An improved class of estimators for nite population variance

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1 Hacettepe Journal of Mathematics Statistics Volume An improved class of estimators for nite population variance Mazhar Yaqub Javid Shabbir Abstract We propose an improved class of estimators in estimating the nite population variance using the auxiliary information. The expressions for the bias mean squared error of the proposed class of estimators are derived up to the rst order of approximation. Some estimators are also derived from a proposed class by allocating the suitable values of known parameters identied as particular members of the proposed class of estimators. A numerical study is carried out to demonstrate performances of the estimators. It is observed that the proposed class of estimators is more ecient than the usual sample mean estimator the regression estimator suggested by Isaki 1983 Shabbir Gupta 007 Singh Solanki 013b Yadav et al. 013 Yadav Kadilar 014 Singh Malik 014 estimators. Keywords: Auxiliary variable; bias; mean squared error; population variance; percentage relative eciency. 000 AMS Classication: Primary 6D05 Secondary 6P0. Received : Accepted : Doi : /HJMS Introduction In this article an improved class of estimators is proposed in estimating the nite population variance under simple rom sampling. Various elds of life like genetics biology medical studies have been facing the problem in estimating the nite population variance. An agriculturist requires sucient knowledge of climatic variation to devise appropriate plan for cultivating his crop. A fair understing of variability is vitally important for better results in dierent walks of life. Singh et al. [36] Das Tripathi [9] have proposed dierent estimators for nite population variance S y. Corresponding Author. Department of Statistics Quaid-i-Azam University Islamabad Pakistan. javidshabbir@gmail.com Department of Statistics Quaid-i-Azam University Islamabad Pakistan.

2 164 Chaudhury [8] Mukhopadhyay [7 8] have made signicant contributions in estimating the nite population variance under super population models. Srivastava Jhajj [47] Wu [51] have taken the advantage of correlation between the study the auxiliary variables in estimating the nite population variance. Isaki [17] has proposed ratio regression type estimators for population variance. Likewise Singh [37] Searls Intrapanich [31] Prasad Singh [9 30] have given some estimators for population variance. Singh Biradar [39] Garcia Cebrain [10] Cebrain Garcia [5] Singh Joarder [40] Upadhyaya Singh [49] Ahmed et al. [] have paid their attention towards the improved estimation or classes of estimators of Sy. AL-Jaraha Ahmed [3] have given some chain ratio-type as well as chain product-type estimators of Sy using double-sampling scheme. Later on Singh Singh [41] Upadhyaya et al. [50] Chra Singh [7] Arcos et al. [1] Kadilar Cingi [ ] Koyuncu Kadilar [4 5] Singh Vishwakarma [43] Turgut Cingi [48] Gupta Shabbir [ ] Shabbir Gupta [33 34] Grover [11] Singh et al. [ ] Yadav et al. [5 53] Yadav Kadilar [54] Singh Solanki [45 46] Singh Malik [35] have paid their attention towards the improved estimation of population variance Sy. Motivated by these studies the present article focuses on improved class of estimators for Sy using the auxiliary information. The rest of the article is organized as: Section provides the notations symbols. Section 3 gives a brief review of some existing estimators of Sy. Section 4 gives the expressions for the bias mean squared error MSE of the proposed class of estimators. The eciency comparison of dierent estimators is shown in Section 5. A numerical study is presented in Section 6. Conclusion is given in Section 7.. Notations Consider a nite population Ω = 1.. i.. N} having N units. We draw a sample of size n by using simple rom sample without replacement SRSWOR sampling scheme from this population. Let y i x i be the values of the study variable y the auxiliary variable x respectively. Let ȳ = 1 n n i=1 yi x = 1 n n i=1 xi be the sample means respectively corresponding to the population means Ȳ = 1 N N i=1 yi X = 1 N xi. Let N i=1 s y = 1 n n 1 i=1 yi ȳ s x = 1 n n 1 i=1 xi x be the sample variances corresponding to population variances Sy = 1 N N 1 i=1 yi Ȳ Sx = 1 N N 1 i=1 xi X µ respectively. Let θ = 11 µ0 µ0 be the covariance between S y S x. Let β y = µ 40 µ 0 β x = µ 04 µ 0 be the population coecients of kurtosis of y x respectively where µ rs = 1 N 1 N i=1 yi Ȳ r x i X s γ = 1/n. We ignored the nite population correctionfpc term because of ease of computation. In order to get biases MSEs of the considered estimators we use the following relative error terms. Let δ 0 = s y S y δ Sy 1 = s x S x such that E δ Sx i = 0 for i = 0 1. Let E δ0 = γ βy 1 = V0 E δ1 = γ βx 1 = V0 E δ 0δ 1 = γ θ 1 = V 11 where V rs = E s y S y r s x S x s S y r S x s }. 3. Some existing estimators We discuss the following estimators. i The variance of the usual unbiased variance estimator Ŝ y is given by

3 3.1 V ar Ŝ y = SyV 4 0. ii Isaki [17] suggested the following regression estimator for S y given by 3. Ŝ Reg = s y + b s y s x S x s x 1643 where b s y s is the sample regression coecient whose population regression coecient x is β = SyV 11/SxV 0. The variance of ŜReg is given by 3.3 V arŝ Reg = SyV ρ = s y s x MSEŜ Reg where ρ s y s x = V 11/ V 0V 0. iii Singh et al. [38] considered the following dierence type estimator for S y given by 3.4 Ŝ d = k 1s y + k S x s x where k 1 k are suitably chosen constants. The bias minimum MSE of Ŝ d to rst order of approximation at optimum values k opt V 1 = 0 V 0 +V 0 V 0 V11 kopt = S x V 11 Sy V 0 +V 0 V 0 V11 are given by BiasŜ d = k 1 1 S y 3.5 MSE minŝ d V = arŝ Reg 1 + Sy 4 V arŝ Reg. iv Shabbir Gupta [3] suggested the following ratio-type exponential estimator Sy given by 3.6 ŜSG = k 3s y + k 4 S x s } S x exp x s x Sx + s x where k 3 k 4 are suitably chosen constants. The bias of Ŝ SG to rst order of approximation is given by Bias Ŝ SG = S y k k3s yv 0 1 k3s yv k4s xv 0. The minimum MSE of Ŝ SG to rst order of approximation at optimum values of k opt 3 = V 0 8 V 0 k opt 8 V 0 +V 0 V 0 V11 4 = S y 8Sx V 0 +V 0 V 0 V11 is given by 3.7 MSE min Ŝ SG = S 4 y 64 4V0 +V 0 +8V 11 V 11V 0 +4V 0 V 0 4V 11 } V V 0 4 Sy 4 V arŝ Reg 1 Sy 4 V arŝ Reg. v Singh Solanki [46] suggested a dierence-in-ratio type estimator for Sy given by 3.8 ŜSS = k 5s y + k 6Sx s x } asx + b as x + b where k 5 k 6 are suitably chosen constants a 0 b are functions of known parameters of the auxiliary variable x. The bias of Ŝ SS to rst order of approximation is given by Bias Ŝ SS = S y k k 5S yτ V 0 k 5S yτv 11 + k 6S xτv 0

4 1644 where τ = asx/ asx + b. The minimum MSE of Ŝ SS to rst order of approximation at optimum values k opt V 5 = 0 1+τ V 0 V 0 V 0 V 0 +τ V0 +V 11 4 k opt 6 = S y V0 τ+v Sx 0 τ 3 +V 11 V 11 V 0 τ +τv 0 V 0 τv11 V 0 V 0 V 0 +τ V0 +V 11 is given by 3.9 MSE min Ŝ SS = S 4 y where A = 1 V 0τ. ASy 4 V arŝ Reg A + Sy 4 V arŝ Reg vi Yadav et al. [53] suggested a general class of estimators for Sy given by 3.10 ŜY G = k 7s y + k 8Sx s x } } as λ x + b a S x sx + 1 λ exp as x + b a Sx + s x + b where k 7 k 8 are suitably chosen constants λ can takes values 0 or 1 a b be the population parameters of the auxiliary variables. Shabbir Gupta [3] estimator in 3.6 Singh Solanki [46] estimator in 3.8 can be generated from 3.10 by substituting the suitable choices of λ a b. The bias of Ŝ Y G to rst order of approximation is given by [ Bias Ŝ Y G = S y k λ 8 } k 7V 0τ 1 + λ τ + S x S y k 8V 0 k 7V 11 }]. The minimum MSE of Ŝ Y G to rst order of approximation at optimum values is given by k opt 7 = V0 k opt 8 = S y S x 8 V 0τ 1 + 3λ + 4λ } 4V 0 V0 λτ 1 + 3λ + 4V 0V 0 4V11 8V V0λτ 1 + λ 4V 0τ 1 + λ +V0τ λ 3 V 0V 11τ 1 + 3λ + 4λ +4V 0V 0τ 1 + λ 4V λ 4V 0 1 V 0τ λ V0 τ λ 1 + λ + 4V 0V 0 4V λ 4λ 1 + λ λ } V0τ MSE min Ŝ Y G = Sy 4 +16Sy 4 V arŝ Reg 1 + λ V 0τ 4 } λ τ V 0 + λτ V 0 4Sy 4 V arŝ Reg. vii Yadav Kadilar [54] suggested two parameters ratio-product-ratio type estimator for Sy given by [ } }] 1 β1 3.1 ŜY K = s s x + β 1Sx β1s x + 1 β 1 Sx y α α 1 β 1s x + 1 β 1 Sx 1 β 1 s x + β 1Sx

5 1645 where α 1 β 1 are suitably chosen constants. The bias MSE of Ŝ Y K to rst order of approximation are given by Bias Ŝ Y K = S y V0 1 α1 3β 1 + α 1β 1 + β 1 V11 1 α 1 β 1 + 4α 1β 1 } S y MSE } V0 + V 0 V Ŝ Y K = S α 1β 1V 0 1 α 1 β 1 + α 1β 1 y +4V 11 α 1 β 1 + 4V 0 α1 β 1 + α1 + β1. Solving above for minimum MSE of Ŝ Y K to rst order of approximation at α 1 β 1 = 1/ 1/ is 3.13 MSE min Ŝ Y K = V ar Ŝ y at α 1 β 1 = V 0 V 11 /V 0 0} we have 3.14 MSE min Ŝ Y K = V ar Ŝ Reg. viii Recently Singh Malik [35] suggested an improved estimator for Sy given by 3.15 ŜSM = s y k9 + k 10Sx s x } asx + b as x + b exp ψ 1 asx + b + as x + b where k 9 k 10 are suitably chosen constants. Here ψ 1 is the scalar quantity which takes the values +1 1 for ratio product type estimators respectively. The bias of Ŝ SM to rst order of approximation is given by Bias Ŝ SM = S y k S yk 9ψ 1τ V S yk 9γ 1τ V S ys xk 10ψ 1τV 0 1 S ys xk 9ψ 1τV 11 S ys xk 10V 11. The minimum MSE of Ŝ SM to rst order of approximation at optimum values k opt 9 = 1 1ψ1τV 0V ψ1τ V0 + 16V11 8V 0 ψ 1τ V0 4 ψ1 τ V0 4ψ1τV0V11 + 8V11 V0V0 V0 ψ1τ V0 k opt 10 = 1 4Sx given by MSE min Ŝ SM = S 4 y or at τ = ψ 1 = 1 we have 3.16 MSE min Ŝ SM = S 4 y Proposed estimator 6ψ 1 τ V 0 V 11 +ψ 1 τ 3 V 0 +8ψ 1τV 11 4ψ 1τV 0 +8V 11 8V 0 V 11 +4ψ 1 τv 0 V 0 ψ 1 τ V 0 4ψ 1τV 0 V 11 +8V 11 V 0V 0 V 0 ψ 1 τ V 0 3ψ 1 τ V 0 4ψ 1τV 0 V 11 +8V 11 V 0V 0 V 0 ψ 1 τ V 0 is } V0 V 0V 0 + 8V V 0 4V arŝ Reg + 16V 11V 11 V 0 V 01 + V 0 + V V 11 Bahl Tuteja [4] exponential type estimators for population variance Sy are given by 4.1 ŜR = s S x s x yexp Sx + s x 4. ŜP r = s s x Sx yexp. Sx + s x }.

6 1646 Following Haq Shabbir [15 16] the average of the ratio the product-type exponential estimators given in is [ ŜA = s 1 y exp S x s x S x + s x } + exp s x S x S x + s x }]. A generalized form of Ŝ A is given by [ } }] 4.3 ŜA = s 1 a S x sx a s x Sx y exp + exp a s x + Sx + b a s x + Sx + b where a 0 b are functions of known parameters of the auxiliary variable. Motivated by Singh Solanki [46] Haq Shabbir [15 16] replacing s y given in 4.3 by Ŝ SS given in 3.8 we propose the following class of estimators for estimating the nite population variance Sy given by 4.4 ŜP = k 11s y + k 1Sx s x } asx + b Ŝ as A x + 1 b where k 11 [ k 1 are suitably } chosen constants }] ŜA 1 = 1 exp + exp. as x s x as x +S x+b as x S x as x +S x+b Expressing 4.4 in term of δ s keeping terms up to power two we have 4.5 Ŝ P = k 11S y1 + δ 0 k 1S xδ 1 } 1 τδ1 + τ δ 1 Solving 4.5 up to rst order of approximation we get 1 + τ δ Ŝ P S y = k 11S y k 11S yτδ 1 + k 11S yδ 0 k 1S xδ k11s yτ δ 1 k 11S yτδ 0δ 1 + k 1S xτδ 1 S y. Using 4.6 the bias MSE of Ŝ P to rst order of approximation are respectively given by 4.7 BiasŜ P = k 11 1 S y k11s yτ V 0 k 11S yτv 11 + k 1S xτv MSEŜ P = k 11S 4 y k 11S 4 y + k 11S 4 yv 0 + k 11S 4 yτv 11 k 1S ys xτv 0 + k 1S 4 xv 0 + 4k 11k 1S ys xτv 0 k 11k 1S ys xv 11 4k 11S 4 yτv k11s4 yτ V k 11S 4 yτ V 0 + S 4 y. Dierentiating 4.8 with respect to k 11 k 1 we get the optimum values of k 11 k 1 as k opt 11 = V A V0 τ + 4V 0A + 4V 0V 0 4V11 k opt 1 = S y S x V11 + 7V 11A 8V 0τ A + 8V 0V 0 8V V 0 τ + 4V 0A + 4V 0V 0 4V 11 11

7 1647 where A is dened earlier. Substituting the optimum values of k 11 k 1 in 4.8 we get MSE minŝ P as } 64AS 4 y V arŝ Reg V0τ MSE minŝ P = S4 y 16 V 0τ + 4A + 4S 4 y V arŝ Reg 4.1. Some members of the proposed class of estimators. Dierent estimators can be generated from the proposed estimator given in 4.4 by substituting the suitable choices of a b. Some generated estimators are listed in Table 1. Table 1. Some members of proposed class of estimators Ŝ P j j = a b Estimator 1 0 Ŝ P 1 N S x Ŝ P N X Ŝ P 3 n S x Ŝ P 4 n X Ŝ P 5 n S x Ŝ P Eciency comparisons In this section we compare the propose estimator with existing estimators. Condition i: By V ar Ŝ y MSE min Ŝ P > 0 if Sy 4 V 0τ 16V 0 + V 0τ + 64Sy 4 V arŝ Reg + 64AV 11 V A + V 0τ + 4Sy 4 V arŝ Reg > 0. Condition ii: By [3.3 or 3.14] 4.9 [ ] V ar Ŝ Reg or MSE min Ŝ Y K MSE min Ŝ P > 0 if 16V0τ Sy 4 V arŝ yreg + V0τ 3 4 Sy 4 +64V 0 Sy 4 V arŝ yreg A + 4Sy 4 V arŝ yreg > 0. Condition iii: By MSE min Ŝ d MSE min Ŝ P > 0 if S 4 yτ 16 S 4 y V arŝ yreg V V0τ + 64V 0S 4 y } V arŝ yreg + V 1 + S 4 y V arŝ yreg 4 + 4S 4 y V arŝ yreg + V0τ Condition iv: By MSE min Ŝ SG MSE min Ŝ P > 0 if 0τ > 0.

8 1648 S 4 yv 0 64 } 3V 0τ V Sy 4 V arŝ Reg } +4V Sy 4 V arŝ Regτ + 1τ 1 } 64Sy 4 V arŝ Reg 1 τ + 1 4τ S 4 y V arŝ Reg 1 + 3A + 4Sy 4 V arŝ Reg 1 + 4S 4 y V arŝ Reg > 0. Condition v: By MSE min Ŝ SS MSE min Ŝ P > 0 if Syτ 4 V0 AV 0τ A S 4 y V arŝ Reg A + S 4 y V arŝ Reg 1 + 3A + 4S 4 Condition vi: By when using λ = 1. MSE min Ŝ Y G MSE min Ŝ P > 0 if S 4 y 3 Syτ 4 V0 AV 0τ A S 4 y y V arŝ Reg > 0. V arŝ Reg A + S 4 y V arŝ Reg 1 + 3A + 4S 4 Condition vii: By when using τ = ψ 1 = 1. MSE min Ŝ SM MSE min Ŝ P > 0 if y V arŝ Reg > 0. V0 + V11 } V 0V 11 + V 0 V 11 Sy 4 V arŝ Reg 3 V V0V V 11 4 V 0 > 0. 51V 11 V 0 V V 0V V 0V 0 60V 0V V 0V 11S 4 y V arŝ Reg + 56V 0V 11 } } Sy 4 V arŝ Reg V11 3 V 11 V 0 4Sy 4 V arŝ Reg + 4 3V0 Note that the proposed estimator Ŝ P is more ecient than the existing estimators Ŝ i i = y Reg d SG SS Y G Y K SM when above conditions are satised. 6. Numerical study In this section we consider the following data sets for numerical comparisons. Population I: [Source: Kadilar Cingi [0]] Let y = Level of apple production 1 unit=100 tones x = Number of apple trees 1 unit=100 trees. N = 104 n = 0 γ = 0.05 Ȳ = X = Sy = S x = ρ yx = ρ S y S x = β y = β x = θ = V 0 = V 0 = V 11 =

9 1649 Population II: [Source: Cochran [6] p.34] Let y = The weekly expenditure on food x = The weekly family income. N = 33 n = 10 γ = 0.1 Ȳ = X = Sy = S x = ρ yx = ρ S y S x = β y = β x = θ = V 0 = V 0 = V 11 = Population III: [Source: Murthy [6] p.399] Let y = Area of wheat in 1964 x = Area of wheat in N = 80 n = 10 γ = 0.1 Ȳ = X = Sy = S x = ρ yx = ρ S y S x = β y =.665 β x = θ =.3338 V 0 = V 0 = V 11 = We use the following expression for Percentage Relative Eciency PRE the Absolute Bias AB. P REŜ y Ŝ i = V arŝ y MSE min Ŝ i or MSE Ŝ i AB = Bias Ŝ i for i=reg d SG SS Y G Y K SM P. MSE PRE AB values based on Populations I II III are given in Tables 4.

10 1650 Table. MSE PRE AB values of dierent estimators for Population I. a b Ŝ y Ŝ Reg Ŝ d MSE PRE AB MSE E+9 PRE AB MSE E+9 PRE AB MSE E+9 PRE AB MSE E+9 PRE AB MSE E+9 PRE AB Bold numbers indicate least values. Ŝ SG Ŝ SS Ŝ Y G Ŝ Y K Ŝ SM Ŝ P

11 1651 Table 3. MSE PRE AB values of dierent estimators for Population II. a b Ŝ y Ŝ Reg Ŝ d Ŝ SG MSE PRE AB MSE PRE AB MSE PRE AB MSE PRE AB MSE PRE AB MSE PRE AB Ŝ SS Ŝ Y G Ŝ Y K Ŝ SM Ŝ P

12 165 Table 4. MSE PRE AB values of dierent estimators for Population III. Ŝ P Ŝ SM Ŝ Y K Ŝ Y G Ŝ SS Ŝ SG Ŝ d Ŝ Reg a b Ŝ y MSE E E E E E E E E E PRE AB MSE E E E E E E E E E PRE AB MSE E E E E E E E E E PRE AB MSE E E E E E E E E E PRE AB MSE E E E E E E E E E PRE AB MSE E E E E E E E E E PRE AB

13 Conclusion In this paper we have suggested an improved class of estimators for estimating the nite population variance using information on the auxiliary variable. Expressions for bias MSE of the proposed class of estimators have been derived to rst order of approximation. The proposed class of estimators Ŝ P is compared with existing estimators both theoretically numerically. From Tables 4 it is observed that the MSE of Ŝ P is smaller as compared to MSE of existing estimators for all dierent choices of a b considered here. Also bias of Ŝ P is smaller as compared to all other considered estimators in all three populations except Ŝ Y K in population II. Among three populations the maximum PRE gained by proposed estimator is in Population I. So it is preferable to use the estimator Ŝ P. References [1] Arcos A. Rueda M. Martinez M. D. Gonzalez S. Roman Y. Incorporating the auxiliary information available in variance estimation Applied Mathematics Computation [] Ahmed M. S. Rahman M. S. Hossain M. I. Some competitive estimators of nite population variance using multivariate auxiliary information International Journal of Information Management Sciences [3] AL-Jaraha J. Ahmed M. S. A class of chain estimators of a nite population variance using double-sampling International Journal of Information Management Sciences [4] Bahl S. Tuteja R. K. Ratio product type exponential estimators Journal of Information Optimization Sciences [5] Cebrian A. A. Garcia M. R. Variance estimation using auxiliary information an almost unbiased multivariate ratio estimator Metrika [6] Cochran W. G. Sampling Techniques: Third Edition. John Wiley & Sons [7] Chra P. Singh H. P. A family of estimators for population variance using knowledge of kurtosis of an auxiliary variable in sample survey Statistics in Transition [8] Chaudhury A. On estimating the variance of a nite population Metrika [9] Das A. K. Tripathi T. P. Use of auxiliary information in estimating the nite population variance Sankhya C [10] Garcia M. R. Cebrian A. A. Repeated substitution method-the ratio estimator for population variance Metrika [11] Grover L. K. A correction note on improvement in variance estimation using auxiliary information Communication in Statistics Theory Methods [1] Gupta S. Shabbir J. On the use of transformed auxiliary variables in estimating population mean Journal of Statistical Planning Inference [13] Gupta S. Shabbir J. On improvement in estimating the population mean in simple rom sampling Journal of Applied Statistics [14] Gupta S. Shabbir J. Variance estimation in simple rom sampling using uxiliary information Hacettepe Journal of Mathematics Statistics [15] Haq A. Shabbir J.Improved family of ratio estimators in simple stratied rom sampling Communication in Statistics Theory Methods [16] Haq A. Shabbir J. Gupta S. Improved exponential ratio type estimators in stratied sampling Pakistan Journal of Statistics [17] Isaki C. T. Variance estimation using auxiliary information Journal of the American Statistical Association [18] Kadilar C. Cingi H. Ratio estimator in stratied sampling Biometrical Journal [19] Kadilar C. Cingi H. Ratio estimators in simple rom sampling Applied Mathematics Computations

14 1654 [0] Kadilar C. Cingi H. A new estimator using two auxiliary variables Applied Mathematics Computations [1] Kadilar C. Cingi H. Ratio estimators for the population variance in simple stratied rom sampling Applied Mathematics Computations [] Kadilar C. Cingi H. Improvement in variance estimation using auxiliary information Hacettepe Journal of Mathematics Statistics [3] Kadilar C. Cingi H. Improvement in variance estimation in simple rom sampling Communication in Statistics Theory Methods [4] Koyuncu N. Kadilar C. Ratio product estimators in stratied rom sampling Journal of Statistical Planning Inference [5] Koyuncu N. Kadilar C. On improvement in estimating population mean in stratied rom sampling Journal of Applied Statistics [6] Murthy M. N. Sampling Theory Methods Statistical Publishing Society. Calcutta India [7] Mukhopadhyay P. Estimating a nite population variance under a super population model Metrika [8] Mukhopadhyay P. Optimum strategies for estimating the variance of a nite population under a super population model Metrika [9] Parsad B. Singh H. P. Some improved ratio-type estimators of nite population variance using auxiliary information in sample surveys Communication in Statistics -Theory Methods [30] Parsad B. Singh H. P.Unbiased estimators of nite population variance using auxiliary information in sample survey Communication in Statistics -Theory Methods [31] Searl D. T Intarapanich P. A note on an estimator for the variance that utilizes the kurtosis American Statistician [3] Shabbir J. Gupta S. On improvement in variance estimation using auxiliary information Communication in Statistics -Theory Methods [33] Shabbir J. Gupta S. Some estimators of nite population variance of stratied sample mean Communication in Statistics -Theory Methods [34] Shabbir J. Gupta S. On estimating the nite population mean in simple stratied rom sampling Communication in Statistics - Theory Methods [35] Singh R. Malik S. Improved estimation of population variance using information on auxiliary attribute in simple rom sampling Applied Mathematics Computation [36] Singh et al. On the utilization of known coecient of kurtosis in estimation procedure of variance Annals of the Institute of Statistical Mathematics [37] Singh H. P. A note on the estimation of variance of sample mean using the knowledge of coecient of variation in normal population Communication in Statistics - Theory Methods [38] Singh H. P. Upadhyaya L. N. Namjoshi U. D. Estimation of nite population variance Current Science [39] Singh H. P. Biradar R. S. Estimation of nite population variance using auxiliary information Journal of the Indian Society of Statistics Operation Research [40] Singh S. Joarder A. H. Estimation of nite population variance using rom nonresponse in survey sampling Metrika [41] Singh H. P. Singh R. Improved ratio type estimators for variance using auxiliary information. Journal of the Indian Social Agriculture Statistics [4] Singh H. P. Tailor R. Use of known correlation coecient in estimating the nite population mean Statistics in Transition [43] Singh H. P. Vishvakarama G. K. Some families of estimators of variance of stratied rom sample mean using auxiliary information Journal of Statistical Theory Practice

15 1655 [44] Singh H. P. Vishvakarama G. K. A general procedure for estimating the population mean in stratied rom sampling using auxiliary information Metron [45] Singh H. P. Solanki R. S. A new procedure for variance estimation in simple rom sampling Statistical Papers a. [46] Singh H. P. Solanki R. S. Improved estimation of nite population variance using auxiliary information Communication in Statistics Theory Methods b. [47] Srivastava S. K. Jhajj H. S. A class of estimator using auxiliary information for estimating nite population variance SankhyaC [48] Turgut Y. Cingi H. New generalized estimators for the population variance using auxiliary information Hacettepe Journal of Mathematics Statistics [49] Upadhyaya L. N. Singh H. P. An estimator of population variance that utilises the kurtosis of an auxiliary variable in sample survey Vikram Math Journal [50] Upadhyaya L. N. Singh H. P. Singh S. A class of estimators for estimating the variance of the ratio estimator Journal of the Japan Statistical Society [51] Wu C. F. Estimation of variance of the ratio estimator. Biometrika [5] Yadav C. F. Upadhyaya L. N. Singh H. P. Chatterjee S. A generalized family of transformed ratio-product estimators for variance in sample surveys Communication in Statistics Theory Methods [53] Yadav S. K. Kadilar C. Shabbir J. Gupta S. Improved family of estimators of population variance in Simple Rom Sampling Journal of Statistical Theory Practice [54] Yadav S. K. Kadilar C. A two parameter variance estimator using auxiliary information Applied Mathematics Computation

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