STATISTIS I TASITIO-new series December 0 587 STATISTIS I TASITIO-new series December 0 Vol. o. 3 pp. 587 594 A GEEAL FAMILY OF DUAL TO ATIO-UM- ODUT ESTIMATO I SAMLE SUVEYS ajesh Singh Mukesh Kumar ankaj hauhan irmala Sawan Florentin Smarandache 3 ABSTAT This paper presents a famil of dual to ratio-cum-product estimators for the finite population mean. Under simple random sampling without replacement (SSWO) scheme epressions of the bias and mean-squared error (MSE) up to the first order of approimation are derived. We show that the proposed famil is more efficient than usual unbiased estimator ratio estimator product estimator Singh estimator (967) Srivenkataramana (980) and Bandopadhaa estimator (980) and Singh et al. (005) estimator. An empirical stud is carried out to illustrate the performance of the constructed estimator over others. Ke words: Famil of estimators auiliar variables bias mean-squared error.. Introduction It is common practice to use the auiliar variable for improving the precision of the estimate of a parameter. Out of man ratio and product methods of estimation are good eamples in this contet. When the correlation between the stud variate and auiliar variates is positive (high) ratio method of estimation is quite effective. On the other hand when this correlation is negative (high) product method of estimation can be emploed effectivel. Let U be a finite population consisting of units U U U. Let and ( ) denote the stud variate and auiliar variates taking the values i and ( i i ) respectivel on the unit U i ( ) where is positivel correlated with and is negativel correlated with. We wish to estimate the population mean Y = i of Department of Statistics BHU Varanasi(U..) India (rsinghstat@ahoo.com). School of Statistics DAVV Indore (M..) India. 3 Department of Mathematics Universit of ew Meico Gallup USA (smarand@unm.edu).
588. Singh M. Kumar. hauhan. Sawan F. Smarandache: A general X of ( ) are known. Assume that a simple random sample of sie n is drawn without replacement from U. The classical ratio and product estimators for estimating Y are: = X (.) and = (.) Z n n n respectivel where = i = i and = i are the sample n n n means of and respectivel. assuming that the population means ( Z) ( X n ) i (Z ni ) Using the transformation i = and i = ( ) ( n) ( n) Srivenkataramana (980) and Bandopadhaa (980) suggested a dual to ratio and product estimator as: = X (.3) and Z = (.4) X n where Z n = and =. ( n) ( n) In some surve situations information on a secondar auiliar variate correlated negativel with the stud variate is readil available. Let Z be the known population mean of. For estimating Y Singh (967) considered a ratiocum-product estimator X = (.5) Z Using a simple transformation i ( ) i with = + g X g and i = ( + g)z gi g = n Singh et al. (005) proposed a dual to usual ratiocum-product estimator ( n)
STATISTIS I TASITIO-new series December 0 589 Z = (.6) X where = ( + g)x g and = ( + g)z g. To the first degree of approimation ( ) θ V = Y (.7) MSE = θy + K (.8) ( ) [ ( ( ) θy [ + ( K ( ) θy [ + ( + K ) + ( K) ] ( ) θy [ + g ( g K ( ) θy [ + g ( g K ( ) θy [ + g ( g + K ) + g ( g gk K MSE = + (.9) MSE MSE = (.0) = (.) MSE = + (.) MSE = (.3) where MSE (.) stands for mean square error (MSE) of (.). f n S θ = f = n = Y S S = X = Z K = ρ K = ρ S S S K = ρ K = K + K ρ = ρ = ρ = S S S S S S S ( i X) = (i Z) ( ) ( ) ( ) = (i Y)( i X) S = (i Y)(i Z) ( ) ( ) S = ( i X)( i Z). ( ) S = (i Y) = S and S In this paper under SSWO we present a famil of dual to ratio-cumproduct estimator for estimating the population mean Y. We obtain the first order approimation of the bias and the MSE for this famil of estimators. umerical illustrations are given to show the performance of the constructed estimator over other estimators.
590. Singh M. Kumar. hauhan. Sawan F. Smarandache: A general. The suggested famil of estimators We define a famil of estimators of Y as Z t = X (.) where i s () are unknown constants to be suitabl determined. To obtain the bias and MSE of t we write = Y( + e 0 ) X( + e ) = Z + e such that E (e o )= E (e )= E (e )= 0 and E e 0 = θ E( e ) = θ E( e ) = θ E ( ) = ( ) 0e θρ ( e0e ) = θρ E( e ) = E( ee ) θρ =. Epressing t in terms of e s we have t = Y( + e0 )( ge) ( ge ) (.) We assume that ge < ge < so that ( ge ) and ( ge ) are epandable. Epanding the right hand side of (.) and retaining terms up to second powers of e s (up to the first order of approimation) we have ( ) t = Y[ + e0 ge ge0e + g e + ge + ge0e ( ) g ee + g e ] (.3) Taking epectations of both sides in (.3) and then subtracting Y from both sides we get the bias of the estimator t up to the first order of approimation as B(t) = E(t Y) ( ) Yg [( K K ) = θ + g where K = ρ. ( ) g { K }] (.4)
STATISTIS I TASITIO-new series December 0 59 From (.3) we have Y Y e g( e e ) (.5) ( ) [ ] t 0 Squaring both sides of (.5) we have ( t Y) = Y [e0 + g { e + e ee} g( e0e e0e (.6) Taking epectations of both sides of (.6) we get the MSE of t to the first degree of approimation as MSE(t) = θy [ + g (g + K ) + g(g gk K (.7) as Minimiation of (.7) with respect to and ields their optimum values K K = g K ( ρ ) (K K K = g( ρ ) ) (.8) Substitution of (.8) in (.7) ields the minimum value of MSE (t) as min.mse(t) = θy ( ρ ) (.9). ρ + ρ ρ ρ ρ where ρ. = is the multiple correlation coefficient of ( ρ ) on and. emark.: For ( )=(0) the estimator t reduces to the dual to ratio estimator = X While for ) =(0) it reduces to the dual to product estimator ( Z = It coincides with the estimator in Singh et al. (005) when ) (
59. Singh M. Kumar. hauhan. Sawan F. Smarandache: A general emark.: The optimum values of io s () are functions of unknown population parameters such as K K K. The values of these unknown population parameters can be guessed quite accuratel from the past data or eperiences gathered in due course of time for instance see Srivastava (967) edd (973) rasad (989p.39) Murth (967pp96-99) Singh et. al. (007) and Singh et. al. (009). Also the prior values of K K K ma be either obtained on the basis of the information from the most recent surve or b conducting a pilot surve see Lui (990p.3805). From (.7) to (.3) and (.9) it can be shown that the proposed estimator t at (.) is more efficient than usual unbiased estimator usual ratio estimator product estimator Singh (967) ratio-cum product estimator Srivenkataramana (980) and Bandopadhaa (980) dual to ratio estimator dual to product estimator and Singh et al. (005) dual to ratio-cumproduct estimator at its optimum conditions. 3. Empirical stud In this section we illustrate the performance of the constructed estimator t over various other estimators through natural data earlier used b Singh (969 p.377). : number of females emploed : number of females in service : number of educated females Y = 7.46 X = 5. 3 Z = 79. 00 = 0. 5046 = 0. 5737 = 0. 0633 ρ = 0.7737 ρ = 0. 070 ρ = 0.0033 =6 and n=0. Table 3.: ange of and for which t is better than 0.94.5.39( (opt) ).5.7.85.7 00.8 04.67 6.46 8.5 7.4 09.7 00.6 3. 0.0 05.0 6.8 8.9 7.8 0.3 00.9 3.34( (opt) ) 0.4 05.08 6.96 9.03 7.8 0.9 0.0 3.4 0. 05.08 6.95 9.0 7.8 0.9 0.04 3.8 0.0 04.8 6.7 8.76 7.6 09.9 00.8 4.5 00.04 03.8 5.38 7.38 6. 08.7 99.8 5.0 98.83 0.5 3.77 5.7 4.6 07.3 98.6
STATISTIS I TASITIO-new series December 0 593 The percent relative efficiencies (E s) of the different estimators with respect to the proposed estimator t are computed b the formula MSE(.) E (t.) = 00 MSE(t) and presented in table 3.. Table 3.: E s of various estimators of Y with respect to Estimator E 00 03.43 3.78 3.64 4.78 04.35 35.68 t opt 78. 4. onclusion (i) Table 3. shows that there is a wide scope of choosing and for which our proposed estimator t performs better than. (ii) Table 3. shows clearl that the proposed estimator t is more efficient than all other estimators and with considerable gain in efficienc. I this paper we have presented a famil of dual to ratio-cum-product estimators for the finite population mean. For future research the famil suggested here can be adapted to double sampling scheme using Kumar and Bahl (006) estimator.
594. Singh M. Kumar. hauhan. Sawan F. Smarandache: A general EFEEES BADYOADHYAYA S. (980): Improved ratio and product estimators Sankha Series 445-49. KUMA M. BAHL S. 006 lass of dual to ratio estimators for double sampling Statistical apers 47 39-36. LUI K. J. (990): Modified product estimators of finite population mean in finite sampling. ommunications in Statistics Theor and Methods 9(0) 3799-3807. MUTHY M.. (967): Sampling Theor and Methods. Statistical ublishing Societ alcutta India. ASAD B. (989): Some improved ratio tpe estimators of population mean and ratio in finite population sample surves. ommunications in Statistics Theor and Methods 8 379-39. EDDY V.. (973): On ratio and product methods of estimation. Sankha Series B 35(3) 307-36. SIGH H.. SIGH. ESEJO M.. IED D.M. ADAAJAH S. (005): On the efficienc of a dual to ratio-cum-product estimator in sample surves. Mathematical roceedings of the oal Irish Academ 05 A() 5-56. SIGH M.. (967): atio-cum-product method of estimation. Metrika 34-7. SIGH M.. (969): omparison of some ratio-cum-product estimators. Sankha Series B 3 375-378. SIGH. AUHA. SAWA. and SMAADAHE F. (007): Auiliar Information and a rior Values in onstruction of Improved Estimators. enaissance High ress. SIGH. SIGH J. and SMAADAHE F. (009): Some Studies in Statistical Inference Sampling Techniques and Demograph. roquest Information & Learning. SIVASTAVA S. K. (967): An estimator using auiliar information. al. Stat. Assoc. Bull. 6-3. SIVEKATAAMAA T (980): A dual to ratio estimator in sample surves. Biometrika 67 99-04.