Improved Exponential Type Ratio Estimator of Population Variance

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1 vista Colombiana de Estadística Junio 2013, volumen 36, no. 1, pp. 145 a 152 Improved Eponential Type Ratio Estimator of Population Variance Estimador tipo razón eponencial mejorado para la varianza poblacional Subhash Kumar Yadav 1,a, Cem Kadilar 2,b 1 Department of Mathematics & Statistics (A Centre of Ecellence), Dr. Rml Avadh University, Faizabad, India 2 Department of Statistics, Hacettepe University, Ankara, Turkey Abstract This article considers the problem of estimating the population variance using auiliary information. An improved version of Singh s eponential type ratio estimator has been proposed and its properties have been studied under large sample approimation. It is shown that the proposed eponential type ratio estimator is more efficient than that considered by the Singh estimator, conventional ratio estimator and the usual unbiased estimator under some realistic conditions. An empirical study has been carried out to judge the merits of the suggested estimator over others. Key words: Auiliary variable, Bias, Efficiency, Mean squared error. sumen Este artículo considera el problema de estimar la varianza poblacional usando información auiliar. Una versión mejorada de un estimador eponencial tipo razón de Singh ha sido propuesta y sus propiedades han sido estudiadas bajo aproimaciones de grandes muestras. Se muestra que el estimador eponencial tipo razón propuesto es más eficiente que el estimador de Singh, el estimador de razón convencional y el estimador insesgado usual bajo algunas condiciones realísticas. Un estudio empírico se ha llevado a cabo con el fin de juzgar los méritos del estimador sugerido sobre otros disponibles. Palabras clave: eficiencia, error cuadrático medio, sesgo, variable auiliar. a Assistant professor. drskystats@gmail.com b Professor. kadilar@hacettepe.edu.tr 145

2 146 Subhash Kumar Yadav & Cem Kadilar 1. Introduction The auiliary information in sampling theory is used for improved estimation of parameters thereby enhancing the efficiencies of the estimators. Let ( i, y i ), i = 1, 2,..., n be the n pair of sample observations for the auiliary and study variables, respectively, drawn from the population of size N using Simple Random Sampling without placement. Let X and Y be the population means of auiliary and study variables, respectively, and let and y be the respective sample means. Ratio estimators are used when the line of regression of y on passes through the origin and the variables and y are positively correlated to each other, while product estimators are used when and y are negatively correlated to each other; otherwise, regression estimators are used. The sample variance estimator of the population variance is defined as t 0 = s 2 y (1) which is an unbiased estimator of population variance Sy 2 = 1 N N 1 (Y i Y ) 2 and its variance is V (t 0 ) = γ Sy 4 (λ 40 1) (2) where λ rs = µrs µ r/2 20 µs/2 02, µ rs = 1 N 1 N (Y i Y ) r (X i X) s, and γ = 1 n. Isaki (1983) proposed the ratio type estimator for estimating the population variance of the study variable as ( ) S t R = s 2 2 y (3) where s 2 y = 1 n 1 s 2 n (y i y) 2, s 2 = 1 n ( i ) 2, S 2 = 1 N (X i X) 2 n 1 N 1 X = 1 N N X i, Y = 1 N N Y i, = 1 n n i, y = 1 n n y i The Bias (B) and Mean Squared Error (MSE) of the estimator in (3), up to the first order of approimation, are given, respectively, as B(t R ) = γs 2 y (λ 40 1) (λ 22 1)] (4) MSE(t R ) = γs 4 y (λ 40 1) + (λ 04 1) 2 (λ 22 1)] (5) Singh, Chauhan, Sawan & Smarandache (2011) proposed the eponential ratio estimator for the population variance as S t = s 2 2 y ep s 2 ] S 2 + s 2 (6) vista Colombiana de Estadística 36 (2013)

3 Improved Eponential Type Ratio Estimator of Population Variance 147 The bias and MSE, up to the first order of approimation, respectively, are 3 B(t ) = γ Sy 2 8 (λ 04 1) 1 ] 2 (λ 22 1) (7) MSE(t ) = γ S 4 y (λ 40 1) + (λ ] 04 1) (λ 22 1) 4 The usual linear regression estimator for population variance is (8) Ŝ 2 lr = s 2 y + b(s 2 s 2 ) (9) where b = s2 y ( λ 22 1) s 2 ( λ 04 1) is the sample regression coefficient. The MSE of Ŝ2 lr, to the first order of approimation, is (λ 40 1) (λ 22 1) 2 MSE(Ŝ2 lr) = γ S 4 y λ 04 1 ] (10) Many more authors, including Singh & Singh (2001, 2003), Nayak & Sahoo (2012), among others, have contributed to variance estimation. 2. Improved Eponential Type Ratio Estimator Motivated by Upadhyaya, Singh, Chatterjee & Yadav (2011) and following them, we propose the improved ratio eponential type estimator of the population variance as follows: The ratio eponential type estimator due to Singh et al. (2011) is given by S t = s 2 2 y ep s 2 ] ] S 2 + s 2 = s 2 y ep 1 2s2 S 2 + s 2 which can be generalized by introducing a positive real constant α (i.e. α 0) as t (α) = αs 2 ] s2 y ep 1 S 2 + (α 1)s 2 = s 2 S 2 s 2 ] y ep S 2 + (α 1)s 2 (11) Here, we note that: (i) For α = 0, t (α) in (11) reduces to t (0) = s2 y ep 1] (12) which is a biased estimator with the MSE larger than s 2 y utilizing no auiliary information as the value of e is always greater than unity. (ii) For α = 1, t (α) in (11) reduces to t (1) S 2 = s2 y ep s 2 ] S 2 (13) vista Colombiana de Estadística 36 (2013)

4 148 Subhash Kumar Yadav & Cem Kadilar (iii) For α = 2, t (α) estimator in (11) reduces to Singh et al. (2011) ratio eponential type S t = s 2 2 y ep s 2 ] S 2 + s 2 (14) Then, we have investigated for which value of α, the proposed estimator, t (α), is more efficient than the estimators, t 0, t R, and t. 3. The First Degree Approimation to the Bias and Mean Squared Error of the Suggested Estimator In order to study the large sample properties of the proposed class of estimator, t (α), we define s2 y = Sy 2 (1 + ε 0 ) and s 2 = S 2 (1 + ε 1 ) such that E (ε i ) = 0 for (i = 0, 1) and E ( ε0) 2 = γ (λ40 1), E ( ε1) 2 = γ (λ04 1), E (ε 0 ε 1 ) = γ (λ 22 1). t (α) To the first degree of approimation, the bias and the MSE of the estimator,, are respectively given by where λ = λ22 1 λ B(t (α) ) = (λ 04 1) γs2 y 2α 2 2α(1 λ) 1] (15) MSE(t (α) ) = γ S4 y (λ 40 1) + (λ ] 04 1) α 2 (1 2αλ) The MSE(t (α) ) is minimum for (16) α = 1 λ = α 0 (say) (17) Substituting α = 1 λ into (11), we get the asymptotically optimum estimator (AOE) in the class of estimators (t (α) ) as (t (α0) ) = λ(s s2 2 s 2 ] y ep ) λs 2 + (1 λ)s 2 (18) The value of λ can be obtained from the previous surveys or the eperience gathered in due course of time, for instance, see Murthy (1967), ddy (1973, 1974) and Srivenkataramana & Tracy (1980), Singh & Vishwakarma (2008), Singh & Kumar (2008) and Singh & Karpe (2010). The mean square error of AOE (t (α0) ), to the first degree of approimation, is given by MSE(t (α0) ) = γ S4 y (λ 40 1) (λ 22 1) 2 ] (19) λ 04 1 which equals to the approimate MSE of the usual linear regression estimator of population variance. vista Colombiana de Estadística 36 (2013)

5 Improved Eponential Type Ratio Estimator of Population Variance 149 In case the practitioner fails to guess the value of λ by utilizing all his resources, it is worth advisable to replace λin (18) by its consistent estimate λ = λ 22 1 λ 04 1 (20) as Thus, the substitution of λ in (18) yields an estimator based on estimated λ ] (t ( α0) ) = λ(s s2 2 s 2 y ep ) λs 2 + (1 λ)s (21) 2 It can be shown to the first degree of approimation that MSE(t (α0) ) = MSE(t( α0) ) = γ S4 y (λ 40 1) (λ 22 1) 2 ] λ 04 1 (22) Thus, the estimator t ( α0), given in (21), is to be used in practice as an alternative to the usual linear regression estimator. 4. Efficiency Comparisons of the Proposed Estimator with the Mentioned Eisting Estimators if 1 α or From (16) and (2), we have MSE(t 0 ) MSE(t (α) ) = γ S4 (λ 04 1) y α (1 2αλ) > 0, 2 α > 1 2λ (23) From (16) and (5), we have MSE(t R ) MSE(t (α) ) = γ S4 y(λ 04 1)(1 1 α ) (1+ 2λ) > 0, if either { min 1, } { 1 < α < ma 1, 2λ 1 } 1, λ > 1 2λ 1 2 (24) α > 1, 0 λ 1 2. (25) From (16) and (8), we have MSE(t ) MSE(t (α) ) = γ S4 y(λ 04 1)( α ) ( λ) > 0, if either 1 α or { min 2, } { 2 < α < ma 2, 4λ 1 α > 2, 0 λ 1 4 } 2, λ > 1 4λ 1 4 (26) (27) vista Colombiana de Estadística 36 (2013)

6 150 Subhash Kumar Yadav & Cem Kadilar From (16) and (10), we have 2 MSE(t (α) ) MSE(Ŝ2 lr) = γ Sy(λ 4 (λ04 1) 04 1) (λ 22 1)] < 0 α if λ 04 < 1 (28) 5. Numerical Illustrations The appropriateness of the proposed estimator has been verified with the help of the four data sets, given in Table 1 (Subramani & Kumarapandiyan 2012). In Table 2, which gives the range of α and also the optimal value, α 0, for the efficiency condition of the proposed estimator, we see that (t (α) ), is quite wide as t ; whereas, from Table 3, which provides the Percent lative Efficiencies (PREs) of different estimators of the population variance with respect to the sample variance, we observe that the proposed estimator is more efficient than t. Table 1: Parameters of populations. Parameters Population 1 Population 2 Population 3 Population 4 N n Y X ρ S y C y S C λ λ λ λ Table 2: Range of α for (t (α) ) to be more efficient than different estimators of the population variance. Populations Estimators t 0 α > 0.50 α > 0.52 α > 0.65 α > 0.64 t R α (1.00, 1.01) α (1.00, 1.08) α (1.00, 1.82) α (1.00, 1.76) t α (0.67, 2.00) α (0.70, 2.00) α (0.95, 2.00) α (0.94, 2.00) Common Range of α ( for t (α) ) to be more α (0.67, 2.00) α (0.70, 2.00) α (0.95, 2.00) α (0.94, 2.00) efficient than t 0, t R, t Optimum value of α α 0 = α 0 = α 0 = α 0 = vista Colombiana de Estadística 36 (2013)

7 Improved Eponential Type Ratio Estimator of Population Variance 151 Table 3: Percent relative efficiencies (PREs) of different estimators of population variance with respect to sample variance t 0 = s 2 y. Populations Estimators t 0 = s 2 y t R 93, t t ( α 0) 94, Conclusion We have suggested an improved eponential ratio estimator for estimating the population variance. From theoretical discussions, given in Section 4 and results in Table 3, we infer that the proposed estimator is better than the mentioned eisting estimators in literature, the usual sample variance, traditional ratio estimator due to Isaki (1983) and Singh et al. (2011) eponential ratio estimator in the sense of having lesser mean squared error. We have also given the range of α along with its optimum value for to be more efficient than different estimators. Hence, the proposed estimator is recommended for its practical use for estimating the population variance when the auiliary information is available. In future articles, we hope to adapt the proposed estimator here to the combined and separate methods in the stratifed random sampling. Acknowledgements The authors are grateful to the referees and the Editor-in-Chief for providing some useful comments on an earlier draft of the paper. cibido: agosto de 2012 Aceptado: mayo de 2013 ] ferences Isaki, C. (1983), Variance estimation using auiliary information, Journal of the American Statistical Association 78, Murthy, M. (1967), Sampling Theory and Methods, Calcutta Statistical Publishing Society, Kolkatta, India. Nayak, R. & Sahoo, L. (2012), Some alternative predictive estimators of population variance, vista Colombiana de Estadistica 35(3), ddy, V. (1973), On ratio and product methods of estimation, Sankhya Serie B 35(3), ddy, V. (1974), On a transformed ratio method of estimation, Sankhya Serie C 36, vista Colombiana de Estadística 36 (2013)

8 152 Subhash Kumar Yadav & Cem Kadilar Singh, H. & Karpe, N. (2010), Estimation of mean, ratio and product using auiliary information in the presence of measurement errors in sample surveys, Journal of Statistical Theory and Practice 4(1), Singh, H. & Kumar, S. (2008), A general family of estimators of finite population ratio, product and mean using two phase sampling scheme in the presence of non-response, Journal of Statistical Theory and Practice 2(4), Singh, H. & Singh, R. (2001), Improved ratio-type estimator for variance using auiliary information, Journal of Indian Society of Agricultural Statistics 54(3), Singh, H. & Singh, R. (2003), Estimation of variance through regression approach in two phase sampling, Aligarh Journal of Statistics 23, Singh, H. & Vishwakarma, G. (2008), Some families of estimators of variance of stratified random sample mean using auiliary information, Journal of Statistical Theory and Practice 2(1), Singh, R., Chauhan, P., Sawan, N. & Smarandache, F. (2011), Improved eponential estimator for population variance using two auiliary variables, Italian Journal of Pure and Applied Mathematics 28, Srivenkataramana, T. & Tracy, D. (1980), An alternative to ratio method in sample surveys, Annals of the Institute of Statistical Mathematics 32, Subramani, J. & Kumarapandiyan, G. (2012), Variance estimation using median of the auiliary variable, International Journal of Probability and Statistics 1(3), Upadhyaya, L., Singh, H., Chatterjee, S. & Yadav, R. (2011), Improved ratio and product eponential type estimators, Journal of Statistical Theory and Practice 5(2), vista Colombiana de Estadística 36 (2013)