Improved Class of Ratio Estimators for Finite Population Variance

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1 Global Journal of Scence Fronter Research: F Mathematcs and Decson Scences Volume 6 Issue Verson.0 Year 06 Tpe : Double lnd Peer Revewed Internatonal Research Journal Publsher: Global Journals Inc. (USA) Onlne ISSN: 9-66 & Prnt ISSN: Improved Class of Rato Estmators for Fnte Populaton Varance Audu Ahmed, Adedao Amos Adewara, Ran Vja Kumar Sngh Usmanu Danfodo Unverst Sokoto, Ngera Abstract- In ths paper, we have suggested a class of mproved rato estmators for fnte populaton varance. The proposed class of estmators s obtaned b transformng both the sample varances of stud and aular varables. The MSE of the proposed estmators have been obtaned and the condtons for ther effcenc over some estng varance estmators have been establshed. The present faml of fnte varance estmator, havng obtanng the optmal values of the constants, ehbt sgnfcant mprovement over the estmators consdered n the stud. The emprcal stud s also conducted to corroborate the theoretcal results and the results show that the proposed class of estmators s more effcent. Kewords: effcenc, mean square error, rato estmator, fnte populaton, varance. GJSFR-F Classfcaton : MSC 00: 00A05 ImprovedClassofRatoEstmatorsforFntePopulatonVarance Strctl as per the complance and regulatons of : 06. Audu Ahmed, Adedao Amos Adewara, Ran Vja Kumar Sngh. Ths s a research/revew paper, dstrbuted under the terms of the Creatve Commons Attrbuton-Noncommercal 3.0 Unported Lcense permttng all non commercal use, dstrbuton, and reproducton n an medum, provded the orgnal work s properl cted.

2 Ref 8. R. Sngh, P. Chauhan, N. Sawan, F. Smarandache, A general faml of estmators for estmatng populaton varance usng known value of some populaton parameters, Far East Journal of Theoretcal Statstcs (007). Improved Class of Rato Estmators for Fnte Populaton Varance Audu Ahmed α, Adedao Amos Adewara σ & Ran Vja Kumar Sngh ρ Abstract- In ths paper, we have suggested a class of mproved rato estmators for fnte populaton varance. The proposed class of estmators s obtaned b transformng both the sample varances of stud and aular varables. The MSE of the proposed estmators have been obtaned and the condtons for ther effcenc over some estng varance estmators have been establshed. The present faml of fnte varance estmator, havng obtanng the optmal values of the constants, ehbt sgnfcant mprovement over the estmators consdered n the stud. The emprcal stud s also conducted to corroborate the theoretcal results and the results show that the proposed class of estmators s more effcent. Kewords: effcenc, mean square error, rato estmator, fnte populaton, varance. I. Introducton The varaton of produce or elds n the manufacturng ndustres and pharmaceutcal laboratores are sometme a matter of concern to researchers (Ahmed et al. []). The use of supplementar (aular) nformaton, beng constant wth unt (e.g. populaton mean, populaton standard devaton, e.t.c.) or unt free constant (e.g. Coeffcent of varaton, Kurtoss, e.t.c.), can enhance the effcenc at the estmaton stage. In recent past, ths concept has been utlzed b several authors to mprove the effcenc of rato and product tpe estmators for estmatng populaton mean as well populaton varance of stud varable. In ths paper, an mproved class of rato estmators for estmatng fnte populaton varance has been proposed wth objectve to produce effcent estmators and ther propertes have establshed. Let Ω = (, 3 NN) be a populaton of sze NN and YY, XX be two real valued functons havng values(yy, XX ) R + > 0 on the tth unt of UU( NN). We assume postve correlaton ρρ > 0 between the stud varable YY and aular varable XX. Let S and S be the fnte populaton varance of YY and XX respectvel and s and s be respectve sample varances based on the random sample of sze n drawn wthout replacement. Sngh et al. [8] defned the general faml of estmators for estmatng fnte populaton varance of the stud varable YY as S Global Journal of Scence Fronter Research ( F ) Volume XVI Issue II V erson I Year 06 7 Author α: Department of Mathematcs, Usmanu Danfodo Unverst Sokoto, Ngera. e-mal: ahmed.audu@udusok.edu.ng Author σ: Department of Statstcs, Unverst of Ilorn, Kwara State, Ngera. e-mal: aaadewara@gmal.com Author ρ: Department of Mathematcs, Kebb State Unverst of Sc. and Tech. Alero, Ngera. e-mal: snghrvk3@gmal.com 06 Global Journals Inc. (US)

3 Improved Class of Rato Estmators for Fnte Populaton Varance Global Journal of Scence Fronter Research ( F ) Volume XVI Issue II V erson I Year 06 8 η = s α as + b ( as b) + ( α)( as b) (.) where a and b are constants based on aular varable X lke coeffcent of skewness, kurtoss and correlaton coeffcent etc. α s the constant that mnmzes the mean square error (MSE) of the estmator. Table shows some members of η -faml for dfferent values of a, b and α. The MSEs/Varance of the estmators n Table are gven below: where MSE ( η ) ( η ) γ ( ψ ) Var = S (.) 0 0 ( ) ( ) ( ) Sγ ψ0 + ψ0 ψ = = Sγ ( ψ0 ) h ( ψ0 ) h( ψ ) + =,3,,5, 6 S S Sβ ( ) SC S h h h h h S C S ( ) S ( ) C SC ( ) S ( ) =, 3 =, =, 5 =, 6 = β β β + β s s S Y Y S X X n n n n = ( ), = ( ), = ( ), = ( ) n = n = N = N = γ N N n n X = X, Y = Y, =, = N N n n = = = = λ, rs ψ n = = and ( ) r λ ( ) rs = Y Y X X rs r/ s/ λ0 λ0 N N The MSE of η to second order appromaton s gven below: = ( η ) = γ ( ψ ) + α θ ( ψ ) αθ ( ψ ) MSE S s (.3) 0 0 (.) The MSE ( η ) epresson s mnmzed for the optmum values of α gven b equaton (.5). Ths s obtaned b partal dfferentaton of equaton (.) wth respect to α. MSE ( η) = γs ( 0 ) mn ψ ( ψ ) ( ψ ) 0 (.5) Man other researchers ncludng Kadlar and Cng [6], Yadav and Kadlar [5], Sngh and Solank [0], Gupta and Shabr [3], Subraman and Kumarapandan [3], Sngh and Vshwakarma [], Sngh, et al. [9], Sanaullah, et al. [7] and Solank and Sngh [] have sgnfcantl contrbuted to the mprovement of both rato and product mean & varance estmators n samplng surve. Ref 3. S. Gupta, J. Shabbr, Varance estmaton n smple random samplng usng aular nformaton, Hacettepe Journal of Mathematcs and Statstcs 37 (008) Global Journals Inc. (US)

4 Improved Class of Rato Estmators for Fnte Populaton Varance Ref. A. A. Adewara, R. Sngh, M. Kumar, Effcenc of some modfed rato and product estmators usng known value of some populaton parameters, Internatonal Journal of Appled Scence and Technolog () (0) II. Proposed Estmator After studng the related fnte populaton varance estmators stated n secton and motvated b the work of Yadav and Kadlar [6] and Adewara et al.[] estmators for populaton mean n whch the former (.e. Yadav and Kadlar [6]) equals the latter (.e. Adewara et al.[]) when k = k = k 3 = k = k 5 = k 6 =. Ther estmators are defned as * X η = k, * * X + C η = k, η3 = k 3, X + C * C = k + X + C η * X + ρ, η5 = k 5 + ρ, * + ρ η6 = k 6 X + ρ Where and are the respectve sample means of the aular and stud varables, havng the relatonshp: () X= f+ ( f) () ( ) Y= f+ f where n f = s fnte populaton correcton, k, =,,3,,5, 6, are real constants. N We proposed the followng class of rato estmator * * S *, s S C * * * s C * * S β( ) * * Sβ( ) C, 3 3 *, *, s β( ) s β( ) C * * SC β( ) * * S + β( ) 5 5 *, 6 6 * s C β( ) s + β( ) Where s and s are the respectve sample fnte varances of the aular and S = fs + f s S fs f s = + wth condton that n< N. stud varables, havng the relatonshp: () ( ) () ( ) In order to obtan the MSE, we defned e s S = and e 0 S ( 0 ) ( ) 0, ( 0 ) γ ( ψ0 ) ( ) γ ( ψ0, ) E( ee 0 ) γ ( ψ ) s S = such that S E e = E e = E e = (.) E e = = Epressng, =,,3,,5, 6, n terms of e 0 and e, we have ( )( ) k S ve ve = k S ve vh e 0 = (.) ( 0)( ) =,3,,5, 6 Global Journal of Scence Fronter Research ( F ) Volume XVI Issue II V erson I Year 06 9 We now assume that ve < and vh e < so that ( ve ) and ( vh e ) are epandable. Epandng the rght hand sde of (.) up to second degree appromaton, 06 Global Journals Inc. (US)

5 Improved Class of Rato Estmators for Fnte Populaton Varance subtract S from ts both sdes, square the both sdes and takng epectaton usng the results n equaton (.), we obtan the MSEs of the proposed estmators as: Global Journal of Scence Fronter Research ( F ) Volume XVI Issue II V erson I Year 06 0 MSE ( ) ( ψ ) + ( 3 )( ψ ) ( k k)( ψ ) k 0 k k 0 S γ v + ( k ) = = k ( 0 ) ( 3 ) ( 0 ) ψ k k h ψ + S γ v + ( k ),3,,5,6 = ( k k) h( ψ ) (.3) The MSE ( η ), =,,3,,5, 6 epressons are mnmzed for the optmum values of k gven b where k Replacng v γ ( ψ ) ( ψ ) ( ) ( ) ( ) 0 + A k = =, = v γ 3 ψ0 ψ + ψ0 + γ ( ψ ) ( ψ ) + ( ) ( ) ( ) v h 0 h A = =, =,3,,5,6 v γ 3h ψ0 h ψ + ψ0 + n v = N n k b k, =,,3,,5,6 n equaton (.3), we obtan the mnmum MSE as MSE III. mn ( ) A S, = = A S, =,3,,5, 6 Effcenc Comparsons (.) (.5) (.6) In ths secton effcences of the proposed estmators are compared wth effcences of some estmators n the lterature The faml of estmators of the populaton varance s more effcent than η 0 f, η faml f, ( ) ( η ) MSEmn < Var 0 =,,3,,5,6 A < γ ( 0 ) ψ = A γ ( ψ0 ),3,,5, 6 < = (3.) The faml of estmators of the populaton varance s more effcent than Notes 06 Global Journals Inc. (US)

6 Improved Class of Rato Estmators for Fnte Populaton Varance Ref The A ( ) ( η) MSEmn < MSE =,,3,,5,6 < γ ( 0 ) ( 0 ) ( ) ψ + ψ ψ = A γ ( ψ0 ) h ( ψ0 ) h( ψ ),3,,5, 6 < + = faml of estmators of the populaton varance s more effcent than η f, (3.) 3. J. Subraman, G. Kumarapandan, Varance estmaton usng quartles and ther functons of an aular varable, Internatonal Journal of Statstcs and Applcatons (5) (0) ( ) ( η) MSEmn < MSEmn =,,3,,5, 6 A < γ ( 0 ) ( 0 ) ψ ψ = A γ ( ψ0 ) ( ψ0 ),3,,5, 6 < = (3.3) When condtons (3.), (3.) and (3.3) are satsfed, we can conclude that the faml of proposed class of estmators s more effcent than the η faml estmator. IV. Emprcal Stud In order to nvestgate the merts of the proposed estmators, we have consdered the followng two real populatons gven as: Data : Subraman and Kumarapandan [3] N = 9, n= 0, Y = 6.633, X = , ρ = 0.690, S = S = , C = , C =.035, ψ =.95, ψ = , ψ =.6977 Data : Subraman and Kumarapandan [3] 0 0 N = 80, n= 0, Y = 5.86, X =.66, ρ = 0.93, S = S = 8.563, C = 0.35, C = , ψ =.667, ψ =.866, ψ = The numercal demonstraton to justf the approprateness of the suggested class of varance estmator has been conducted usng the two data sets. Table and Table 3 show the mean square errors of proposed estmators and that of some estng estmators and ther percentage relatve effcences of dfferent estmators wth respect to s respectvel. V. Concluson From secton 3, the theoretcal condtons obtaned for the effcences of the proposed estmators supported b the numercal llustraton n secton gven n the Table and 3. From the result, we nfer that the suggested varance estmators produce a better estmate of fnte populaton varance than the estng estmators n the sense of havng hgher percentage relatve effcenc whch mples lesser mean square error. Global Journal of Scence Fronter Research ( F ) Volume XVI Issue II V erson I Year Global Journals Inc. (US)

7 Improved Class of Rato Estmators for Fnte Populaton Varance Global Journal of Scence Fronter Research ( F ) Volume XVI Issue II V erson I Year 06 References Références Referencas. A. A. Adewara, R. Sngh, M. Kumar, Effcenc of some modfed rato and product estmators usng known value of some populaton parameters, Internatonal Journal of Appled Scence and Technolog () (0) M.S. Ahmed, W.A. Daeh, A.A.O. Hurarah, Some estmators for fnte populaton varance under two-phase samplng, Statstcs n Transton 6 () (003) S. Gupta, J. Shabbr, Varance estmaton n smple random samplng usng aular nformaton, Hacettepe Journal of Mathematcs and Statstcs 37 (008) Notes. C.T. Isak, Varance estmaton usng aular nformaton, Journal of the Amercan Statstcal Assocaton 78 (983) C. Kadlar, H. Cng, Rato estmators for populaton varance n smple and stratfed samplng, Appled Mathematcs and Computaton 73 (006) C. Kadlar, H. Cng, Improvement n varance estmaton n smple random samplng, Communcatons n Statstcs Theor and Methods 36 (007) A. Sanaullah, H. Khan, A. Al, R. Sngh, Improved rato-tpe estmators n surve samplng, Journal of Relablt and Statstcal Studes 5 () (0) R. Sngh, P. Chauhan, N. Sawan, F. Smarandache, A general faml of estmators for estmatng populaton varance usng known value of some populaton parameters, Far East Journal of Theoretcal Statstcs (007). 9. R. Sngh, P. Chauhan, N. Sawan, F. Smarandache, Improved eponental estmator for populaton varance usng two aular varables, Italan Journal of Pure and Appled Mathematcs 8 (0) H. P. Sngh, R. S. Solank, A new procedure for varance estmaton n smple random samplng usng aular nformaton, Statstcal Papers 5() (03) H. P. Sngh, G. Vshwakarma, Some famles of estmators of varance of stratfed random sample mean usng aular nformaton, Journal of Statstcal Theor and Practce () (008) 3.. R. S. Solank, H. P. Sngh, An mproved class of estmators for the populaton varance, Model Asssted Statstcs and Applcatons 8(3) (03) J. Subraman, G. Kumarapandan, Varance estmaton usng quartles and ther functons of an aular varable, Internatonal Journal of Statstcs and Applcatons (5) (0) L.N. Upadhaa, H. P. Sngh, An estmator for populaton varance that utlzes the kurtoss of an aular varable n sample surves, Vkram Mathematcal Journal, 9 (999) S. K. Yadav, C. Kadlar, Improved eponental tpe rato estmator of populaton varance, Revsta Colombana de Estadístca 36() (03) S. K. Yadav, C. Kadlar, Improved class of rato and product estmators, Appled Mathematcs and Computaton 9 (03) Global Journals Inc. (US)

8 Improved Class of Rato Estmators for Fnte Populaton Varance Table : Some Member of η -faml for dfferent values of a, b andα Ref. L.N. Upadhaa, H. P. Sngh, An estmator for populaton varance that utlzes the kurtoss of an aular varable n sample surves, Vkram Mathematcal Journal, 9 (999) -7. Estmator a b α η 0 = s Sample varance S η = s s Isak [] S C η = s s C Kadlar and Cng [5] S β( ) η3 = s s β( ) Sβ( ) C η = s sβ( ) C SC β( ) η5 = s sc β( ) S + β( ) η6 = s s + β( ) Upadhaa and Sngh [] β ( ) C β ( ) C C β ( ) Table : MSE of dfferent estmators β ( ) Estmator Data Data Estmator Data Data η η faml (Sngh et al. [7] estmators) η η η η η η η opt faml (Newl proposed estmators) Global Journal of Scence Fronter Research ( F ) Volume XVI Issue II V erson I Year Global Journals Inc. (US)

9 Improved Class of Rato Estmators for Fnte Populaton Varance Table 3 : PRE of dfferent estmators wth respect to s Estmator Data Data Estmator Data Data Global Journal of Scence Fronter Research ( F ) Volume XVI Issue II V erson I Year 06 η η faml (Sngh et al. [7] estmators) η η η η η η η opt faml (Newl proposed estmators) Notes 06 Global Journals Inc. (US)