Revista Colombiana de Estadística Junio 2013, volumen 36, no. 1, pp. 145 a 152 Improved Exponential Type Ratio Estimator of Population Variance Estimador tipo razón exponencial mejorado para la varianza poblacional Subhash Kumar Yadav1;a, Cem Kadilar2;b 1Department of Mathematics & Statistics (A Centre of Excellence), Dr. Rml Avadh University, Faizabad, India 2Department of Statistics, Hacettepe University, Ankara, Turkey Abstract This article considers the problem of estimating the population variance using auxiliary information. An improved version of Singh’s exponential type ratio estimator has been proposed and its properties have been studied under large sample approximation. It is shown that the proposed exponential 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: Auxiliary variable, Bias, Efficiency, Mean squared error. Resumen Este artículo considera el problema de estimar la varianza poblacional usando información auxiliar. Una versión mejorada de un estimador expo- nencial tipo razón de Singh ha sido propuesta y sus propiedades han sido estudiadas bajo aproximaciones de grandes muestras. Se muestra que el esti- mador exponencial 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 auxiliar. aAssistant professor. E-mail: [email protected] bProfessor. E-mail: [email protected] 145 146 Subhash Kumar Yadav & Cem Kadilar 1. Introduction The auxiliary information in sampling theory is used for improved estimation of parameters thereby enhancing the efficiencies of the estimators. Let (xi; yi); i = 1; 2; : : : ; n be the n pair of sample observations for the auxil- iary and study variables, respectively, drawn from the population of size N using Simple Random Sampling without Replacement. Let X and Y be the population means of auxiliary and study variables, respectively, and let x and y be the re- spective sample means. Ratio estimators are used when the line of regression of y on x passes through the origin and the variables x and y are positively correlated to each other, while product estimators are used when x and y are negatively correlated to each other; otherwise, regression estimators are used. The sample variance estimator of the population variance is defined as 2 t0 = sy (1) 2 1 PN 2 which is an unbiased estimator of population variance Sy = N−1 i=1(Yi − Y ) and its variance is 4 V (t0) = γ Sy (λ40 − 1) (2) µrs 1 PN r s 1 where λrs = r=2 s=2 , µrs = N−1 i=1(Yi − Y ) (Xi − X) , and γ = n . µ20 µ02 Isaki (1983) proposed the ratio type estimator for estimating the population variance of the study variable as 2 2 Sx tR = sy 2 (3) sx where n n N 1 X 1 X 1 X s2 = (y − y)2; s2 = (x − x)2;S2 = (X − X)2 y n − 1 i x n − 1 i x N − 1 i i=1 i=1 i=1 N N n n 1 X 1 X 1 X 1 X X = X ; Y = Y ; x = x ; y = y N i N i n i n i i=1 i=1 i=1 i=1 The Bias (B) and Mean Squared Error (MSE) of the estimator in (3), up to the first order of approximation, are given, respectively, as 2 B(tR) = γSy [(λ40 − 1) − (λ22 − 1)] (4) 4 MSE(tR) = γSy [(λ40 − 1) + (λ04 − 1) − 2 (λ22 − 1)] (5) Singh, Chauhan, Sawan & Smarandache (2011) proposed the exponential ratio estimator for the population variance as 2 2 2 Sx − sx tRe = sy exp 2 2 (6) Sx + sx Revista Colombiana de Estadística 36 (2013) 145–152 Improved Exponential Type Ratio Estimator of Population Variance 147 The bias and MSE, up to the first order of approximation, respectively, are 3 1 B(t ) = γ S2 (λ − 1) − (λ − 1) (7) Re y 8 04 2 22 (λ − 1) MSE(t ) = γ S4 (λ − 1) + 04 − (λ − 1) (8) Re y 40 4 22 The usual linear regression estimator for population variance is 2 2 2 2 Sblr = sy + b(Sx − sx) (9) 2 sy (λb22−1) where b = 2 is the sample regression coefficient. sx(λb04−1) 2 The MSE of Sblr, to the first order of approximation, is 2 2 4 (λ22 − 1) MSE(Sblr) = γ Sy (λ40 − 1) − (10) λ04 − 1 Many more authors, including Singh & Singh (2001, 2003), Nayak & Sahoo (2012), among others, have contributed to variance estimation. 2. Improved Exponential Type Ratio Estimator Motivated by Upadhyaya, Singh, Chatterjee & Yadav (2011) and following them, we propose the improved ratio exponential type estimator of the population variance as follows: The ratio exponential type estimator due to Singh et al. (2011) is given by 2 2 2 2 Sx − sx 2 2sx tRe = sy exp 2 2 = sy exp 1 − 2 2 Sx + sx Sx + sx which can be generalized by introducing a positive real constant ‘α’ (i.e. α ≥ 0) as 2 2 2 (α) 2 αsx 2 Sx − sx tRe = sy exp 1 − 2 2 = sy exp 2 2 (11) Sx + (α − 1)sx Sx + (α − 1)sx (α) Here, we note that: (i) For α = 0, tRe in (11) reduces to (0) 2 tRe = sy exp [1] (12) 2 which is a biased estimator with the MSE larger than sy utilizing no auxiliary information as the value of ‘e’ is always greater than unity. (α) (ii) For α = 1, tRe in (11) reduces to 2 2 (1) 2 Sx − sx tRe = sy exp 2 (13) Sx Revista Colombiana de Estadística 36 (2013) 145–152 148 Subhash Kumar Yadav & Cem Kadilar (α) (iii) For α = 2, tRe in (11) reduces to Singh et al. (2011) ratio exponential type estimator 2 2 2 Sx − sx tRe = sy exp 2 2 (14) Sx + sx (α) Then, we have investigated for which value of α, the proposed estimator, tRe , is more efficient than the estimators, t0, tR, and tRe. 3. The First Degree Approximation 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, (α) 2 2 2 2 tRe , we define sy = Sy (1 + "0) and sx = Sx (1 + "1) such that E ("i) = 0 for 2 2 (i = 0; 1) and E "0 = γ (λ40 − 1), E "1 = γ (λ04 − 1), E ("0"1) = γ (λ22 − 1). To the first degree of approximation, the bias and the MSE of the estimator, (α) tRe , are respectively given by (λ − 1) B(t(α)) = γS2 04 [2α(1 − λ) − 1] (15) Re y 2α2 (λ − 1) MSE(t(α)) = γ S4 (λ − 1) + 04 (1 − 2αλ) (16) Re y 40 α2 where λ = λ22−1 . λ04−1 (α) The MSE(tRe ) is minimum for 1 α = = α (say) (17) λ 0 1 Substituting α = λ into (11), we get the asymptotically optimum estimator (α) (AOE) in the class of estimators (tRe ) as 2 2 (α0) 2 λ(Sx − sx) (tRe ) = sy exp 2 2 (18) λSx + (1 − λ)sx The value of λ can be obtained from the previous surveys or the experience gathered in due course of time, for instance, see Murthy (1967), Reddy (1973, 1974) and Srivenkataramana & Tracy (1980), Singh & Vishwakarma (2008), Singh & Kumar (2008) and Singh & Karpe (2010). (α0) The mean square error of AOE (tRe ), to the first degree of approximation, is given by 2 (α0) 4 (λ22 − 1) MSE(tRe ) = γ Sy (λ40 − 1) − (19) λ04 − 1 which equals to the approximate MSE of the usual linear regression estimator of population variance. Revista Colombiana de Estadística 36 (2013) 145–152 Improved Exponential Type Ratio Estimator of Population Variance 149 In case the practitioner fails to guess the value of ‘λ’ by utilizing all his re- sources, it is worth advisable to replace λin (18) by its consistent estimate λb22 − 1 λb = (20) λb04 − 1 Thus, the substitution of λb in (18) yields an estimator based on estimated ‘λ’ as " 2 2 # (α ) λb(S − s ) (t b0 ) = s2 exp x x (21) Re y 2 2 λSb x + (1 − λb)sx It can be shown to the first degree of approximation that (λ − 1)2 (α0) (αb0) 4 22 MSE(tRe ) = MSE(tRe ) = γ Sy (λ40 − 1) − (22) λ04 − 1 (αb0) Thus, the estimator tRe , given in (21), is to be used in practice as an alter- native to the usual linear regression estimator. 4. Efficiency Comparisons of the Proposed Estimator with the Mentioned Existing Estimators (α) 4 (λ04−1) From (16) and (2), we have MSE(t0)−MSE(tRe ) = γ Sy α2 (1−2αλ) > 0, if 1 α > (23) 2λ (α) 4 1 From (16) and (5), we have MSE(tR)−MSE(tRe ) = γ Sy (λ04 −1)(1− α ) (1+ 1 α − 2λ) > 0, if either 1 1 1 min 1; < α < max 1; ; λ > (24) 2λ − 1 2λ − 1 2 or 1 α > 1; 0 ≤ λ ≤ : (25) 2 (α) 4 1 1 1 From (16) and (8), we have MSE(tRe) − MSE(tRe ) = γ Sy (λ04 − 1)( 2 − α )( 2 + 1 α − 2λ) > 0, if either 2 2 1 min 2; < α < max 2; ; λ > (26) 4λ − 1 4λ − 1 4 or 1 α > 2; 0 ≤ λ ≤ (27) 4 Revista Colombiana de Estadística 36 (2013) 145–152 150 Subhash Kumar Yadav & Cem Kadilar From (16) and (10), we have 2 (α) 2 4 (λ04 − 1) MSE(t ) − MSE(Sb ) = γ S (λ04 − 1) − (λ22 − 1) < 0 Re lr y α if λ04 < 1 (28) 5.