A Note on Generalized Exponential Type Estimator for Population Variance in Survey Sampling

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1 Revista Colombiana de Estadística July 015, Volume 38, Issue, pp 385 to 397 DOI: A Note on Generalized Exponential Type Estimator for Population Variance in urvey ampling Una nota sobre el estimador exponencial generalizado para la varianza poblacional en muestreo de encuestas Javid habbir 1,a, at Gupta,b 1 Department of tatistics, Quaid-i-Azam University, Islamabad, Pakistan Department of Mathematics and tatistics, The University of North Carolina at Greensboro, Greensboro, United tates Abstract Recently a new generalized estimator for population variance using information on the auxiliary variable has been introduced by Asghar, anaullah & Hanif 01 In that paper there was some inaccuracy in the bias and ME expressions In this paper, we provide the correct expressions for bias and ME of the Asghar et al 01 estimator, up to the first order of approximation We also propose a new generalized exponential type estimator for population variance which performs better than the existing estimators Four data sets are used for numerical comparison of efficiencies Key words: Auxiliary Variable, Bias, Efficiency, Mean quare Error, Variance Resumen Recientemente, un nuevo estimador generalizado de varianza de la población utilizando información sobre la variable auxiliar ha sido introducida por Asghar et al 01 En ese documento había alguna inexactitud en las expresiones de sesgo y ECM En este trabajo, proporcionamos las expresiones correctas de sesgo y ECM de Asghar et al 01 hasta el primer orden de aproximación También proponemos un nuevo estimador tipo exponencial generalizado de la varianza de la población que se comporta mejor que los estimadores existentes Cuatro conjuntos de datos se utilizan para la comparación numérica de la eficiencia Palabras clave: error cuadrático medio, sesgo, variable auxiliar, varianza, eficiencia a Professor javidshabbir@gmailcom b Professor sngupta@uncgedu 385

2 386 Javid habbir & at Gupta 1 Introduction In applications, the auxiliary information is frequently used to increase precision of the estimators by taking advantage of the correlation between the study variable and the auxiliary variable ingh, Upadhyaya & Namjoshi 1988 introduced several estimators for population variance using the known population mean and population variance of the auxiliary variable Isaki 1983 introduced the traditional ratio and regression estimator for population variance ome related work in this direction is also due to Jhajj, harma & Grover 005, Kadilar & Cingi 006, Gupta & habbir 008, Bansal, Javed & Khanna 011, ingh, Chauhan, wan & marandache 011, Upadhyaya, ingh, Chatterjee & Yadav 011, ubramani & Kumarapandiyan 01, Nayak & ahoo 01 and Yadav & Kadilar 013, 01 The following notations will be needed to discuss various estimators Consider a finite population U = {1,, i, N} of N identifiable units Let y i and x i i = 1,,, N be the values on the ith population unit for the study variable y and the auxiliary variable x respectively Let Y = 1 N N i=1 y i and X = 1 N N i=1 x i be the N i=1 y i Y and x = 1 N 1 N i=1 x i X be population means and y = 1 N 1 the population variances for y and x respectively The focus here is on estimating the finite population variance based on a sample of size n from the underlying population by using simple random sampling without replacement RWOR scheme Let y = 1 n n i=1 y i and x = 1 n n i=1 x i be the sample means and s y = 1 n n 1 i=1 y i y and s x = 1 n n 1 i=1 x i x be the corresponding sample variances for y and x respectively To obtain the biases and ME s expressions for various estimators, we define the following error terms: Let φ 0 = s y 1, φ y 1 = x 1, φ X = s x 1, such that Eφ x i = 0 for i = 0, 1, Eφ 0 = γδ0, Eφ 1 = γcx, Eφ = γδ0, Eφ 0 φ 1 = γδ 1 C x, Eφ 0 φ = γδ, Eφ 1 φ = γδ 03 C x, where δ0 = δ 0 1, δ0 = δ 0 1, δ = δ 1, and γ = 1 n as finite population correction factor term is ignored Also δ rs = µrs, where µ rs = 1 N 1 N y i Y r x i X s i=1 The variance of the usual variance estimator Ŝ 0 = s y, is given by µ r/ 0 µs/ 0 V arŝ 0 = γ yδ 0 1 The traditional ratio estimator Ŝ R1 by Isaki 1983, when x is known, is given by Ŝ R1 = s y x s x Revista Colombiana de Estadística

3 Generalized Exponential Type Estimator for Population Variance 387 The bias and ME respectively of Ŝ R1, to first order of approximation, are given by and BiasŜ R1 = yγδ 0 δ 3 MEŜ R1 = yγδ 0 + δ 0 δ Another form of ratio estimator Ŝ R when X is known, is given by X ŜR = s y 5 x The bias and ME respectively of Ŝ R, to first order of approximation, are given by and BiasŜ R = yγ C x δ 1 C x MEŜ R = yγ δ 0 + C x δ 1 C x 7 Yadav & Kadilar 013 suggested the following exponential type estimator for population variance: where a is a constant ŜY K = y exp x s x x + a 1s x 6, 8 The bias and minimum ME of Ŝ Y K, to first order of approximation, at optimum value of a ie a opt = δ 0 δ, is given by BiasŜ Y K 1 = yγ a 1 a + 1 δ0 1 a a δ 9 and MEŜ Y K min = y γ δ 0 δ δ 0 10 The usual regression estimator Ŝ Reg1 by Isaki 1983, is given by Ŝ Reg1 = s y + b 1 x s x, 11 where b 1 is the sample regression coefficient Corresponding population regression coefficient is given by β 1 = y δ x δ 0 Revista Colombiana de Estadística

4 388 Javid habbir & at Gupta The ME of Ŝ Reg1, is given by MEŜ Reg1 = yγ δ 0 δ Another regression estimator Ŝ Reg, is given by δ 0 1 Ŝ Reg = s y + b X x, 13 where b is the sample regression coefficient The corresponding population regression coefficient is given by β = y δ1 XC x The ME of Ŝ Reg, is given by MEŜ Reg = yγ δ 0 δ 1 1 Asghar Estimator Recently Asghar et al 01 introduced the following generalized exponential type estimator for population variance Other work for exponential type estimators for finite population means can be found at habbir, Haq & Gupta 01 { } c X x Ŝ A = λs y exp where 0 < λ 1, < c < and d X + d 1 x, 15 Asghar et al 01 reported the bias and minimum ME of ŜA, to first order of approximation at optimum values of ω= c d opt = δ 1 C x 1 and λ opt = 1, δ0 δ1 given by and BiasŜ A = yγ λ {1 + 1 } ω C x ωδ 1 C x y 16 MEŜ A min = y γ 1 1 δ 0 δ 1 17 However, the bias and minimum ME of Ŝ A expressions reported in 16 and 17 require some revisions and so does their study based on efficiency and numerical comparisons say instead of Ŝ A, to first degree of ap- The revised bias and ME of Ŝ A proximation are derived below Rewriting 15 in error terms, we have = λy 1 + φ 0 exp c d φ 1 Ŝ A Ŝ A y = λ 1 y + λ y { 1 + φ 1 1 c φ 1 } 1 φ 0 c c d φ 1 + d + c d c d φ 1 cd φ 0φ 1 18 or Revista Colombiana de Estadística

5 Generalized Exponential Type Estimator for Population Variance 389 From 18, the bias of Ŝ A, to first degree of approximation, is given by BiasŜ A = y { c λ 1 + λγ d + c d c d C x cd } δ 1C x 19 Also from 18, the ME of Ŝ A, to first degree of approximation, is given by { MEŜ A = ye λ 1 + λ φ 0 c c d φ 1 + d + c d c d φ 1 c } d φ 0φ 1 or MEŜ A = y { c 1 + λ 1 + γδ0 + d + c d c d { c λ 1 + d + c d c d γcx c x} d γδ1c γc x cd γδ 1C x } 0 The minimum ME of Ŝ A λ opt = MEŜ A min = y 1 + at optimum value of λ ie c d + c d c d γc x c d γδ 1C x 1 + γδ0 + c d + c d c d γc x c d γδ is given by 1C x 1 { } c 1 + d + c d c d γc x c d γδ 1C x { 1 + γδ 0 + c d + c d c d γc x c d γδ } 1 1C x Note In expression 0, the principal constant is λ The roles of other constants c and d are to provide a wider family of estimators Clearly one can obtain many estimators by choosing the different values of c and d If c = 1 and d = in 15, 19 and 1, we get the estimator Ŝ A1 say, with the bias and minimum ME, given by X x ŜA1 = λs y exp X + x { 3 BiasŜ A1 = y λ 1 + λγ 8 C x 1 } δ 1C x 3 { MEŜ A1 min = 1 + γ 3 8 y 1 C x 1 δ } 1C x 1 + γ δ0 + C x δ 1 C x Alternatively one can think of a different exponential type estimators as described in the following section Revista Colombiana de Estadística

6 390 Javid habbir & at Gupta 3 Proposed Estimator Motivated by Yadav & Kadilar 01 and Asghar et al 01, we propose the following general class of exponential type estimators: c X x e x sx Ŝ P = λs y exp X + d 1 x exp x + f 1 s x, 5 where < c <, < e <, d > 0 and f We can generate many more estimators from 5, for λ = 1, as described below: i For c = e = 0 in 5, we get the usual variance estimator Ŝ 0 = s y ii For c = 1, d = and e = 0 in 5, we get Ŝ 1 = s y exp X x X+x iii For c = d = 1 and e = 0 in 5, we get Ŝ = s X x y exp X iv For c = 0, f = and e = 1 in 5, we get Ŝ 3 = s y exp v For c = 0 and e = f = 1 in 5, we get Ŝ = s y exp vi For c = 1 and e = 0 in 5, we get Ŝ 5 = s y exp vii For c = 0 and e = 1 in 5, we get Ŝ 6 = s y exp x s x x +s x x s x x X x X+d 1x x s x x +f 1s x viii For d = and e = 0 in 5, we get Ŝ 7 = s cx x y exp X+x ix For c = 0 and f = 1 in 5, we get Ŝ 8 = s e y exp x s x x +s x x For c = e = 1 and d = f = in 5, we get Ŝ 9 exp x s x x +s x xi For c = d = e = f = 1 in 5, we get Ŝ 10 = s X x y exp exp X = s y exp x s x x X x X+x Revista Colombiana de Estadística

7 Generalized Exponential Type Estimator for Population Variance 391 The bias and ME expressions for the estimators Ŝ i, i = 1,,, 10 to first order of approximation, are given by BiasŜ 1 3 = yγ 8 C x 1 δ 1C x 6 BiasŜ 1 = yγ C x δ 1 C x 7 BiasŜ 3 3 = yγ 8 δ 0 1 δ 8 BiasŜ = yγ δ0 δ 9 BiasŜ 5 1 = yγ d 1 d + 1 C x 1d d δ 1C x 30 BiasŜ 6 1 = yγ f 1 f + 1 δ0 1 f f δ 31 BiasŜ 7 1 = yγ 8 c + 1 c δ0 1 cδ BiasŜ 8 1 = yγ 8 e + 1 e C x 1 eδ 1C x BiasŜ 9 3 = yγ C 8 x + δ0 1 δ 1C x + δ + 1 δ 03C x BiasŜ 10 = yγ Cx + δ0 δ1 C x + δ + δ 03 C x MEŜ 1 = yγ δ Cx δ 1 C x MEŜ = yγ δ0 + Cx δ 1 C x MEŜ 3 = yγ δ0 + 1 δ 0 δ MEŜ = yγ δ0 + δ0 δ 39 MEŜ 5 min = y γ δ0 δ1 for d opt = C x 0 δ 1 MEŜ 6 min = y γ δ 0 δ δ0 for f opt = δ 0 δ 1 MEŜ 7 min = y γ δ0 δ1 for c opt = δ 1 C x MEŜ 8 min = y γ δ 0 δ δ0 for e opt = δ δ0 3 MEŜ 9 = yγ δ0 + 1 C x + δ0 δ1 C x + δ + 1 δ 03C x MEŜ 10 = yγ δ0 + Cx + δ0 δ1 C x + δ + δ 03 C x 5 Revista Colombiana de Estadística

8 39 Javid habbir & at Gupta For the general class of estimators given in 5, we can rewrite the proposed estimator in error terms as follows: Ŝp = y cφ 1 eφ 1 + φ 0 exp exp or 1 + d 11 + φ f 11 + φ Ŝ p = y 1 + φ0 c d φ 1 e f φ c d φ 0φ 1 e f φ 0φ + α 1 φ 1 + α φ + α 3 φ 1 φ 3, 6 where α 1 = c d c d + c d, α = e f e f + e f and α 3 = ce df Also from 6, the bias of Ŝ p, to first order of approximation, is given by BiasŜ p = yγ α 1 Cx + α δ0 + α 3 δ 03 C x c d δ 1C x e f δ 7 From 6, the ME of Ŝ p, to first order of approximation, is given by MEŜ p = yγ δ 0 + ω 1C x + ω δ 0 ω 1 δ 1 C x ω δ + ω 1 ω δ 03 C x, 8 where ω 1 = c d and ω = e f From 8, the optimum values of ω 1 and ω, are given by ω 1opt = δ 0δ 1 δ 03 δ C x δ0 δ 03 and ω opt = δ δ 03 δ 1 δ0 δ 03 ubstituting the optimum values of ω 1 and ω in 8, we can get the corresponding minimum ME of Ŝ p as, given by MEŜ p min = y γ δ 0 δ 1 δ δ 1 δ 03 δ 0 δ 03 9 Comparison of Estimators Now we compare the proposed estimator Ŝ p with existing estimators i By 1 and 9 MEŜ p min<me Ŝ 0 if δ 1 + δ δ 1 δ 03 δ 0 δ 03 Revista Colombiana de Estadística

9 Generalized Exponential Type Estimator for Population Variance 393 ii By and 9 MEŜ p min < MEŜ R1 if δ 0 δ + δ 1 + δ δ 1 δ 03 δ 0 δ 03 iii By 7 and 9 MEŜ p min < MEŜ R if C x δ 1 + δ δ 1 δ 03 δ 0 δ 03 iv By and 9 MEŜ p min < MEŜ A1 min if 1 { 1 + γ 3 8 C x 1 δ 1C x } 1 + γ δ 0 + C x δ 1 C x δ 0 + δ 1 + δ δ 1 δ 03 v By 36 and 9 MEŜ p min < MEŜ 1 if vi By 37 and 9 MEŜ p min < MEŜ if Cx δ 1 + δ δ 1 δ 03 δ0 δ 03 δ 0 δ 03 C x δ 1 + δ δ 1 δ 03 δ 0 δ 03 vii By 38 and 9 MEŜ p min < MEŜ 3 if viii By 39 and 9 MEŜ p min < MEŜ if δ 0 + δ + δ1 + δ δ 1 δ 03 δ0 δ 03 δ 0 δ + δ 1 + δ δ 1 δ 03 δ 0 δ 03 ix By 1, 0, and 9 MEŜ p min < MEŜ Reg, MEŜ 5 min, MEŜ 7 min if δ δ 1 δ 03 δ 0 δ 03 Revista Colombiana de Estadística

10 39 Javid habbir & at Gupta x By 10, 1, 1, 3 and 9 MEŜ p min < MEŜ Y K min, MEŜ Reg1, MEŜ 6 min MEŜ 8 min if δ δ 03 δ 0 δ 1 δ 0 xi By and 9 MEŜ p min < MEŜ 9 if Cx δ δ δ + δ 03C x + δ δ 1 δ 03 δ0 δ 03 xii By 5 and 9 MEŜ p min < MEŜ 10 if C x δ 1 + δ 0 δ + δ 03 C x + δ δ 1 δ 03 δ 0 δ 03 The proposed estimator will perform better than the competing estimators when Conditions i-xii are satisfied 5 Numerical Examples We use the following four data sets for a numerical comparison of estimators Population 1: source: Murthy 1967, p 6 Let y be the output and x be the number of workers in a factory N = 80, n = 5, Y = , X = 8515, C y = 035, C x = 0985, ρ yx = 0910, δ 0 = 383, δ 1 = 057, δ 0 = 35360, δ = 93, δ 03 = 1680 Population : source: Murthy 1967, p 8 Let y be the output and x be the fixed capital N = 80, n = 5, Y = , X = 11663, C y = 035, C x = 07507, ρ yx = 0913, δ 0 = 38, δ 1 = 0518, δ 0 = 8306, δ = 1930, δ 03 = 1037 Population 3: source: Cochran 1977, p 03 Let y be the actual weight of peaches on each tree and x be the eye estimate of weight of peaches on each tree N = 10, n = 5, Y = 569, X = 5961, C y = 0180, C x = 0161, ρ yx = 0937, δ 0 = 199, δ 1 = 01873, δ 0 = 593, δ = 119, δ 03 = 0956 Revista Colombiana de Estadística

11 Generalized Exponential Type Estimator for Population Variance 395 Population : source: arndal, wensson & Wretman 199, p y =P population in thousand, x =RMT85-Revenues from the 1985 municipal taxation in millions kronor N = 8, n = 35, Y = 936, X = 5088, C y = 176, C x = 3, ρ yx = 0961, δ 0 = 889, δ 1 = 787, δ 0 = 8888, δ = 7879, δ 03 = 877 The results based on Populations 1- are given in Table 1 Table 1: The percentage relative efficiency of different estimators with respect to Ŝ 0 for Populations 1- Estimator Pop1 Pop Pop3 Pop Ŝ ŜR ŜR ŜA Ŝ Ŝ Ŝ Ŝ Ŝ5, Ŝ 7, Ŝ Reg Ŝ6, Ŝ 8, Ŝ Reg1, Ŝ Y K Ŝ Ŝ Ŝp We use following expression to obtain the percent relative efficiency P RE of different estimators with respect to 0 PRE = V ar 0 MEi 100, where i = 0, R1, R, A1, 1 10, Reg1, Reg, Y K, P 6 Conclusion We have developed a new generalized class of exponential type estimators for population variance which is more efficient than many of the existing estimators We have also improved the bias and ME expressions of the Asghar et al 01 estimator While we note that the improved Asghar et al 01 estimator Ŝ A1 performs better than several estimators This estimator can be improved further as shown by the performance of Ŝ p We also note that the estimators Ŝ 6, Ŝ 8, ŜReg1, Ŝ Y K and Ŝ 5, Ŝ7, Ŝ Reg have the same ME for all four populations Received: July 01 Accepted: January 015 Revista Colombiana de Estadística

12 396 Javid habbir & at Gupta References Asghar, A, anaullah, A & Hanif, M 01, Generalized exponential type estimator for population variance in survey sampling, Revista Colombiana de Estadística 71, 11 Bansal, M, Javed, M & Khanna, N 011, A class of estimators of variance of the regression estimator, International Journal of Agriculture and tatistical ciences 71, Cochran, W G 1977, ampling Techniques, 3 edn, Jhon Wiley & ons, New York Gupta, & habbir, J 008, Variance estimation in simple random sampling using auxiliary information, Hacettepe Journal of Mathematics and tatistics 371, Isaki, C T 1983, Variance estimation using auxiliary information, Journal of the American tatistical Association 78381, Jhajj, H, harma, M K & Grover, L K 005, An efficient class of chain estimators of population variance under sub-sampling scheme, Journal of the Japan tatistical ociety 35, Kadilar, C & Cingi, H 006, Ratio estimators for the population variance in simple and stratified random sampling, Applied Mathematics and Computation 173, Murthy, M N 1967, ampling Theory and Methods, tatistical Publishing ociety, Calcutta Nayak, R & ahoo, L 01, ome alternative predictive estimators of populationvariance, Revista Colombiana de Estadística 353, arndal, C E, wensson, B & Wretman, J H 199, Model Asssisted urvey ampling, pringer Verlag, New York habbir, J, Haq, A & Gupta, 01, A new difference-cum-exponential type estimator of finite population mean in simple random sampling, Revista Colombiana de Estadística 371, ingh, H P, Upadhyaya, L N & Namjoshi, U D 1988, Estimation of finite population variance, Current cience 57, ingh, R, Chauhan, P, wan, N & marandache, F 011, Improved exponential estimator for population variance using two auxiliary variables, Italian Journal of Pure and Applied Mathematics 8, ubramani, J & Kumarapandiyan, G 01, Variance estimation using median of the auxiliary variable, International Journal of Probability and tatistics 13, 36 0 Revista Colombiana de Estadística

13 Generalized Exponential Type Estimator for Population Variance 397 Upadhyaya, L N, ingh, H P, Chatterjee, & Yadav, R 011, Improved ratio and product exponential type estimators, Journal of tatistical Theory and Practice 5, Yadav, K & Kadilar, C 013, Improved exponential type ratio estimator of population variance, Revista Colombiana de Estadística 361, Yadav, K & Kadilar, C 01, A two parameter variance estimator using auxiliary information, Applied Mathematics and Computation 6, Revista Colombiana de Estadística