Departmet of athematcs, Uversty of e exco, Gallup, USA orrespodg author Abstract I ths paper e have proposed a meda based estmator usg ko value of some populato parameter(s) smple radom samplg. Varous exstg estmators are sho partcular members of the proposed estmator. The bas ad mea squared error of the proposed estmator s obtaed up to the frst order of approxmato uder smple radom samplg thout replacemet. A emprcal study s carred out to judge the superorty of proposed estmator over others. Keyords: Bas, mea squared error, smple radom samplg, meda, rato estmator.. Itroducto osder a fte populato U {U, U,..., U } of dstct ad detfable uts. Let Y be the study varable th value Y measured o U,,,3...,. The problem s to estmate the populato mea Y Y. The smplest estmator of a fte populato mea s the sample mea obtaed from the smple radom samplg thout replacemet, he there s o auxlary formato avalable. Sometmes there exsts a auxlary varable X hch s postvely correlated th the study varable Y. The formato avalable o the auxlary varable X may be utlzed to obta a effcet estmator of the populato mea. The samplg theory descrbes a de varety of techques for usg auxlary formato to obta more effcet estmators. The rato estmator ad the regresso estmator are the to mportat estmators avalable the lterature hch are usg the auxlary formato. To ko more about the rato ad regresso estmators ad other related results oe may refer to [-3]. Whe the populato parameters of the auxlary varable X such as populato mea, coeffcet of varato, kurtoss, skeess ad meda are ko, a umber of modfed rato estmators are proposed the lterature, by extedg the usual rato ad Expoetal- rato type estmators.

2 Before dscussg further about the modfed rato estmators ad the proposed meda based modfed rato estmators the otatos ad formulae to be used ths paper are descrbed belo: - Populato sze - Sample sze Y - Study varable X - Auxlary varable 3 r r (x X), 3 Where oeffcet of skeess of the auxlary varable ρ - orrelato o-effcet betee X ad Y X, Y - Populato meas x, y - Sample meas, - Average of sample medas of Y m - Sample meda of Y β - Regresso coeffcet of Y o X B (.) - Bas of the estmator V (.) - Varace of the estmator SE (.) - ea squared error of the estmator SE(e) PRE(e,p) - Percetage relatve effcecy of the proposed estmator p th SE(e) respect to the exstg estmator e. The formulae for computg varous measures cludg the varace ad the covarace of the SRSWOR sample mea ad sample meda are as follos: f f (y Y) Sy,V(x) (x X) Sx,V(m) (m ) V (y), ov(y, x) f (x X)(y Y) (x X)(y Y), ov(y, m) (m )(y Y), xx V(x), X mm V(m), ym ov(y, m), Y yx ov(y, x) XY

3 y (y Y), Sx (x X) Where f ; S, I the case of smple radom samplg thout replacemet (SRSWOR), the sample mea y s used to estmate the populato mea Y. That s the estmator of Y Yr y th the varace V(Y f (.) r ) S y The classcal Rato estmator for estmatg the populato mea Y of the study varable Y s defed y asyr X. The bas ad mea squared error of Y R are gve as belo: x xx yx B(YR ) Y (.) SE(Y R ) V(y) R V(x) R ov(y, x) here Y R (.3) X. Proposed estmator Suppose ( m ) t y, t y, t yexp here a b, m am b m ( ) m Such that t, t, t, here deotes theset of all possble rato typeestmatorsfor estmatg the populato mea Y. By defto theset s a lear varety, f t y t for here t R W, g (=,, ) deotes the statstcal costats ad R deotes the set of real umbers. (.) (.) Also, t y m ( ) g, t ( m ) yexp m ad a b, m am b. To obta the bas ad SE expressos of the estmator t, e rte y Y( e), m ( e) such that E (e) E(e ),

4 E(e Vm ov y,m V y ), E(e ) mm, E(e e) ym Y Expressg the estmator t terms of e s, e have Y t Y( e) a here a ( e ). b g e exp e (.3) Expadg the rght had sde of equato(.3) up to the frst order of approxmato, e get g(g ) t Y e e e ee (.4) 4 8 here g. (.5) Takg expectatosof both sdes of (.4)ad the subtractg Y from both sdes,eget thebases of estmators,up to thefrst order of g(g ) B(t) Y (g ) B(t) Yg B(t ) Y 4 8 From (.4), e have 4 8 mm mm approxmato as ym ym mm ym (.6) (.7) (.8) the t Y Y(e e ) (.9) Squarg both sdes of (.9) ad the takg expectatos, e get the SE of the estmator t, up to the frst order of approxmato as SE(t) V y herer R Vm Rov y,m Y. SE(t) ll be mmum, he ov y,m R V m k(say) (.) (.)

5 Puttg the value of (=k) (.), e get the mmum SE of the estmator t, as m. SE(t) V y The mmum SE of the estmator t s same as that of tradtoal lear regresso estmator. From (.5) ad (.), e have g k (.) (.3) From (.) ad (.3), e have oly to equatos three ukos. It s ot possble to fd the uque values of s (=,, ). I order to get uque values for s, e shall mpose the lear restrcto Bt Bt B y (.4) Equatos (.), (.) ad (.4) ca be rtte matrx form as g B(t ) k B(t ) (.5) Usg (.5) e get the uque value of s (=,, ) as r r r her e r gb(t ) B(t) B(t )( g k) B(t) k kb(t ) kb(t ) (.6) Table.: Some members of the proposed estmator a b g Estmators q y - q y m

6 - q3 y m - q 4 y m ( m) - - q 5 yexp m q q q q yexp yexp ( m) m ( m) m y yexp m y yexp m ( m) q y yexp m m ( m) m ( m) m 3. Emprcal Study For umercal llustrato e cosder: the populato ad take from [4] pageo.77, the populato 3 s take from [5] page o.4. The parameter values ad costats computed for the above populatos are gve the Table 3.. SE for the proposed ad exstg estmators computed for the three populatos are gve the Table 3. hereas the PRE for the proposed ad exstg estmators computed for the three populatos are gve the Table 3.3. Parameters Table: 3. Parameter values ad costats for 3 dfferet populatos For sample sze =3 For sample sze =5 Popl- Popl- Popl-3 Popl- Popl- Popl

7 Y X R V (y) V (x) V (m) ov (y,m) ov (y, x) Table: 3.. Varace / ea squared error of the exstg ad proposed estmators Estmators For sample sze =3 For sample sze =5 Populato- Populato- Populato-3 Populato- Populato- Populato-3 q q q q q q q q q q t(opt) Table: 3.3. Percetage Relatve Effcecy of estmators th respect to y Estmators For sample sze =3 For sample sze =5 Populato- Populato- Populato-3 Populato- Populato- Populato-3

8 q q q q q q q q q q t(opt) ocluso From emprcal study e coclude that the proposed estmator uder optmum codtos perform better tha other estmators cosdered ths paper. The relatve effceces ad SE of varous estmators are lsted Table 3. ad 3.3. Refereces. urthy.. (967). Samplg theory ad methods. Statstcal Publshg Socety, alcutta, Ida.. ochra, W. G. (977): Samplg Techques. Wley Easter Lmted. 3. Khare B.B. ad Srvastava S.R. (98): A geeral regresso rato estmator for the populato mea usg to auxlary varables. Alg. J. Statst.,: Ssoda, B.V.S. ad Dved, V.K. (98): A modfed rato estmator usg co-effcet of varato of auxlary varable. Joural of the Ida Socety of Agrcultural Statstcs 33(), Sgh G.. (3): O the mprovemet of product method of estmato sample surveys. Joural of the Ida Socety of Agrcultural Statstcs 56 (3), Sgh H.P. ad Talor R. (3): Use of ko correlato co-effcet estmatg the fte populato meas. Statstcs Trasto 6 (4), Sgh H.P., Talor R., Talor R. ad Kakra.S. (4): A mproved estmator of populato mea usg Poer trasformato. Joural of the Ida Socety of Agrcultural Statstcs 58(), 3-3.

9 8. Sgh, H.P. ad Talor, R. (5): Estmato of fte populato mea th ko co-effcet of varato of a auxlary. STATISTIA, ao LXV,.3, pp Kadlar. ad g H. (4): Rato estmators smple radom samplg. Appled athematcs ad omputato 5, Koyucu. ad Kadlar. (9): Effcet Estmators for the Populato mea. Hacettepe Joural of athematcs ad Statstcs, Volume 38(), Sgh R., Kumar. ad Smaradache F. (8): Almost ubased estmator for estmatg populato mea usg ko value of some populato parameter(s). Pak.j.stat.oper.res., Vol.IV, o., pp Sgh, R. ad Kumar,. (): A ote o trasformatos o auxlary varable survey samplg. od. Asss. Stat. Appl., 6:, 7-9. do.333/as Sgh R., alk S., haudhary.k., Verma H.K., ad Adeara A.A. (): A geeral famly of rato-type estmators systematc samplg. Jour. Relab. Stat. Ssc., 5(): Sgh, D. ad haudhary, F. S. (986): Theory ad aalyss of survey desgs. Wley Easter Lmted. 5. ukhopadhyay, P. (998): Theory ad methods of survey samplg. Pretce Hall.

Comparison of Dual to Ratio-Cum-Product Estimators of Population Mean

Comparison of Dual to Ratio-Cum-Product Estimators of Population Mean Research Joural of Mathematcal ad Statstcal Sceces ISS 30 6047 Vol. 1(), 5-1, ovember (013) Res. J. Mathematcal ad Statstcal Sc. Comparso of Dual to Rato-Cum-Product Estmators of Populato Mea Abstract