JAMSI, 11 (015, No. 1 91 A Class of Modified Ratio Estimators for Estimation of Population Variance J. SUBRAMANI AND G. KUMARAPANDIYAN Abstract In this paper we have proposed a class of modified ratio type variance estimators for estimation of population variance of the study variable using the known parameters of the auxiliary variable. The bias and mean squared error of the proposed estimators are obtained and also derived the conditions for which the proposed estimators perform better than the traditional ratio type variance estimator and existing modified ratio type variance estimators. Further we have compared the proposed estimators with that of the traditional ratio type variance estimator and existing modified ratio type variance estimators for certain natural populations. Mathematics Subject Classification 000: 6 D05 General Terms: Auxiliary variable, Bias, Mean squared error Additional Key Words and Phrases: Coefficient of variation, Kurtosis, Median, Natural populations, Simple random sampling, Skewness 1. INTRODUCTION It is common practice to use the auxiliary variable for improving the precision of the estimate of a parameter. When the information on an auxiliary variable X is known, a number of estimators such as ratio, product and linear regression estimators are available in the literature. When the correlation between the study variable and the auxiliary variable is positive, ratio method of estimation is quite effective. Consider a finite population U = {U 1, U,, U N } of N distinct and identifiable units. Let Y be a real variable with value Y i measured on U i, i = 1,,3,, N giving a vector Y = {Y 1, Y,, Y N }. The problem is to estimate the population mean Y = 1 N Y N i=1 i on the basis of a random sample selected from the population U or its variance S y = 1 (Y (N 1 i Y. When there is no additional information on the auxiliary 1 This research was supported by the University Grants Commission, New Delhi through Major Research Project. N i=1 variable available, the simplest estimator of population mean or variance is the simple random sample mean or variance without replacement. As stated earlier, if an auxiliary variable X closely related to the study variable Y is available and X is easy 10.1515/jamsi-015-0006 University of SS. Cyril and Methodius in Trnava Download Date 8/18/18 7:9 AM
9 J. G. to obtain then one can use ratio, product and regression estimators to improve the performance of the estimator of the study variable. Estimating the finite population variance has great significance in various fields such as Industry, Agriculture, Medical and Biological sciences. In this paper, we consider the problem of estimation of the population variance and use the auxiliary information to improve the efficiency of the estimator of population variance. Before discussing further about the traditional ratio type variance estimator, existing modified ratio type variance estimators and the proposed modified ratio type variance estimators, the notations to be used in this paper are described below: N Population size n Sample size γ = (1 f n Y Study variable X Auxiliary variable X, Y Population means x, y Sample means S y, S x Population variances s y, s x Sample variances C X, C y Coefficient of variations λ rs = μ rs r/ s/ μ μ0 0 μ rs = 1 N (y N i=1 i Y r (x i X s β 1(x = μ 03 μ3 0, Skewness of the auxiliary variable M d Median of the auxiliary variable Q 1 First (lower quartile of the auxiliary variable Q 3 Third (upper quartile of the auxiliary variable Q r Inter-quartile range of the auxiliary variable Q d Semi-quartile range of the auxiliary variable Q a Semi-quartile average of the auxiliary variable D i i th Decile of the auxiliary variable B(. Bias of the estimator MSE(. Mean squared error of the estimator Ŝ R Traditional ratio type variance estimator of S y Ŝ i Existing modified ratio type variance estimator of S y Download Date 8/18/18 7:9 AM
JAMSI, 11 (015, No. 1 93 Ŝ pi Proposed modified ratio type variance estimator of S y Isaki (1983 suggested a ratio type variance estimator for the population variance S y when the population variance S x of the auxiliary variable X is known together with its bias and mean squared error are given below: Ŝ R = s S x y s (1.1 x B(Ŝ R = γs y [(β (x 1 (λ 1 (1. MSE(Ŝ R = γs y 4 [(β (y 1 + (β (x 1 (λ 1 (1.3 where β (y = μ 40, β μ (x = μ 04, λ 0 μ = μ 0 μ 0 μ 0 The ratio type variance estimator given in (1.1 is used to improve the precision of the estimate of the population variance compared to simple random sampling when there exists a positive correlation between X and Y. Further improvements are also achieved on the ratio estimator by introducing a number of modified ratio estimators with the use of known parameters like Coefficient of Variation, Coefficient of Kurtosis, Median, Quartiles and Deciles. The problem of constructing efficient estimators for the population variance has been widely discussed by various authors such as Agarwal and Sithapit (1995, Ahmed et al. (000, Al-Jararha and Al-Haj Ebrahem (01, Arcos et al (005, Cochran (1977, Das and Tripathi (1978, Garcia and Cebrain (1997, Gupta and Shabbir (008, Isaki (1983, Kadilar and Cingi (006a,b, Murthy (1967, Prasad and Singh (1990, Reddy (1974, Shabbir and Gupta (006, Singh and Solanki (013, Singh and Chaudhary (1986, Singh et al. (1988, (01a,b,c,013, tailor and Shrama (01, Upadhyaya and Singh (1999, 001, 006, Wolter (1985 and Yadav and Kadilar (013a,b. The following table contains all modified ratio type variance estimators using known population parameters of the auxiliary variable in which some of the estimators are already suggested in the literature, remaining estimators are introduced in this paper Download Date 8/18/18 7:9 AM
94 J. G. Table 1: Modified ratio type estimators for estimating population variance with the bias and mean squared error Estimator Bias - B(. Mean squared error MSE(. Ŝ 1 = s y [ S x + s x + Kadilar and Cingi (006b γs y δ 1 [δ 1 (β (x 1 (λ 1 γs y 4 [(β (y 1 + δ 1 (β (x 1 δ 1 (λ 1 Ŝ = s y [ S x + β (x s x + β (x Upadhyaya and Singh (1999 γs y δ [δ (β (x 1 (λ 1 γs y 4 [(β (y 1 + δ (β (x 1 δ (λ 1 Ŝ 3 = s y [ S x + β 1(x s x + β 1(x γs y δ 3 [δ 3 (β (x 1 (λ 1 γs y 4 [(β (y 1 + δ 3 (β (x 1 δ 3 (λ 1 Ŝ 4 = s y [ S x + ρ s x + ρ γs y δ 4 [δ 4 (β (x 1 (λ 1 γs y 4 [(β (y 1 + δ 4 (β (x 1 δ 4 (λ 1 Ŝ 5 = s y [ S x + S x s x + S x γs y δ 5 [δ 5 (β (x 1 (λ 1 γs y 4 [(β (y 1 + δ 5 (β (x 1 δ 5 (λ 1 Ŝ 6 = s y [ S x + M d s x + M d (01a γs y δ 6 [δ 6 (β (x 1 (λ 1 γs y 4 [(β (y 1 + δ 6 (β (x 1 δ 6 (λ 1 Ŝ 7 = s y [ S x + Q 1 s x + Q 1 (01b γs y δ 7 [δ 7 (β (x 1 (λ 1 γs y 4 [(β (y 1 + δ 7 (β (x 1 δ 7 (λ 1 Download Date 8/18/18 7:9 AM
JAMSI, 11 (015, No. 1 95 Ŝ 8 = s y [ S x + Q 3 s x + Q 3 (01b γs y δ 8 [δ 8 (β (x 1 (λ 1 γs y 4 [(β (y 1 + δ 8 (β (x 1 δ 8 (λ 1 Ŝ 9 = s y [ S x + Q r s x + Q r (01b γs y δ 9 [δ 9 (β (x 1 (λ 1 γs y 4 [(β (y 1 + δ 9 (β (x 1 δ 9 (λ 1 Ŝ 10 = s y [ S x + Q d s x + Q d (01b γs y δ 10 [δ 10 (β (x 1 (λ 1 γs 4 y [(β (y 1 + δ 10 (β (x 1 δ 10 (λ 1 Ŝ 11 = s y [ S x + Q a s x + Q a (01b γs y δ 11 [δ 11 (β (x 1 (λ 1 γs 4 y [(β (y 1 + δ 11 (β (x 1 δ 11 (λ 1 Ŝ 1 = s y [ S x + D 1 s x + D 1 (01c γs y δ 1 [δ 1 (β (x 1 (λ 1 γs 4 y [(β (y 1 + δ 1 (β (x 1 δ 1 (λ 1 Ŝ 13 = s y [ S x + D s x + D (01c γs y δ 13 [δ 13 (β (x 1 (λ 1 γs 4 y [(β (y 1 + δ 13 (β (x 1 δ 13 (λ 1 Ŝ 14 = s y [ S x + D 3 s x + D 3 (01c γs y δ 14 [δ 14 (β (x 1 (λ 1 γs 4 y [(β (y 1 + δ 14 (β (x 1 δ 14 (λ 1 Ŝ 15 = s y [ S x + D 4 s x + D 4 (01c γs y δ 15 [δ 15 (β (x 1 (λ 1 γs 4 y [(β (y 1 + δ 15 (β (x 1 δ 15 (λ 1 Ŝ 16 = s y [ S x + D 5 s x + D 5 (01c γs y δ 16 [δ 16 (β (x 1 (λ 1 γs 4 y [(β (y 1 + δ 16 (β (x 1 δ 16 (λ 1 Download Date 8/18/18 7:9 AM
96 J. G. Ŝ 17 = s y [ S x + D 6 s x + D 6 (01c γs y δ 17 [δ 17 (β (x 1 (λ 1 γs 4 y [(β (y 1 + δ 17 (β (x 1 δ 17 (λ 1 Ŝ 18 = s y [ S x + D 7 s x + D 7 (01c γs y δ 18 [δ 18 (β (x 1 (λ 1 γs 4 y [(β (y 1 + δ 18 (β (x 1 δ 18 (λ 1 Ŝ 19 = s y [ S x + D 8 s x + D 8 (01c γs y δ 19 [δ 19 (β (x 1 (λ 1 γs 4 y [(β (y 1 + δ 19 (β (x 1 δ 1 (λ 1 Ŝ 0 = s y [ S x + D 9 s x + D 9 (01c γs y δ 0 [δ 0 (β (x 1 (λ 1 γs 4 y [(β (y 1 + δ 0 (β (x 1 δ 0 (λ 1 Ŝ 1 = s y [ S x + D 10 s x + D 10 (01c Ŝ = s y [ β (x S x + β (x s x + Kadilar and Cingi (006b Ŝ 3 = s y [ S x + β (x s x + β (x Kadilar and Cingi (006b γs y δ 1 [δ 1 (β (x 1 (λ 1 γs 4 y [(β (y 1 + δ 1 (β (x 1 δ 1 (λ 1 γs y δ [δ (β (x 1 (λ 1 γs 4 y [(β (y 1 + δ (β (x 1 δ (λ 1 γs y δ 3 [δ 3 (β (x 1 (λ 1 γs 4 y [(β (y 1 + δ 3 (β (x 1 δ 3 (λ 1 Ŝ 4 = s y [ β 1(x S x + β 1(x s x + γs y δ 4 [δ 4 (β (x 1 (λ 1 γs 4 y [(β (y 1 + δ 4 (β (x 1 δ 4 (λ 1 Ŝ 5 = s y [ S x + β 1(x s x + β 1(x γs y δ 5 [δ 5 (β (x 1 (λ 1 γs 4 y [(β (y 1 + δ 5 (β (x 1 δ 5 (λ 1 Download Date 8/18/18 7:9 AM
JAMSI, 11 (015, No. 1 97 Ŝ 6 = s y [ ρs x + γs ρs x + C y δ 6 [δ 6 (β (x 1 (λ 1 γs 4 y [(β (y 1 + δ 6 (β (x 1 δ 6 (λ 1 x Ŝ 7 = s y [ S x + ρ s x + ρ γs y δ 7 [δ 7 (β (x 1 (λ 1 γs 4 y [(β (y 1 + δ 7 (β (x 1 δ 7 (λ 1 Ŝ 8 = s y [ S xs x + γs S x s x + C y δ 8 [δ 8 (β (x 1 (λ 1 γs 4 y [(β (y 1 + δ 8 (β (x 1 δ 8 (λ 1 x Ŝ 9 = s y [ S x + S x γs s x + S y δ 9 [δ 9 (β (x 1 (λ 1 γs 4 y [(β (y 1 + δ 9 (β (x 1 δ 9 (λ 1 x Ŝ 30 = s y [ M ds x + γs M d s x + C y δ 30 [δ 30 (β (x 1 (λ 1 γs 4 y [(β (y 1 + δ 30 (β (x 1 δ 30 (λ 1 x Ŝ 31 = s y [ S x + M d s x + M d (013 γs y δ 31 [δ 31 (β (x 1 (λ 1 γs 4 y [(β (y 1 + δ 31 (β (x 1 δ 31 (λ 1 Ŝ 3 = s y [ β 1(x S x + β (x β 1(x s x + β (x γs y δ 3 [δ 3 (β (x 1 (λ 1 γs 4 y [(β (y 1 + δ 3 (β (x 1 δ 3 (λ 1 Ŝ 33 = s y [ β (x S x + β 1(x β (x s x + β 1(x γs y δ 33 [δ 33 (β (x 1 (λ 1 γs 4 y [(β (y 1 + δ 33 (β (x 1 δ 33 (λ 1 Ŝ 34 = s y [ ρs x + β (x ρs x + β (x γs y δ 34 [δ 34 (β (x 1 (λ 1 γs 4 y [(β (y 1 + δ 34 (β (x 1 δ 34 (λ 1 Download Date 8/18/18 7:9 AM
98 J. G. Ŝ 35 = s y [ β (x S x + ρ β (x s x + ρ γs y δ 35 [δ 35 (β (x 1 (λ 1 γs 4 y [(β (y 1 + δ 35 (β (x 1 δ 35 (λ 1 Ŝ 36 = s y [ S xs x + β (x S x s x + β (x γs y δ 36 [δ 36 (β (x 1 (λ 1 γs 4 y [(β (y 1 + δ 36 (β (x 1 δ 36 (λ 1 Ŝ 37 = s y [ β (x S x + S x β (x s x + S x γs y δ 37 [δ 37 (β (x 1 (λ 1 γs 4 y [(β (y 1 + δ 37 (β (x 1 δ 37 (λ 1 Ŝ 38 = s y [ M ds x + β (x M d s x + β (x γs y δ 38 [δ 38 (β (x 1 (λ 1 γs 4 y [(β (y 1 + δ 38 (β (x 1 δ 38 (λ 1 Ŝ 39 = s y [ β (x S x + M d β (x s x + M d γs y δ 39 [δ 39 (β (x 1 (λ 1 γs 4 y [(β (y 1 + δ 39 (β (x 1 δ 39 (λ 1 Ŝ 40 = s y [ ρs x + β 1(x ρs x + β 1(x γs y δ 40 [δ 40 (β (x 1 (λ 1 γs 4 y [(β (y 1 + δ 40 (β (x 1 δ 40 (λ 1 Ŝ 41 = s y [ β 1(x S x + ρ β 1(x s x + ρ γs y δ 41 [δ 41 (β (x 1 (λ 1 γs 4 y [(β (y 1 + δ 41 (β (x 1 δ 41 (λ 1 Ŝ 4 = s y [ S xs x + β 1(x S x s x + β 1(x γs y δ 4 [δ 4 (β (x 1 (λ 1 γs 4 y [(β (y 1 + δ 4 (β (x 1 δ 4 (λ 1 Ŝ 43 = s y [ β 1(x S x + S x β 1(x s x + S x γs y δ 43 [δ 43 (β (x 1 (λ 1 γs 4 y [(β (y 1 + δ 43 (β (x 1 δ 43 (λ 1 Download Date 8/18/18 7:9 AM
JAMSI, 11 (015, No. 1 99 Ŝ 44 = s y [ M ds x + β 1(x M d s x + β 1(x γs y δ 44 [δ 44 (β (x 1 (λ 1 γs 4 y [(β (y 1 + δ 44 (β (x 1 δ 44 (λ 1 Ŝ 45 = s y [ β 1(x S x + M d β 1(x s x + M d γs y δ 45 [δ 45 (β (x 1 (λ 1 γs 4 y [(β (y 1 + δ 45 (β (x 1 δ 45 (λ 1 Ŝ 46 = s y [ S xs x + ρ S x s x + ρ γs y δ 46 [δ 46 (β (x 1 (λ 1 γs 4 y [(β (y 1 + δ 46 (β (x 1 δ 46 (λ 1 Ŝ 47 = s y [ ρs x + S x γs ρs x + S y δ 47 [δ 47 (β (x 1 (λ 1 γs 4 y [(β (y 1 + δ 47 (β (x 1 δ 47 (λ 1 x Ŝ 48 = s y [ M ds x + ρ S x s x + ρ γs y δ 48 [δ 48 (β (x 1 (λ 1 γs 4 y [(β (y 1 + δ 48 (β (x 1 δ 48 (λ 1 Ŝ 49 = s y [ ρs x + M d γs ρs x + M y δ 49 [δ 49 (β (x 1 (λ 1 γs 4 y [(β (y 1 + δ 49 (β (x 1 δ 49 (λ 1 d Ŝ 50 = s y [ M ds x + S x γs M d s x + S y δ 50 [δ 50 (β (x 1 (λ 1 γs 4 y [(β (y 1 + δ 50 (β (x 1 δ 50 (λ 1 x Ŝ 51 = s y [ S xs x + M d γs S x s x + M y δ 51 [δ 51 (β (x 1 (λ 1 γs 4 y [(β (y 1 + δ 51 (β (x 1 δ 51 (λ 1 d where δ i = S x S ; i = 1,,3,,51; ω x + ω 1 =, ω = β (x, ω 3 = β 1(x, ω 4 = ρ, i ω 5 = S x, ω 6 = M d, ω 7 = Q 1, ω 8 = Q 3, ω 9 = Q r, ω 10 = Q d, ω 11 = Q a, ω 1 = D 1, ω 13 = D, ω 14 = D 3, ω 15 = D 4, ω 16 = D 5, ω 17 = D 6, ω 18 = D 7, ω 19 = D 8, ω 0 = D 9, Download Date 8/18/18 7:9 AM
100 J. G. ω 1 = D 10, ω = β (x, ω 3 = β (x, ω 4 = β 1(x, ω 5 = β 1(x, ω 6 = ρ, ω 7 = ρ ω 8 =, ω S 9 = S x, ω x C 30 =, ω x M 31 = M d, ω d C 3 = β (x, ω x β 33 = β 1(x, ω 1(x β 34 = β (x (x ρ ω 35 = ρ, ω β 36 = β (x, ω (x S 37 = S x, ω x β 38 = β (x, ω (x M 39 = M d,ω d β 40 = β 1(x, ω (x ρ 41 = ρ β 1(x ω 4 = β 1(x S x, ω 43 = S x β 1(x, ω 44 = β 1(x = β 1(x M d, M d ω 49 = M d, ω β 50 = S x and ω 1(x M 51 = M d d S x, ω 45 = M d β 1(x, ω 46 = ρ S x, ω 47 = S x ρ, ω 48 Upadhyaya and Singh (001 have suggested the following modified ratio type variance estimator using the population mean of the auxiliary variable together with its bias and mean squared error are given below: Ŝ 5 = s y [ X x (1.4 B(Ŝ 5 = γs y [C x λ 1 (1.5 MSE(Ŝ 5 = γs 4 y [(β (y 1 + C x λ 1 where λ 1 = μ 1 (1.6 μ 0 μ 0 The modified ratio type variance estimators discussed above are biased but have smaller mean squared errors compared to the traditional ratio type variance estimator suggested by Isaki (1983 under certain conditions. The list of estimators given in Table 1 uses the known values of the parameters and their linear combinations and improved the traditional ratio type estimator. In this paper an attempt has been made to modify the ratio type variance estimator suggested by Upadhyaya and Singh (001 using known parameters of the auxiliary variable and its linear combination. The materials of the present study are arranged as given below. The proposed estimators using known parameters of the auxiliary variable are presented in section where as the conditions in which the proposed estimators perform better than the traditional and existing modified estimators are derived in section 3. The Download Date 8/18/18 7:9 AM
JAMSI, 11 (015, No. 1 101 performance of the proposed estimators with that of the traditional and existing modified estimators are assessed for certain natural populations in section 4 and the conclusion is presented in section 5.. PROPOSED ESTIMATORS In this section we have suggested a class of modified ratio type variance estimators using the known parameters of the auxiliary variable for estimating the population variance of the study variable Y. The proposed class of modified ratio type variance estimators Ŝ pi, i = 1,,,51 for estimating the population variance S y is given below: Ŝ pi = s y [ X +ω i ; i = 1,,3,,51 (.1 x +ω i The bias and mean squared error of the proposed estimators Ŝ pi, i = 1,,,51 have been derived (see Appendix A and are given below Bias(Ŝ pi = (1 f S y (θ pi C x θ pi λ 1 (. n MSE(Ŝ pi = (1 f S y 4 ((β (y 1 + θ pi C x θ pi λ 1 ; i = 1,, 3,,51 (.3 where θ pi = n X X +ω i REMARK.1 When the study variable Y and auxiliary variable X are negatively correlated and the population parameters of the auxiliary variable are known, the following modified product type variance estimators can be proposed: Ŝ pi = s y [ x +ω i i = 1,,3,,51 (.4 X +ω i 3. EFFICIENCY OF THE PROPOSED ESTIMATORS As we mentioned earlier the mean squared error of the traditional ratio type variance estimator S r is given below: MSE(Ŝ R = γs 4 y [(β (y 1 + (β (x 1 (λ 1 (3.1 The mean squared error of the modified ratio type variance estimators S i given in table 1 are represented in single class as given below: Download Date 8/18/18 7:9 AM
10 J. G. MSE(Ŝ i = γs y 4 [(β (y 1 + δ i (β (x 1 δ i (λ 1 ; i = 1,,3,,51 (3. The mean squared error of the modified ratio type variance estimators S 5 suggested by upadhyaya and singh (001 is given below: MSE(Ŝ 5 = γs 4 y [(β (y 1 + C x λ 1 (3.3 The mean squared errors of the proposed modified ratio type variance estimators are given below: MSE(Ŝ pi = (1 f S y 4 ((β (y 1 + θ pi C x θ pi λ 1 ; i = 1,, 3,,51 (3.4 n From the expressions given in (3.1 and (3.4 we have derived (see Appendix B the condition for which the proposed estimators S pi, i = 1,, 3,, 51 are more efficient than the traditional ratio type variance estimator and it is given below: 1 MSE(Ŝ pi < MSE(Ŝ R if θ pi λ 1+((β (x 1 (λ 1+λ 1 (3.5 From the expressions given in (3. and (3.4 we have derived (see Appendix C the conditions for which the proposed estimators S pi, i = 1,, 3,, 51 are more efficient than the modified ratio type variance estimators given in table 1, S i ; i = 1,, 3,, 51 and are given below: 1 MSE(Ŝ pi < MSE(Ŝ i if θ pi λ 1+(δ i (β(x 1 δ i (λ 1+λ 1 (3.6 From the expressions given in (3.3 and (3.4 we have derived (see Appendix D the conditions for which the proposed estimators S pi, i = 1,, 3,, 51 are more efficient than the modified ratio type variance estimator S 5 and are given below: MSE(Ŝ pi MSE( Ŝ 5 either λ 1 1 θ C pi 1(or 1 θ pi λ 1 1 (3.7 x Download Date 8/18/18 7:9 AM
JAMSI, 11 (015, No. 1 103 4. NUMERICAL STUDY The performance of the proposed modified ratio type variance estimators listed in table are assessed with that of traditional ratio type estimator and existing modified ratio type variance estimators listed for certain natural populations. The populations 1 and are taken from Singh and Chaudhary (1986 given in page 177. The population parameters and the constants computed from the above populations are given below: Population 1: Singh and Chaudhary (1986 N = 34 n = 0 Y = 85.641 λ = 1.155 S x = 15.0506 = 0.705 β 1(x λ 1 X = 0.888 S y = 73.3141 β (y = 13.3666 β (x =.913 = 0.873 = 0.3104 M d =15 Q 1 = 9.45 Q 3 = 5.475 Q r = 16.05 Q d = 16.05 Q a = 17.45 D 1 = 7.03 D = 7.68 D 3 = 10.8 D 4 = 1.94 D 5 = 15 D 6 =.7 D 7 = 5.04 D 8 = 33.56 D 9 = 43.61 D 10 = 56.4 Population : Singh and Chaudhary (1986 N = 34 n = 0 Y = 85.641 X = 19.9441 S y = 73.3141 β λ = 1.44 S x = 15.015 = 0.753 (y β (x = 13.3666 = 3.757 β 1(x = 1.758 λ 1 = 0.946 M d = 14.5 Q 1 = 9.95 Q 3 = 7.8 Q a Q r = 17.875 Q d = 8.9375 = 18.865 D 1 = 6.06 D = 8.3 D 3 = 10.7 D 4 = 11.1 D 5 = 14.5 D 6 = 1.0 D 7 = 6.45 D 8 = 30.44 D 9 = 37.3 D 10 = 63.4 5. The mean squared error of the existing and proposed modified ratio type variance estimators for the above population given below: Download Date 8/18/18 7:9 AM
104 J. G. Table MSE(. of the existing and proposed modified ratio type variance estimators for population 1 Existing Proposed Bias(. MSE(. Estimator Estimator Bias(. MSE(. Ŝ 1 193.4567 8305040.947 Ŝ p1 77.614 7901409.589 Ŝ 189.6191 88577.533 Ŝ p 65.9785 78703.643 Ŝ 3 193.1857 8303644.97 Ŝ p3 76.6955 7895569.07 Ŝ 4 193.9398 830759.556 Ŝ p4 79.931 791069.8376 Ŝ 5 170.47 8185653.1155 Ŝ p5 33.7963 761460.8707 Ŝ 6 170.319 8186036.5850 Ŝ p6 33.8714 7615115.3167 Ŝ 7 194.999 8309384.718 Ŝ p7 80.580 79030.8059 Ŝ 8 187.6845 875317.6357 Ŝ p8 61.0748 7795375.086 Ŝ 9 193.710 8304084.3017 Ŝ p9 76.989 7897396.779 Ŝ 10 19.5865 8300558.7975 Ŝ p10 74.713 7883000.6155 Ŝ 11 191.8956 897000.4445 Ŝ p11 7.5377 7869069.1599 Ŝ 1 193.696 8305931.417 Ŝ p1 78.088 7905186.948 Ŝ 13 194.657 83115.1070 Ŝ p13 81.8881 798514.0560 Ŝ 14 161.9435 814307.587 Ŝ p14 6.7408 756587.9971 Ŝ 15 194.6569 83113.6155 Ŝ p15 81.8870 798507.658 Ŝ 16 16.0399 814351.8877 Ŝ p16 6.8101 7566357.506 Ŝ 17 188.8916 881531.8967 Ŝ p17 64.0680 7814716.766 Ŝ 18 194.06 8308901.9409 Ŝ p18 80.4 7918089.0587 Ŝ 19 183.5978 85485.7975 Ŝ p19 5.346 7738509.3354 Ŝ 0 194.4667 831043.705 Ŝ p0 81.1867 794073.4637 Ŝ 1 194.3964 8309881.4741 Ŝ p1 80.9301 79448.0103 Ŝ 185.784 865536.87 Ŝ p 56.7674 7767404.116 Ŝ 3 194.395 8309875.4575 Ŝ p3 80.959 7941.0747 Ŝ 4 185.8133 865686.6605 Ŝ p4 56.899 7767811.3938 Ŝ 5 191.3000 89393.9140 Ŝ p5 70.7307 7857516.3340 Ŝ 6 193.836 8306930.9379 Ŝ p6 78.8843 7909475.933 Ŝ 7 194.6390 8311131.4101 Ŝ p7 81.807 798087.7466 Ŝ 8 167.068 816938.3846 Ŝ p8 30.8108 759413.3077 Ŝ 9 194.6386 831119.607 Ŝ p9 81.8194 798079.58 Ŝ 30 167.1514 8169755.4807 Ŝ p30 30.8840 759467.716 Ŝ 31 194.6895 8311391.6909 Ŝ p31 8.0080 79973.47 Ŝ 3 145.8871 8060755.641 Ŝ p3 17.9756 750330.8713 Ŝ 33 194.6893 8311390.7610 Ŝ p33 8.0073 79968.985 Ŝ 34 146.000 8061435.6153 Ŝ p34 18.0305 75037.1084 Download Date 8/18/18 7:9 AM
JAMSI, 11 (015, No. 1 105 Ŝ 35 19.9551 830457.393 Ŝ p35 75.968 7890677.815 Ŝ 36 19.9670 830518.7994 Ŝ p36 75.966 789099.0386 Ŝ 37 178.8485 89857.373 Ŝ p37 44.3384 7685618.941 Ŝ 38 155.858 8111656.3586 Ŝ p38.8150 7538189.735 Ŝ 39 168.783 817819.8387 Ŝ p39 3.3710 7604849.6035 Ŝ 40 181.087 841370.6859 Ŝ p40 47.870 7709041.6847 Ŝ 41 166.7609 8167750.7848 Ŝ p41 30.546 75973.871 Ŝ 4 18.703 849683.3804 Ŝ p4 50.6830 777586.6369 Ŝ 43 181.645 84440.615 Ŝ p43 48.8148 771579.6791 Ŝ 44 176.6576 818593.1860 Ŝ p44 41.404 7664933.877 Ŝ 45 173.40 801861.6358 Ŝ p45 37.1913 7637676.776 Ŝ 46 170.319 8186036.5850 Ŝ p46 33.8714 7615115.3167 Ŝ 47 159.459 813084.646 Ŝ p47 5.0396 7553934.3985 Ŝ 48 156.3915 8114555.73 Ŝ p48 3.145 7540518.755 Ŝ 49 145.8315 8060471.0773 Ŝ p49 17.957 7503153.4053 Ŝ 50 134.6559 8003341.091 Ŝ p50 14.0431 7474185.547 1.1457 7939536.8330 Ŝ p51 10.8871 7450090.5703 Ŝ 51 Ŝ 5 8.071 7930533.085 Ŝ R 15.5580 8311667.4056 Download Date 8/18/18 7:9 AM
106 J. G. Table 3 MSE(. of the existing and proposed modified ratio type variance estimators for population Existing Proposed Bias(. MSE(. Estimator Estimator Bias(. MSE(. Ŝ 1 74.8748 8700066.3664 Ŝ p1 81.9543 793335.0831 Ŝ 67.4800 866043.5448 Ŝ p 65.611 7817638.5301 Ŝ 3 73.5533 869369.999 Ŝ p3 78.5355 790186.3797 Ŝ 4 75.6578 8704093.433 Ŝ p4 84.0855 7936710.50 Ŝ 5 41.8679 853054.595 Ŝ p5 34.4309 7616001.5114 Ŝ 6 43.501 8538919.0760 Ŝ p6 35.6790 764408.394 Ŝ 7 76.783 870785.444 Ŝ p7 85.8340 7947666.4656 Ŝ 8 64.567 8646864.7198 Ŝ p8 59.9815 7783806.406 Ŝ 9 75.885 870194.113 Ŝ p9 83.070 7930341.3699 Ŝ 10 7.508 8687867.1404 Ŝ p10 75.9674 788566.4956 Ŝ 11 7.5089 8687898.6943 Ŝ p11 75.980 7885718.985 Ŝ 1 75.863 87018.9335 Ŝ p1 83.0643 7930304.040 Ŝ 13 76.6674 870986.6597 Ŝ p13 86.9579 7954701.1868 Ŝ 14 31.818 8479004.965 Ŝ p14 7.978 7571999.8359 Ŝ 15 76.6604 870950.887 Ŝ p15 86.9377 7954574.911 Ŝ 16 33.8546 8489465.8908 Ŝ p16 9.1347 7579983.967 Ŝ 17 69.4550 867196.406 Ŝ p17 69.1858 784651.487 Ŝ 18 75.900 870544.4919 Ŝ p18 84.8179 7941301.8389 Ŝ 19 56.5008 8605634.933 Ŝ p19 48.4540 7709070.5888 Ŝ 0 76.4897 870837.89 Ŝ p0 86.440 795147.719 Ŝ 1 76.1611 870668.545 Ŝ p1 85.4996 794557.489 Ŝ 66.7349 865813.4456 Ŝ p 63.8659 7808719.175 Ŝ 3 76.168 8706506.0880 Ŝ p3 85.401 7944961.8077 Ŝ 4 67.385 866080.54 Ŝ p4 64.8041 7814718.1590 Ŝ 5 69.5908 867894.5544 Ŝ p5 69.4685 7844448.5663 Ŝ 6 75.903 8705356.1073 Ŝ p6 84.7708 7941006.146 Ŝ 7 76.578 870888.800 Ŝ p7 86.6986 7953078.3599 Ŝ 8 48.8531 8566375.7830 Ŝ p8 40.619 7655051.18 Ŝ 9 76.5665 8708767.7148 Ŝ p9 86.6644 795864.459 Ŝ 30 50.1860 857316.487 Ŝ p30 41.536 7663513.406 Ŝ 31 76.7199 8709556.8046 Ŝ p31 87.1113 7955660.834 Ŝ 3 06.6709 8350388.1091 Ŝ p3 17.7903 7500315.7683 Ŝ 33 76.7158 8709535.6494 Ŝ p33 87.0993 7955585.595 Ŝ 34 09.6060 8365383.3370 Ŝ p34 18.684 7506746.177 Download Date 8/18/18 7:9 AM
JAMSI, 11 (015, No. 1 107 Ŝ 35 74.117 8696146.6470 Ŝ p35 79.9563 7910773.1951 Ŝ 36 74.3795 8697518.944 Ŝ p36 80.6475 791511.084 Ŝ 37 5.961 8587461.1638 Ŝ p37 44.3855 76835.0994 Ŝ 38 16.9786 840307.861 Ŝ p38 1.11 75480.39 Ŝ 39 35.9633 850071.5781 Ŝ p39 30.4083 7588699.7565 Ŝ 40 55.1960 8598934.734 Ŝ p40 46.899 769884.0318 Ŝ 41 33.9686 8490049.77 Ŝ p41 9.015 758044.3507 Ŝ 4 61.8741 863334.991 Ŝ p4 55.7610 7756593.1089 Ŝ 43 56.6540 860641.48 Ŝ p43 48.64 7710301.95 Ŝ 44 5.187 8583487.6588 Ŝ p44 43.568 767693.4831 Ŝ 45 50.947 8573774.1884 Ŝ p45 41.648 766419.6974 Ŝ 46 43.501 8538919.0760 Ŝ p46 35.6790 764408.394 Ŝ 47 9.6949 8468156.743 Ŝ p47 6.836 7564156.365 Ŝ 48 19.477 8415600.513 Ŝ p48.1574 7531487.3776 Ŝ 49 1.304 8379163.5680 Ŝ p49 19.5567 7513015.7817 Ŝ 50 00.781 830314.8836 Ŝ p50 16.167 748850.58 164.4345 813586.8030 Ŝ p51 9.4709 7438144.1980 Ŝ 51 Ŝ 5 87.3340 7957053.4638 Ŝ R 10.6359 8709947.6355 From the values of table and table 3, it is observed that the bias of the proposed modified ratio type variance estimators are less than the bias of the traditional and existing modified ratio type variance estimators. Similarly, it is observed that the mean squared error of the proposed modified ratio type variance estimators are less than the mean squared error of the traditional and existing modified ratio type variance estimators. 6. CONCLUSION In this paper a class of modified ratio type variance estimators has been proposed using the known parameters of the auxiliary variable. The bias and mean squared error of the proposed modified ratio type variance estimators are derived. Further we have derived the conditions for which the proposed estimators are more efficient than the traditional and existing modified ratio type variance estimators. We have also assessed the performances of the proposed estimators with that of the existing estimators for two natural populations. It is observed from the numerical comparison Download Date 8/18/18 7:9 AM
108 J. G. that the bias and mean squared error of the proposed estimators are less than the bias and mean squared error of the traditional and existing estimators. Hence we strongly recommend that the proposed modified ratio type variance estimators may be preferred over the traditional ratio type variance estimator and modified ratio type variance estimators for the use of practical applications. ACKNOWLEDGMENT The first author wishes to record his gratitude and thanks to UGC-MRP, New Delhi, for the financial assistance. REFERENCES [1 AGARWAL, M.C. and SITHAPIT, A.B. 1995. Unbiased ratio type estimation. Statistics and Probability Letters, 5: 361-364 [ AHMED, M.S., RAMAN, M.S. and HOSSAIN, M.I. 000. Some competitive estimators of finite population variance multivariate auxiliary information. Information and Management Sciences, 11 (1: 49-54 [3 AL-JARARHA, J. and AL-HAJ EBRAHEM, M. 01. A ratio estimator under general sampling design. Austrian Journal of Statistics, 41(: 105-115 [4 ARCOS, A., RUEDA, M., MARTINEZ, M.D., GONZALEZ, S. and ROMAN, Y. 005. Incorporating the auxiliary information available in variance estimation. Applied Mathematics and Computation, 160: 387-399 [5 COCHRAN, W. G. 1977: Sampling techniques, Third Edition, Wiley Eastern Limited [6 DAS, A.K. and TRIPATHI, T.P. 1978. Use of auxiliary information in estimating the finite population variance. Sankhya, 40: 139-148 [7 GARCIA, M.K. and CEBRAIN, A.A. 1997. Variance estimation using auxiliary information: An almost unbiased multivariate ratio estimator. Metrika, 45: 171-178 [8 GUPTA, S. and SHABBIR, J. 008. Variance estimation in simple random sampling using auxiliary information. Hacettepe Journal of Mathematics and Statistics, 37: 57-67 [9 ISAKI, C.T. 1983. Variance estimation using auxiliary information. Journal of the American Statistical Association, 78: 117-13 [10 KADILAR, C. and CINGI, H. 006a. Improvement in variance estimation using auxiliary information. Hacettepe Journal of Mathematics and Statistics, 35 (1: 111-115 [11 KADILAR, C. and CINGI, H. 006b. Ratio estimators for population variance in simple and stratified sampling. Applied Mathematics and Computation, 173: 1047-1058 [1 MURTHY, M.N. 1967. Sampling theory and methods. Statistical Publishing Society, Calcutta, India [13 PRASAD, B. and SINGH, H.P. 1990. Some improved ratio type estimators of finite population variance in sample surveys. Communication in Statistics: Theory and Methods, 19: 117-1139 [14 REDDY, V.N. 1974. On a transformed ratio method of estimation, Sankhya C, 36: 59-70 [15 SHABBIR, J. and GUPTA, S. 006. On estimation of finite population variance. Journal of Interdisciplinary Mathematics, 9(, 405-419 [16 SINGH, D. and CHAUDHARY, F.S. 1986. Theory and analysis of sample survey designs. New Age International Publisher Download Date 8/18/18 7:9 AM
JAMSI, 11 (015, No. 1 109 [17 SINGH, H.P. and SOLANKI, R.S. 013. A new procedure for variance estimation in simple random sampling using auxiliary information. Statistical Papers, 54, 479-497 [18 SINGH, H.P., UPADHYAYA, U.D. and NAMJOSHI, U.D. 1988. Estimation of finite population variance. Current Science, 57: 1331-1334 [19 SISODIA, B.V.S. and DWIVEDI, V.K. 1981. A modified ratio estimator using coefficient of variation of auxiliary variable. Journal of the Indian Society of Agricultural Statistics, 33(1: 13-18 [0 SUBRAMANI, J. and KUMARAPANDIYAN, G. 01a. Variance estimation using median of the auxiliary variable. International Journal of Probability and Statistics, Vol. 1(3, 36-40 [1 SUBRAMANI, J. and KUMARAPANDIYAN, G. 01b. Variance estimation using quartiles and their functions of an auxiliary variable, International Journal of Statistics and Applications, 01, Vol. (5, 67-4 [ SUBRAMANI, J. and KUMARAPANDIYAN, G. 01c. Estimation of variance using deciles of an auxiliary variable. Proceedings of International Conference on Frontiers of Statistics and Its Applications, Bonfring Publisher, 143-149 [3 SUBRAMANI, J. and KUMARAPANDIYAN, G. 013. Estimation of variance using known coefficient of variation and median of an auxiliary variable. Journal of Modern Applied Statistical Methods, Vol. 1(1, 58-64 [4 TAILOR, R. and SHARMA, B. 01. Modified estimators of population variance in presence of auxiliary information. Statistics in Transition-New series, 13(1, 37-46 [5 UPADHYAYA, L. N. and SINGH, H.P. 006. Almost unbiased ratio and product-type estimators of finite population variance in sample surveys. Statistics in Transition, 7 (5: 1087 1096 [6 UPADHYAYA, L.N. and SINGH, H.P. 1999. An estimator for population variance that utilizes the kurtosis of an auxiliary variable in sample surveys. Vikram Mathematical Journal, 19, 14-17 [7 UPADHYAYA, L.N. and SINGH, H.P. 001. Estimation of population standard deviation using auxiliary information. American Journal of Mathematics and Management Sciences, 1(3-4, 345-358 [8 WOLTER, K.M. 1985. Introduction to Variance Estimation. Springer-Verlag [9 YADAV, S.K. and KADILAR, C. 013a. A class of ratio-cum-dual to ratio estimator of population variance. Journal of Reliability and Statistical Studies, 6(1, 9-34 [30 YADAV, S.K. and KADILAR, C. 013b. Improved Exponential type ratio estimator of population variance. Colombian Journal of Statistics, 36(1, 145-15 Download Date 8/18/18 7:9 AM
110 J. G. Appendix-A We have derived the expression for the bias and mean squared error of the proposed estimators Ŝ pi : i = 1,,3,,51 to first order of approximation with the following notations: Let e 0 = s y S y x X and e S 1 = y X. Further we can write s y = S y (1 + e 0 and x = X (1 + e 1 and from the definition of e 0 and e 1 we obtain: E[e 0 = E[e 1 = 0 (1 f E[e 0 = (β n (y 1 (1 f E[e 1 = C n x (1 f E[e 0 e 1 = λ n 1 The bias of the proposed estimators Ŝ pi : i = 1,,3,,51 is derived as given below: X + η Ŝ pi = s y [ i ; i = 1,,3,,51 x + η i Ŝ pi = Ŝ pi = Ŝ pi s y (X + e 1 X + η i (X + η i s y (X (X + η i (1 + e + η i 1X X + η i s y = (1 + θ pi e 1 where θ p i = X Ŝ pi = s y (1 + θ pi e 1 1 X + η i Ŝ pi = s y (1 θ pi e 1 + θ pi e 1 θ 3 pi e 3 1 + Neglecting the terms more than nd order, we will get Ŝ pi = s y (1 θ pi e 1 + θ pi e 1 Ŝ pi = (S y (1 + e 0 (1 θ pi e 1 + θ pi e 1 Ŝ pi = (S y + S y e 0 (1 θ pi e 1 + θ pi e 1 Ŝ pi = S y + S y e 0 S y θ pi e 1 S y θ pi e 0 e 1 + S y θ pi e 1 + S y θ pi e 0 e 1 Neglecting the terms more than 3 rd order, we will get Download Date 8/18/18 7:9 AM
JAMSI, 11 (015, No. 1 111 Ŝ pi = S y + S y e 0 S y θ pi e 1 S y θ pi e 0 e 1 + S y θ pi e 1 Ŝ pi S y = S y e 0 S y θ pi e 1 S y θ pi e 0 e 1 + S y θ pi e 1 Taking expectation on both sides, we will get E(Ŝ pi S y = S y E(e 0 S y θ pi E(e 1 S y θ pi E(e 0 e 1 + S y θ pi E(e 1 Bias(Ŝ pi = S y θ pi E(e 1 S y θ pi E(e 0 e 1 Bias(Ŝ pi = S y θ pi Bias(Ŝ pi = (1 f C n x S y (1 f λ n 1 θ pi (1 f (S n y θ pi C x S y λ 1 θ pi Bias(Ŝ pi = γs y (θ pi C x θ pi λ 1 where θ pi = X X + η i The mean squared error of the class of proposed estimators Ŝ pi : i = 1,, 3,,51 to first order of approximation is derived as given below: X + η Ŝ pi = s y [ i ; i = 1,,3,,51 x + η i Ŝ pi = Ŝ pi = Ŝ pi s y (X + e 1 X + η i (X + η i s y (X (X + η i (1 + e + η i 1X X + η i s y = (1 + θ pi e 1 where θ p i = X Ŝ pi = s y (1 + θ pi e 1 1 X + η i Ŝ pi = s y (1 θ pi e 1 + θ pi e 1 θ 3 pi e 3 1 + Neglecting the terms more than 1 st order, we will get Ŝ pi = s y (1 θ pi e 1 Ŝ pi = (S y (1 + e 0 (1 θ pi e 1 Ŝ pi = (S y + S y e 0 (1 θ pi e 1 Ŝ pi = S y + S y e 0 S y θ pi e 1 S y θ pi e 0 e 1 Ŝ pi S y = S y e 0 S y θ pi e 1 S y θ pi e 0 e 1 Download Date 8/18/18 7:9 AM
11 J. G. Squaring both sides (Ŝ pi S y = (S y e 0 S y θ pi e 1 S y θ pi e 0 e 1 Neglecting the terms more than nd order, we will get (Ŝ pi S y = S 4 y e 0 + S 4 y θ pi e 1 S 4 y θ pi e 0 e 1 Taking expectation on both sides we will get: E(Ŝ pi S y = S 4 y E(e 0 + S 4 y θ pi E(e 1 S 4 y θ pi E(e 0 e 1 MSE(Ŝ pi = MSE(Ŝ pi = (1 f (S 4 n y (β (y 1 + S 4 y θ pi C 4 x S y θ pi λ 1 (1 f S 4 n y ((β (y 1 + θ pi C x θ pi λ 1 ; i = 1,, 3,,51 MSE(Ŝ pi = γs 4 y ((β (y 1 + θ pi C x θ pi λ 1 ; i = 1,, 3,,51 Appendix-B The conditions for which proposed estimators Ŝ pi perform better than the traditional ratio type variance estimator Ŝ R are derived and are given below: For MSE(S pi MSE( S R γs 4 y [(β (y 1 + θ pi C x θ pi λ 1 γs 4 y [(β (y 1 + (β (x 1 (λ 1 ((β (y 1 + θ pi C x θ pi λ 1 [(β (y 1 + (β (x 1 (λ 1 θ pi C x θ pi λ 1 (β (x 1 (λ 1 θ pi C x θ pi λ 1 + λ 1 λ 1 (β (x 1 (λ 1 (θ pi λ 1 (β (x 1 (λ 1 + λ 1 (θ pi λ 1 ((β (x 1 (λ 1 + λ 1 θ pi λ 1 + ((β (x 1 (λ 1 + λ 1 λ 1 + ((β (x 1 (λ 1 + λ 1 θ pi 1 1 1 λ 1 + ((β (x 1 (λ 1 + λ 1 That is, MSE(Ŝ pi MSE( Ŝ R if θ pi 1 Download Date 8/18/18 7:9 AM
JAMSI, 11 (015, No. 1 113 Appendix-C The conditions for which proposed estimators Ŝ pi perform better than the existing modified ratio type variance estimators Ŝ i are derived and are given below: For MSE(S pi MSE( S i γs 4 y [(β (y 1 + θ pi C x θ pi λ 1 γs 4 y [(β (y 1 + δ i (β (x 1 δ i (λ 1 ((β (y 1 + θ pi C x θ pi λ 1 [(β (y 1 + δ i (β (x 1 δ i (λ 1 θ pi C x θ pi λ 1 δ i (β (x 1 δ i (λ 1 θ pi C x θ pi λ 1 + λ 1 λ 1 δ i (β (x 1 δ i (λ 1 (θ pi λ 1 δ i (β (x 1 δ i (λ 1 + λ 1 (θ pi λ 1 (δ i (β (x 1 δ i (λ 1 + λ 1 θ pi λ 1 + (δ i (β (x 1 δ i (λ 1 + λ 1 θ pi λ 1 + (δ i (β (x 1 δ i (λ 1 + λ 1 1 1 1 That is, MSE(Ŝ pi MSE(Ŝ i if θ pi λ 1 + (δ i (β (x 1 δ i (λ 1 + λ 1 1 Appendix-D The conditions for which proposed estimators Ŝ pi perform better than the existing modified ratio type variance estimator Ŝ 5 are derived and are given below: For MSE(S pi MSE( S 5 γs 4 y [(β (y 1 + θ pi C x θ pi λ 1 γs 4 y [(β (y 1 + C x λ 1 ((β (y 1 + θ pi C x θ pi λ 1 [(β (y 1 + C x λ 1 θ pi C x θ pi λ 1 C x λ 1 θ pi C x θ pi λ 1 C x + λ 1 0 (θ pi 1C x λ 1 (θ pi 1 0 (θ pi + 1(θ pi 1 λ 1 (θ pi 1 0 (θ pi 1[(θ pi + 1 λ 1 0 Condition 1: (θ pi 1 0 and [(θ pi + 1 λ 1 0 θ pi 1 and (θ pi + 1 λ 1 Download Date 8/18/18 7:9 AM
114 J. G. θ pi 1 and θ pi λ 1 1 λ 1 1 θ pi 1 Condition : (θ pi 1 0 and [(θ pi + 1 λ 1 0 θ pi 1 and (θ pi + 1 λ 1 θ pi 1 and θ pi λ 1 1 1 θ pi λ 1 1 That is, MSE(Ŝ pi MSE( Ŝ 5 either λ 1 1 θ C pi 1(or 1 θ pi λ 1 1 x J. Subramani Department of Statistics Ramanujan School of Mathematical Sciences, Pondicherry University R V Nagar, Kalapet, Puducherry 605014, India. Email: drjsubramani@yahoo.co.in G. Department of Statistics Ramanujan School of Mathematical Sciences, Pondicherry University R V Nagar, Kalapet, Puducherry 605014, India. Email: kumarstat88@gmail.com Download Date 8/18/18 7:9 AM