Exponential Estimators for Population Mean Using the Transformed Auxiliary Variables

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1 Appl. Math. Inf. Sci. 9, No. 4, (015) 107 Applied Mathematics & Information Sciences An International Journal Exponential Estimators for Population Mean Usin the Transformed Auxiliary Variables Rahana Rashid 1, Muhammad Noor ul Amin, Muhammad Hanif 1 1 National Collee of Business Administration Economics, Lahore, Pakistan. COMSATS Institute of Information Technoloy, Lahore, Pakistan. Received: 0 Nov. 014, Revised: 0 Feb. 015, Accepted: Feb. 015 Published online: 1 Jul. 015 Abstract: This paper deals with exponential ratio-cum-ratio product-cum-product type estimators usin transformed auxiliary variables under simple rom samplin without replacement. The proposed estimators are useful for estimatin the finite population mean. The development of estimators is based on the information of two transformed auxiliary variables. The eneralized form of the proposed estimator has been developed the special cases are discussed. The bias mean square error expressions of the proposed estimators are derived up to the first order of approximation. An empirical study has been carried out to compare the efficiency of proposed estimators with some available estimators in literature. An improvement has been reflected in terms of mean square error (MSE). Keywords: Transformed auxiliary variable, simple rom samplin, exponential estimators, mean square error, bias. 1 Introduction The role of auxiliary information is of prime importance in samplin theory. In survey samplin auxiliary variables are commonly used in order to obtain improved desins to achieve hiher precision in the estimates of some population parameters such as population total, population mean, population proportion, population ratio, etc. In case when the relation between auxiliary study variable is positive then ratio estimation is often suested. However, product method of estimation is usually considered, when the correlation coefficient between study auxiliary variable is neative. In most of the survey situations the auxiliary information is always available. It may either be promptly available or may be collected without much difficulty by avertin a part of survey resources. Laplace [7] made use of auxiliary information for the population census of France. The work of Neyman [8] may be referred to as the initial work where the use of auxiliary information has been established. The development is continue by usin the exponential form of the ratio product type estimators such as Bahl Tuteja [], Noor ul amin Hanif [9] Sanaullah et al. [1] etc. Consider the finite population S = {x 1,x,...,x N } of size N. Let Y be the study variable with population mean Ȳ while X Z are auxiliary variables with population mean X Z, respectively. The sample means for variables Y, X, Z are denoted byȳ, x, z, respectively. The sample are drawn by the simple rom samplin without replacement (SRSWOR) of size n (n<n). We also have followin assumptions: e y = y Y Y, e x = x X X,etc, E(e y )=E(e x )=0,E(ey )=θc y, E(e x )=θc x E(e xe y )=θρ xy C y C x, E(e z e y )=θρ yz C y C z θ = 1 n 1 N, C i = Si i,k i j = ρ i j C i C j where i=x,y,z, j = x,y,z i j (1) These notations are similar for the other auxiliary variables. Correspondin author nooramin.stats@mail.com c 015 NSP Natural Sciences Publishin Cor.

2 108 R. Rashid et. al. : Exponential Estimators for Population Mean Cochran [4] Robson [11] proposed classical ratio product estimators for estimatin the population mean as t 1 = y X x, () t = y z Z, (3) respectively, the mean square error equations of the estimators t 1 t are MSE(t 1 ) Y θ[c y +C x(1 K yx )], (4) MSE(t ) Y θ[c y +C z(1+k yz )], (5) respectively. Srivenkatramana [13] suested dual to ratio product estimators t 3 t 4 by applyin a transformation x = NX nx on auxiliary variables X Z as t 3 = y x X, (6) t 4 = y Z z, (7) where x = (1 + )X x, z = (1 + )Z z = n Mean square error of t 3 t 4 up to the first order of approximation is MSE(t 3 ) Y θ[c y + C x( K yx )], (8) MSE(t 4 ) Y θ[c y + C x (+K yz)], (9) respectively. Bahl Tuteja [] proposed the followin exponential ratio product type estimators. The exponential estimators are preferable on classical ratio product estimators, when the linear relationship between study variable auxiliary variable is not very stron. t 5 = yexp[ X x ], (10) X+ x t 6 = yexp[ z Z ], (11) Z+ z Mean square error of t 5 t 6 up to the first order of approximation is MSE(t 5 ) Y θ[cy + C x 4 K yxcx ], (1) MSE(t 6 ) Y θ[cy + C z 4 + K yz ], (13) respectively, the exponential dual to ratio type estimator proposed by Sharma Tailor [14] as t 7 = yexp[ x X x ], (14) + X MSE(t 7 ) Y θ[c y + C x( 4 K yx)], (15) the exponential dual to product type estimator is Z z t 8 = yexp[ Z+ z ], (16) MSE(t 8 ) Y θ[c y + C z( 4 + K yz)]. (17) Proposed Estimators Let x = NX nx z = NZ nz then x = (1+)X x z = (1+)Z z, where = n. Usin the transformation x z we propose exponential ratio-cum-ratio exponential product-cum-product estimators in eneralized form, iven by t 9G = yexp[a( x X Z x + X )+b(z z )], (18) + Z X x z t 10G = yexp[c( X+ x )+d(z Z+ z )], (19) respectively, where a, b, c d are positive real numbers. In order to obtain the bias mean square error of the proposed estimators, usin the notations (1), the proposed exponential ratio-cum-ratio type estimator may be written as t 9G = Y(1+e y )exp[ ae x (1 e x ) 1 be z (1 e z ) 1 ], (0) Openin the exponential function in (0) inorin the terms with power three or reater, we et t 9G Y[e y ae x + a e x be z 8 + b e z 8 + abe x e z 4 ae xe y be ze y ], (1) c 015 NSP Natural Sciences Publishin Cor.

3 Appl. Math. Inf. Sci. 9, No. 4, (015) / Takin expectation on both sides of (1) after simplifications, the bias of t 9G is obtained as Bias(t 9G ) Y θ 8 [ (C z(ab + b )+a C x) 4(aK yx C x + bk yz C z)], () For the derivation of mean square error of t 9G, we aain use (0), inorin the terms with power two or hiher, we et t 9G = Y(1+e y )exp[ ae x be z ], (3) Expin exponential terms, inorin hiher order terms, we et t 9G Y Y[e y ae x be z ], (4) Takin square expectations on the both sides of (4), we et c= C y(ρ yx ρ yz ρ zx ) C x (1 ρ xz) d= C y(ρ yz ρ yx ρ xz ) C z (1 ρ xz) respectively, after substitutin the values of c d in mean square error of t 10G, the minimum MSE of t 10G is denoted by (t 10G ) min, is iven by MSE(t 10G ) min = Y θc y [1+ 3(ρyx + ρ yz ρ yxρ xz ρ yz ) ]. (1 ρ xz).1 Special Cases of Proposed Estimators (9) The several estimators may be obtained for the different values of a,b,c d. 1. For a=1 b=1, the followin exponential ratiocum-ratio type estimator may be obtained from proposed estimator (t 9G ). t 9 = yexp[ x X x + X + z Z z ], (30) + Z MSE(t 9G ) Y θ[c y + 4 (a C x +bc z(b+a ) (ak yx C x + bk yz C z))]. (5). For c=1 d=1, the followin exponential product-cum-product type estimator may be obtained from proposed estimator (t 10G ). Partially differentiatin (5) w.r.t. a b, equatin to zero, the optimal values of a b are obtained as a= C y(ρ yx ρ yz ρ zx ) C x (1 ρxz ) b= C y(ρ yz ρ yx ρ zx ) C z (1 ρxz ) respectively, after substitutin the values of a b in mean square error of t 9G the minimum MSE of t 9G is denoted by (t 9G ) min, is iven by MSE(t 9G ) min = Y θc y [1 (ρ yx+ ρ yz ρ yx ρ xz ρ yz ) (1 ρ xz ) ]. (6) Likewise, the mean square error bias of t 10G may be obtained as MSE(t 10G ) Y θ[c y + 4 (c C x + dc z (d+ c)+(ck yx C x + dk yzc z ))]. (7) Bias(t 10G ) Yθ 8 [ (C z(cd + d ) +c C x) 4(cK yx C x + dk yz C z)], (8) Partially differentiatin (7) w.r.t. c d, equatin to zero, the optimal values of c d are obtained as X x Z z t 10 = yexp[ + X+ x Z+ z ], (31) Note that various estimators may be obtained in the family of ratio-cum-ratio type estimator product-cum-product type estimator for the different values of a,b c,d respectively. 3 Efficiency Comparison This section provides theoretical comparison between some existin estimators of population mean the proposed estimator with respect to mean square error. The comparison of exponential ratio-cum-ratio type estimator is made with y,t 1,t 3,t 5 t 7 exponential product-cum-product estimator is made with y,t,t 4,t 6 t 8. In simple rom samplin without replacement the variance of sample mean y is V(y)=θS y. Reddy [10] has proved that in repetitive surveys is stable so, we obtained the conditions (3) - (41) in terms of. Grover et al. [5] has noticed that has rane (, 0) when both variables have positive values neative correlation coefficient(ρ xz ). Similarly, has rane(0, ) for positive correlation coefficient(ρ xz ). Keepin this in mind we can define the preference reion of proposed estimators MSE(t 9 )< V(y)if ε (0,( K yz 1 )+ C x ( K yz 1 )) (3) c 015 NSP Natural Sciences Publishin Cor.

4 110 R. Rashid et. al. : Exponential Estimators for Population Mean MSE(t 9 )<MSE(t 1 ) if ( K yz MSE(t 9 )<MSE(t 3 ) if ε (0,( K yz 1 )+ C x C z 1 )+ C x C z (1 k yx)) (33) MSE(t 10 )<MSE(t 8 ) if ε ( C x C z ( 1 K yx ),0) (41) When the condition (3) to (41) is fulfilled, the suested ratio-cum-ratio product-cum- product exponential estimator will be more competent in comparison with available estimators. MSE(t 9 )<MSE(t 5 ) if ε (0,( 1 + K yz ) C x ( 1 1 C )+ x K yx (1 1 (34) )) MSE(t 9 )<MSE(t 7 ) if C z ε (0,( 1 + K yz ) C x K yx Cx (1 (35) )) MSE(t 10 )< V(y) if MSE(t 10 )<MSE(t ) if MSE(t 10 )<MSE(t 4 ) if ε (0,( 1 + K yz )) (36) ε (( 1 + K yz ) Cx ( 1 + K yx ),0)) ε (( 1 + K yz ) C x ( 1 + k yx ) (K yx 1),0)) ε ( 1 (1 1 ) K yz (1 1 ) Cx ( 1 + K yx ),0) (37) (38) (39) 4 Empirical Study This section includes empirical study for suested estimators. This study has been performed usin three real populations (see appendix for parameters). The description of the populations is iven by Population I [Ahmed [1]] y : No of literate persons x : No of cultivators z : Total population Population II [Population census report of Multan district in 1998, Pakistan] y : Population matric above x : Primary but below matric z : Population of both sexes Population III [Gujarati [6]] y : Averae miles per allon x : Top speed, miles per z : Cubic feet of Cab space The optimum values for the constants, MSE relative efficiencies are iven in Table The comparison of each estimator has been done with mean per unit estimator. Table 4.1: Optimum Values of the Constants Population a b c d * * * * *Data is not applicable For the evaluation, the classical ratio product estimator, dual to classical ratio-type estimator proposed by Srivenkatramana [13], dual to classical product-type estimator by Byopadhyay [3], exponential ratio-type estimators product-type estimators proposed by Bahl-Tuteja [], exponential dual to ratio type estimator by Sharma Tailor [14] Bahl-Tuteja [] usin transformed auxiliary variables have been considered. MSE(t 10 )<MSE(t 6 ) if ε (( C x C z K yz ),0) (40) c 015 NSP Natural Sciences Publishin Cor.

5 Appl. Math. Inf. Sci. 9, No. 4, (015) / Table 4.: Mean Square error (MSE) for the Estimators Estimators 1 3 y t * t * t * t * t * t 9G * t * * 1.85 t 4 * * 1.57 t 6 * * 1.61 t 8 * * 1.59 t 10 * * 1.6 t 10G * * 0.66 The percentae relative efficiencies have been calculated by applyin the followin formula: PRE = VAR(y) 100 (4) MSE(t ) Table 4.3: Percentae Relative Efficiencies for the Estimators Estimators 1 3 y t * t * t * t * t * t 9G * t * * 98.7 t 4 * * t 6 * * t 8 * * t 10 * * t 10G * * Conclusion From Table 4.3, we can see that the suested exponential ratio-cum-ratio type estimator is competent as compare to classical ratio-type estimator, dual to classical ratio-type estimator proposed by Srivenkatramana [13], exponential ratio estimator introduced by Bahl Tuteja [] exponential dual to ratio-type estimator suested by Sharma Tailor [14]. Similarly we can see that the suested exponential product-cum-product type estimator is more efficient than classical product-type estimator, dual to classical product-type estimator proposed by Byopadhyay [3], exponential product estimator iven by Bahl Tuteja [] adapted estimator of Bahl Tuteja [] usin transformed auxiliary variables. Accordin to the results of population 1- we may conclude that suested eneralized exponential ratio-cum-ratio type estimator is efficient on comparison with all considered estimators. Further for population 3, proposed eneralized form of exponential product-cum-product type estimator has shown improved performance as compare to all considered product type estimators. Acknowledement The authors are rateful to the anonymous referee for a careful checkin of the details for helpful comments that improved this paper. Appendix: Table A: Parameters of Populations Population Parameter 1 3 N n Y X Z C y C x C z ρ yx ρ yz ρ xz References [1] Ahmed, M.S. The General Class of Chain Estimators for the Ratio of two Means usin Doublin Samplin. Comm. Statist. Theory Meth. 6(9), (1997). [] Bahl, S. Tuteja, R.K. Ratio product type exponential estimator. Information Optimization Sciences, 1, (1991). [3] Bapdyopadhyaya, S. Improved ratio product estimators. Sankhya, 4(C),45-49 (1980). [4] Cochran, W.G. Samplin theory when the samplin units are of unequal sizes. J. Amer. Statist. Assoc., 37, (194). [5] Grover, L.K., Kaur.P. Vishwakarmma, G.K. Product type exponential estimators of population mean under linear transformation of auxiliary variable in simple rom samplin.applied Mathematics Computations,19, (01). [6] Gujarati, D.N. Basic Econometrics. Tata McGraw Hill.(004). [7] Laplace, P.S. A philosophical essay on probabilities. Enlish translation, Dover, 1951,(180). [8] Neyman, J. Contribution to the theory of samplin human population. J. Amer. Statist. Assoc., 33, (1938). c 015 NSP Natural Sciences Publishin Cor.

6 11 R. Rashid et. al. : Exponential Estimators for Population Mean [9] Noor-ul-Amin, M. Hanif, M. Some exponential estimators in survey samplin. Pak. J. Statist. 8(3), (01). [10] Reddy V.N. A study on the use of prior knowlede on certain population parameters in estimation, Sankhya Series C 40, 9-37 (1978). [11] Robson, D.t. Application of multivariate polykays to the theory of unbiased ratio-type estimation. J. Amer. Statist. Assoc., 5, (1957). [1] Sanaullah, A.,Ali H.A., Noor ul Amin, M., Hanif, M. Generalized exponential chain ratio estimators under stratified two-phase rem samplin. Applied Mathematics Computation, 6, (014). [13] Srivenkatramana, T. A dual to ratio estimator in sample surveys. Biometrika, 67, (1980). [14] Sharma, B. Tailor, R., A new ratio-cum-dual to ratio estimator of finite population mean in simple rom samplin. Global J. of Sci. Frontier Research, 10(1), 7-31(010). Muhammad Hanif completed his master deree from New South Wales University, Australia in Multistae Cluster Samplin. He completed his Ph.D. in Statistics from University of Punjab, Lahore, Pakistan. He has more than 40 years research experience. He is an author of more than 00 researh papers 10 books. He has served as professor in various part of the world i.e. Australia, Libya, Pakistan Saudi Arabia. He is presently a Professor of statistics Vice Rector (Research) in NCBA E, Lahore, Pakistan. Rahana Rashid is the student of PhD at National Collee of Business Administration Economics (NCBA E), Lahore, Pakistan. She has completed her Master in Statistics from NCBA E. Her research area is survey samplin. Muhammad Noor-ul-Amin received his PhD deree from NCBA E, Lahore, Pakistan. He is currently workin as Assistant Professor at COMSATS Institute of Information Technoloy. His research interests include Two-phase samplin desins adaptive cluster samplin desins. c 015 NSP Natural Sciences Publishin Cor.