Regression-Cum-Exponential Ratio Type Estimators for the Population Mean

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1 Middle-East Journal of Scientific Research 19 (1: , 014 ISSN IDOSI Publications, 014 DOI: /idosi.mejsr Regression-um-Eponential Ratio Tpe Estimats f the Population Mean 1 Naureen Ria, Muhammad No-ul-Amin and 3 Muhammad Hanif 1,3 National ollege of Business Administration and Economics, Lahe, Pakistan OMSATS Institute of Infmation and Technolog, Lahe, Pakistan Abstract: This paper proposes regression-cum-eponential ratio tpe estimats under simple random sampling without replacement f estimating the finite population mean of stud variable b using the infmation of two auiliar variables. The mean square err and bias epressions have been derived. The theetical comparison of proposed estimats has been made with eisting estimats. Empirical stud is conducted to evaluate the efficienc of proposed estimats. Ke wds: Regression estimat auiliar variable eponential estimat mean square err bias INTRODUTION The use of auiliar infmation has been studied b various auths in various fms to improve the efficienc of their constructed estimat. In the use of auiliar variables, ratio, product and regression estimats are cner stone s in the estimation of population characteristics. Auiliar infmation has alwas been seems effective in increasing the precision of estimates in surve sampling. Grant [7] was the first who estimated the population of England based on auiliar infmation. The wk of Neman [13] ma be referred as an initial wk where auiliar infmation has been discussed in detail. ochran [3] suggested the classical ratio tpe estimat f the population mean and ochran [4] used auiliar infmation in regression estimats. Robson [14] and Murth [11] investigated; if the crelation is negative the product method of estimation is quite effective. Mohant [10] used two auiliar variables b combining the regression and ratio method. Bahl and Tuteja [] were the first, who used eponential tpe estimats f estimating the population mean. Samiuddin and Hanif [15] introduced ratio and regression estimation procedure b using two auiliar variables and the have provided modification of Mohant [10] estimat. Hanif et al. [9] proposed an estimat which was the modification of Singh and Espejo [17] estimat. The development is continued in the fm of eponential estimats f different situations b man auths such as Singh and Vishwakarma [18], No ul amin and Hanif [1] and Sanaullah et al. [16]. responding Auth: Naureen Ria, National ollege of Business Administration and Economics, Lahe, Pakistan e NOTATIONS AND VARIOUS EXISTING ESTIMATORS Y X Z =, e =, e = Y X Z Ε e = Ε e = Ε e = 0, Ε e = θ, Ε e = θ Ε ee =θρ, Ε ee =θρ ( S S θ=, =, ρ =,H =ρ n N Y S S i ij ij j ochran [3] and Robson [14] suggested the classical ratio and product estimats, respectivel, f estimating the population mean as: X t1 = = t Z (. (.3 The mean square equations (MSE of the estimats of t 1 and t are ( 1 θ + ( Y 1 H θ + ( + Y 1 H (.4 (.5 ochran [4] suggested the classical regression estimat given b

2 Middle-East J. Sci. Res., 19 (1: , 014 t = 3 b X r + (.6 The mean square err of t 3 is given b ( 3 ( = θy 1 ρ (.7 Bahl and Tuteja [] proposed the following eponential ratio and product tpe Estimats X = + t4 ep X Z = + t5 ep Z (.8 (.9 respectivel, the mean square err of the estimats t 4 and t 5 are given b Y 1 4H 4 4 θ + ( + Y 1 4H 4 5 θ + ( (.10 (.11 respectivel, the bias epressions of the estimats t 4 and t 5 are given b Yθ Bias(t 4 3 4H 8 (.1 Yθ Bias(t 5 4H 1 8 (.13 The modification of Mohant [10] have given b Samiuddin and Hanif [15] as Z t 7 = + k1( X (.17 The mean square err and bias epression of t 7 is given b 7 θy ( ρ + ( ρ ( 1 1 ρ ρ ρ Bias t =θy θyρ + b Xρ 7 (.18 Samiuddin and Hanif [15] were using the idea of hand [6] and suggested the estimat given b XZ t8 = (.19 The mean square err and bias epression of t 8 is given b 8 θy ρ + ρ + + ρ ( 8 ( (.0 Biast θ Y + + ρ ρ ρ (.1 respectivel. Hanif et al. [9] proposed an estimat which was the modification of the Singh and Espejo [17] estimat, given as Mohant [10] suggested the following the Regression-um-Ratio Estimat f estimating the finite population mean Z t6 = + b X (.14 The mean square err and bias epressions f t 6 are given b 6 = θy 1 ρ + ρ + ρρ (.15 t = Y+ k X k + 1 k Z { } 9 1 Z (. The mean square err and bias epression of t 9 is given b where ( 9 (. =θy 1 ρ (.3 1 Z Bias ( t Y 4 Y 9 θ β. (.4 Bias t =θy θyρ + b Xρ (.16 respectivel ρ ρ = + ρ ρ ρ. 1 ρ S, β = ( ρ ρ ρ S( 1 ρ.

3 Middle-East J. Sci. Res., 19 (1: , 014 PROPOSED ESTIMATOR In this section, an estimat is developed b combining the concept of Bahl and Tuteja [] eponential tpe estimat and classical regression estimat. The generalied regression-ratio estimat in the eponential fm is given b Now putting the value of (3.6 and (3.7 in equation (3.4 and after some simplifications, we get the minimied mean square err as θy 1 γρ ρ G min 1 ρ γρ + γ ρ ρ ρ (3.8 X tg = +α( Z ep γ X+β ( 1 (3.1 In der to derive the bias of (3.1 we again use (3.3 and simplif as where, α, β are real positive constants and γ ma take the values-1 and 1. In der to obtained the bias and mean square err, rewriting (3.1 b using the notations (.1, we get ( t = Y1+ e +α Z Z1+ e X X1 ( + e ep γ X+β ( 1X1 ( + e G After some simplification, (3. is given b e β 1 tg = Y Y e Ze ep + α γ 1 + e β β 1 (3. (3.3 Epanding the eponential term up to first degree in (3.3, squaring and taking epectations, we ma get mean square err as MSE ( tg θ Y +θα Z +θγy β θzαyρ + θγyαzρ θγyρ β β (3.4 In der to get optimum value of a and ß we partiall differentiating equation (3.4 with respect to α and ß and equating to ero we get optimum value of α and ß e β 1 tg Y + Ye αze ep γ +γ e β β (3.9 Epanding the eponential function up to second degree in (3.9 and after some simplification, we get bias as: θγy Bias ( t G αγθzρ ( β 1 βh β β (3.10 Special cases of proposed estimats: F the γ = 1 and γ = -1, the following regression-cum-eponential ratio tpe and regression-cum-eponential product tpe estimat in generalied fm ma be obtained from proposed estimat and X = +α( tgr Z ep X +β 1 X = +α( + ( β Z ep X 1 respectivel. F the α = b, β =, γ = 1 and α = b, β =, γ = -1, the regression-cum-eponential ratio tpe and regression-cum-eponential product tpe estimat ma be written as Y H α= H γ Z β ( 1 ρ γ β= H H H (3.5 (3.6 and X = + ( + b Z ep X X = + ( b Z ep X + Using (3.6 in equation (3.5, we get Y H H H α= Z γ( 1 ρ ( respectivel. F the α = b, β =, γ = 1 and α = b, β =, γ = -1, the regression-cum-eponential ratio tpe and regression-cum-eponential product tpe estimat ma be written as

4 and X = + ( b Z ep X + X = + ( b Z ep X + F the α = b, γ = 0, the proposed estimat ma take the fm of classical regression estimat. F the α = 0, β =, γ = 1, the proposed estimat ma take the fm of Bhal and Tuteja [1] eponential ratio tpe estimat. F the α = 0, β =, γ = -1, the proposed estimat ma take the fm of Bhal and Tuteja [1] eponential product tpe estimat. EFFIIENY OMPARISONS OF PROPOSED ESTIMATOR OVER OTHER ESTIMATORS In this section, the theetical conditions have been derived, when the deduced estimats perfms better as compare to some eisting estimats. The comparison of t GR and t GP is made with t 3, t 4, t 6, t 7, t 8 and t 3, t 5, t 9, respectivel, using the mean square errs. omparison of MSE (t GR min with classical regression estimat ( 1 ρ < MSE ( t 3 θy 1 ρ ρ ρ + ρ ρ ρ < θ Y(1 ρ. Middle-East J. Sci. Res., 19 (1: , 014 ρ >ρ (4.1 omparison of MSE (t GR min with Bahl and Tuteja s [] Estimat ( 1 ρ < MSE ( t 4 θy 1 ρ ρ ρ + ρ ρ ρ < Y θ H 4. 4 ( ρ > 1 4H (4. ( omparison of MSE (t GR min with Mohant [10] Estimat 1719 < MSE ( t ( 1 ρ 6 θy 1 ρ ρ ρ + ρ ρ ρ ρ >ρ + (4.3. H HH omparison of MSE (t GR min with Samiuddin and Hanif [15] Estimat ( 1 ρ < MSE ( t 7 θy 1 ρ ρ ρ + ρ ρ ρ ρ > ρ ρ ρ ρ ρ (4.4. ( 1 omparison of MSE (t GR min with modification of hand [6] Estimat, which was given Samiuddin and Hanif [15] θ < MSE ( t 8 Y 1 ρ ρ ρ + ρ ρ ρ ( 1 ρ < θ Y ( + + ρ ρ + ρ ρ > + +. H H ( 1 H omparison of t GP with lassical Product Estimat ( 1 ρ < MSE ( t 3 θy 1 ρ ρ ρ + ρ ρ ρ < Y θ + 1+ H. ( (4.5 ρ > 1 + H (4.6 omparison of t GP with Bahl and Tuteja [] Estimat

5 ( 1 ρ < MSE ( t 5 θy 1 ρ ρ ρ + ρ ρ ρ < Y θ + 1+ H 4. 4 ( ( ρ > 1 + 4H (4.7 omparison of t GP with Hanif et al. [9] Estimat < MSE ( t 9 θy 1 ρ ρ ρ + ρ ρ ρ ( 1 ρ <θy( 1 ρ. Middle-East J. Sci. Res., 19 (1: , 014 ρ. > 0 (4.8 If the conditions (4.1-(4.5 are fulfilled, the perfmance of t GP is better than the classical regression estimat, Bahl and Tuteja [] ratio, Mohant [10], Samiuddin and Hanif [15] and modification of hand [6] estimats, respectivel. Similarl, when the conditions (4.6-(4.8 are satisfied, the estimat, t GP, is me efficient than the classical product, Bahl and Tuteja [] product, Hanif et al. [9] estimats, respectivel. NUMERIAL EXAMPLE To show the perfmance of the proposed estimat in comparison of other estimats in single phase sampling, three iginal data set used b others auths in literature has been considered. The descriptions of the population are given below. Population 1: Anderson [1] Y: Head length of second son X: Head length of first son Z: Head breadth of first son N = 5, n 1 = 15, Y = , X = 185.7, Z = 151.1, = , = , = , ρ = 0.693, ρ = , ρ = Table 1: Percentage relative efficiencies f the estimats Populations Estimats t * t * t * t * t * t GR * t 3 * * t 5 * * t 9 * * t GP * * N = 34, n 1 = 15, Y = 4.9, X =.59, Z =.91, = 1.013, = , = , ρ = 0.736, ρ = 0.643, ρ = Population III: Gujrati [8] Y: The number of wild cats drilled. X: Price at the well head in the previous period. Z: Domestic output. N = 30, n 1 = 1, Y = , X = , Z = 7.548, = , = , = , ρ = , ρ = , ρ = The results of percent relative efficiencies have been given in Table 1. The percent relative efficiencies have been computed b using the fmula Population II: ochran [4] REFERENES Y: Number of placebo children X:Number of paraltic polio cases in the not 1. Anderson, T.W., An Introduction to inoculated group. Multivariate Statistical Analsis. John Wile & Z: Number of paraltic polio cases in the placebo group Sons, Inc., New Yk. 170 Var Y PRE = 100 The results from population 1- have shown that the generalied eponential regression-cum-eponential ratio tpe estimat (t GR is me efficient as compared to classical regression estimat, Mohant [10] estimat, eponential ratio-tpe estimat, Samiuddin and Hanif [15] estimat. The results from population 3 have shown that the generalied regression-cumeponential product tpe estimat (t GR is me efficient as compared to, classical regression tpe estimat, eponential product-tpe estimat and Hanif et al. [9] estimat. *

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