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Revista Colombiana de Estadística July 2015, Volume 38, Issue 2, pp. 385 to 397 DOI: http://dx.doi.org/10.15446/rce.v38n2.51667

A Note on Generalized Exponential Type Estimator for Population Variance in Survey Sampling Una nota sobre el estimador exponencial generalizado para la varianza poblacional en muestreo de encuestas Javid Shabbir1,a , Sat Gupta2,b 1 Department 2 Department

of Statistics, Quaid-i-Azam University, Islamabad, Pakistan

of Mathematics and Statistics, The University of North Carolina at Greensboro, Greensboro, United States

Abstract Recently a new generalized estimator for population variance using information on the auxiliary variable has been introduced by Asghar, Sanaullah & Hanif (2014). In that paper there was some inaccuracy in the bias and MSE expressions. In this paper, we provide the correct expressions for bias and MSE of the Asghar et al. (2014) 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 Square 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. (2014). 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. (2014) 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. b Professor.

E-mail: [email protected] E-mail: [email protected]

385

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Javid Shabbir & Sat 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. Singh, 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. Some related work in this direction is also due to Jhajj, Sharma & Grover (2005), Kadilar & Cingi (2006), Gupta & Shabbir (2008), Bansal, Javed & Khanna (2011), Singh, Chauhan, Swan & Smarandache (2011), Upadhyaya, Singh, Chatterjee & Yadav (2011), Subramani & Kumarapandiyan (2012), Nayak & Sahoo (2012) and Yadav & Kadilar (2013, 2014). The following notations will be needed to discuss various estimators. Consider a finite population U = {1, 2 . . . , i, . . . N } of N identifiable units. Let yi and xi (i = 1, 2, . . . , N ) be the values on the ith population unit for the study variable y and the PN PN auxiliary variable x respectively. Let Y = N1 i=1 yi and X = N1 i=1 xi be the P P N N population means and Sy2 = N 1−1 i=1 (yi − Y )2 and Sx2 = N 1−1 i=1 (xi − X)2 be 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 usingP simple random sampling without replacement (SRSWOR) Pn n scheme. Let y = n1 i=1 yi and x = n1 i=1 xi be the sample means and s2y = Pn Pn 1 1 2 2 2 i=1 (yi − y) and sx = n−1 i=1 (xi − x) be the corresponding sample n−1 variances for y and x respectively. To obtain the biases and MSE s expressions for various estimators, we define the following error terms: s2y s2x x 2 − 1, such that E(φi ) = 0 for i = 0, 1, 2. Sy2 − 1, φ1 = X − 1, φ2 = Sx ∗ ∗ ∗ , E(φ0 φ1 ) = γδ21 Cx , E(φ0 φ2 ) = γδ22 , E(φ20 ) = γδ40 , E(φ21 ) = γCx2 , E(φ22 ) = γδ04 ∗ ∗ ∗ E(φ1 φ2 ) = γδ03 Cx , where δ40 = (δ40 − 1), δ04 = (δ04 − 1), δ22 = (δ22 − 1), and µrs γ = n1 as finite population correction factor term is ignored. Also δrs = r/2 s/2 , µ20 µ02 N P where µrs = N 1−1 (yi − Y )r (xi − X)s . i=1

Let φ0 =

The variance of the usual variance estimator Sb02 = s2y , is given by ∗ V ar(Sb02 ) = γSy4 δ40 .

(1)

2 The traditional ratio estimator (SbR1 ) by Isaki (1983), when Sx2 is known, is given by

2 SbR1 = s2y

Sx2 s2x

(2)

Revista Colombiana de Estadística 38 (2015) 385–397

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Generalized Exponential Type Estimator for Population Variance

2 The bias and MSE respectively of SbR1 , to first order of approximation, are given by 2 ∗ ∗ Bias(SbR1 )∼ − δ22 ] = Sy2 γ[δ04

(3)

2 ∗ ∗ ∗ M SE(SbR1 )∼ + δ04 − 2δ22 ]. = Sy4 γ[δ40

(4)

and

2 Another form of ratio estimator (SbR2 ) when X is known, is given by 2 SbR2 = s2y

X x

.

(5)

2 , to first order of approximation, are The bias and MSE respectively of SbR2 given by 2 Bias(SbR2 )∼ (6) = Sy2 γ Cx2 − δ21 Cx

and ∗ 2 M SE(SbR2 )∼ + Cx2 − 2δ21 Cx . = Sy4 γ δ40

(7)

Yadav & Kadilar (2013) suggested the following exponential type estimator for population variance: Sx2 − s2x 2 2 b , (8) SY K = Sy exp Sx2 + (a − 1)s2x where a is a constant. The bias and minimum MSE of SbY2 K , to first order of approximation, at optiδ∗ mum value of a i.e. aopt = δ04 ∗ , is given by 22

Bias(SbY2 K ) ∼ = Sy2 γ

1 1 1 − 2+ 2 2a a a

1 ∗ ∗ δ04 − δ22 a

(9)

and ∗2 δ22 2 4 ∗ ∼ b M SE(SY K )min = Sy γ δ40 − ∗ δ40

(10)

2 The usual regression estimator SbReg1 by Isaki (1983), is given by

2 SbReg1 = s2y + b1 Sx2 − s2x ,

(11)

where b1 is the sample regression coefficient. Corresponding population regression coefficient is given by β1 =

∗ Sy2 δ22 2 δ∗ . Sx 04


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Javid Shabbir & Sat Gupta 2 The MSE of SbReg1 , is given by

∗2 δ22 2 4 ∗ b M SE(SReg1 ) = Sy γ δ40 − ∗ . δ04

(12)

2 Another regression estimator SbReg2 , is given by

2 SbReg2 = s2y + b2 X − x ,

(13)

where b2 is the sample regression coefficient. The corresponding population regression coefficient is given by β2 =

Sy2 δ21 XCx

.

2 The MSE of SbReg2 , is given by

∗ 2 2 M SE(SbReg2 ) = Sy4 γ δ40 − δ21 .

(14)

2. Asghar Estimator Recently Asghar et al. (2014) introduced the following generalized exponential type estimator for population variance. Other work for exponential type estimators for finite population means can be found at Shabbir, Haq & Gupta (2014) ( ) c X − x 2 SbA = λs2y exp , (15) X + (d − 1) x where 0 < λ ≤ 1, −∞ < c < ∞ and d > 0. 2 , to first Asghar et al. (2014) reported the bias and minimum MSE of SbA −1 c order of approximation at optimum values of ω(= d )opt = δ21 (Cx ) and λopt = 2 −1 δ40 − δ21 , given by 1 2 2 2 ∼ 2 b Bias(SA ) = Sy γ λ 1 + ω Cx − ωδ21 Cx − Sy2 (16) 2

and 2 4 ∼ b M SE(SA )min = Sy γ 1 −

1 . 2 δ40 − δ21

(17)

2 However, the bias and minimum MSE of SbA expressions reported in (16) and (17) require some revisions and so does their study based on efficiency and numerical comparisons. ∗2 2 The revised bias and MSE of SbA (say) instead of SbA , to first degree of approximation are derived below.

Rewriting (15) in error terms, we have " −1 # 1 c ∗2 2 ∼ λS (1 + φ0 ) exp − φ1 1 + φ1 − φ1 or SbA = y d c 2 c c c c c ∗2 2 ∼ 2 2 2 b SA − Sy = (λ − 1) Sy + λSy φ0 − φ1 + + − 2 φ1 − φ0 φ1 . (18) d 2d2 d d d



389

∗2 From (18), the bias of SbA , to first degree of approximation, is given by

2 c c c c 2 ∗2 ∼ 2 + C − Bias(SbA − δ C . ) = Sy (λ − 1) + λγ 21 x x 2d2 d d2 d

(19)

∗2 Also from (18), the MSE of SbA , to first degree of approximation, is given by

2 2 c c c c c ∗2 ∼ 4 2 + φ or M SE(SbA ) = Sy E (λ − 1) + λ φ0 − φ1 + − − φ φ 0 1 1 d 2d2 d d2 d

"

2 c c c c 2 ∗2 ∼ 4 2 ∗ b + − 2 γCx − 4 γδ21 Cx M SE(SA ) = Sy 1 + λ 1 + γδ40 + 2 d2 d d d # (20) 2 c c c c 2 + − 2 γCx − γδ21Cx . − 2λ 1 + 2d2 d d d ∗2 at optimum value of λ i.e. The minimum MSE of SbA

1+ λopt =

c2 2d2

∗ +2 1 + γδ40

  ∗2 M SE(SbA )min ∼ = Sy4 1 −

+ c2 d2

c d

−

+

c d

γCx2 − dc γδ21 Cx is given by − dc2 γCx2 − 4 dc γδ21 Cx

c d2

n 2 c 1 + 2d 2 + ∗ +2 1 + γδ40

c d

c2 d2

− +

 o2 γCx2 − dc γδ21 Cx   . (21) c c 2 − d2 γCx − 4 d γδ21 Cx

c d2 c d

Note. In expression (20), 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. 2 (say), with If c = 1 and d = 2 in (15), (19) and (21), we get the estimator SbA1 the bias and minimum MSE, given by X −x 2 SbA1 = λs2y exp X +x 3 2 1 2 Bias(SbA1 ) = Sy2 (λ − 1) + λγ Cx − δ21 Cx 8 2 " 2 # 3 2 1 1 + γ C − δ C 21 x x 2 4 8 2 ∼S 1− M SE(SbA1 )min = y ∗ + C 2 − 2δ C ) . 1 + γ (δ40 21 x x

(22) (23) (24)

Alternatively one can think of a different exponential type estimators as described in the following section. Revista Colombiana de Estadística 38 (2015) 385–397

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3. Proposed Estimator Motivated by Yadav & Kadilar (2014) and Asghar et al. (2014), we propose the following general class of exponential type estimators: ! ! e Sx2 − s2x c X −x 2 2 b exp , (25) SP = λsy exp Sx2 + (f − 1) s2x X + (d − 1) x where −∞ < c < ∞, −∞ < e < ∞, d > 0 and f > 0. We can generate many more estimators from (25), for λ = 1, as described below: (i) For c = e = 0 in (25), we get the usual variance estimator Sb02 = s2y . (ii) For c = 1, d = 2 and e = 0 in (25), we get Sb12 = s2y exp (iii) For c = d = 1 and e = 0 in (25), we get Sb22 = s2y exp

(vi) For c = 1 and e = 0 in (25), we get Sb52 = s2y exp

(vii) For c = 0 and e = 1 in (25), we get Sb62 = s2y exp

(viii) For d = 2 and e = 0 in (25), we get Sb72 = s2y exp (ix) For c = 0 and f = 1 in (25), we get Sb82 = s2y exp

X−x X+x

X−x X

(iv) For c = 0, f = 2 and e = 1 in (25), we get Sb32 = s2y exp (v) For c = 0 and e = f = 1 in (25), we get Sb42 = s2y exp

2 Sx −s2x 2 Sx

2 Sx −s2x 2 +(f −1)s2 Sx x

c(X−x) X+x

.

.

2 Sx −s2x 2 +s2 Sx x

X−x X+(d−1)x

.

.

.

.

.

2 e(Sx −s2x ) 2 Sx +s2x

.

(x) For c = e = 1 and d = f = 2 in (25), we get Sb92 = s2y exp 2 2 S −s exp Sx2 +s2x . x

X−x X+x

x

2 (xi) For c = d = e = f = 1 in (25), we get Sb10 = s2y exp

X−x X

exp

2 Sx −s2x 2 Sx

.



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The bias and MSE expressions for the estimators Sbi2 , (i = 1, 2, . . . , 10) to first order of approximation, are given by 1 3 (26) Bias(Sb12 ) ∼ = Sy2 γ Cx2 − δ21 Cx 8 2 1 Bias(Sb22 ) ∼ (27) = Sy2 γ Cx2 − δ21 Cx 2 3 ∗ 1 ∗ Bias(Sb32 ) ∼ − δ22 (28) = Sy2 γ δ04 8 2 ∗ ∗ Bias(Sb42 ) ∼ − δ22 ] (29) = Sy2 γ [δ04 1 1 1 1 − 2+ Bias(Sb52 ) ∼ Cx2 − δ21 Cx (30) = Sy2 γ 2d2 d d d 1 1 1 ∗ 1 ∗ − 2+ δ04 − δ22 (31) Bias(Sb62 ) ∼ = Sy2 γ 2f 2 f f f

1 2 1 1 ∗ ∗ c + c δ04 − cδ22 8 4 2 1 2 1 1 2 ∼ 2 2 b Bias(S8 ) = Sy γ e + e Cx − eδ21 Cx 8 4 2 1 3 1 2 ∼ 2 2 ∗ ∗ b Bias(S9 ) = Sy γ C + δ04 − (δ21 Cx + δ22 ) + δ03 Cx 8 x 2 4 2 2 2 ∗ ∗ Bias(Sb ) ∼ − (δ21 Cx + δ ) + δ03 Cx =S γ C +δ Bias(Sb72 ) ∼ = Sy2 γ

10

y

x

04

22

1 2 2 ∼ 4 ∗ b M SE(S1 ) = Sy γ δ40 + Cx − δ21 Cx 4 ∗ 2 ∼ 4 b M SE(S2 ) = Sy γ δ40 + Cx2 − 2δ21 Cx 1 ∗ ∗ ∗ − δ22 M SE(Sb32 ) ∼ + δ04 = Sy4 γ δ40 4 2 ∼ 4 ∗ b M SE(S ) = S γ [δ + δ ∗ − 2δ ∗ ] 4

y

40

04

y

40

04

(34) (35)

(37) (38) (39)

22

x

(33)

(36)

∗ Cx 2 M SE(Sb52 )min ∼ − δ21 for dopt = = Sy4 γ δ40 δ21 ∗ δ ∗2 δ04 ∗ M SE(Sb62 )min ∼ − 22 for f = = Sy4 γ δ40 opt ∗ ∗ δ04 δ22 ∗ 2δ21 2 M SE(Sb72 )min ∼ − δ21 for copt = = Sy4 γ δ40 Cx ∗2 δ 2δ ∗ ∗ M SE(Sb82 )min ∼ − 22 for eopt = ∗22 = Sy4 γ δ40 ∗ δ04 δ04 1 1 ∗ 2 ∼ 4 ∗ 2 ∗ b M SE(S9 ) = Sy γ δ40 + C + δ04 − (δ21 Cx + δ22 ) + δ03 Cx 4 x 2 2 4 ∗ 2 ∗ ∗ M SE(Sb ) ∼ − 2 (δ21 Cx + δ ) + 2δ03 Cx =S γ δ + C +δ 10

(32)

22

(40) (41) (42) (43) (44) (45)


392


For the general class of estimators given in (25), we can rewrite the proposed estimator in error terms as follows: −eφ2 −cφ1 2 2 b exp or Sp = Sy (1 + φ0 ) exp 1 + (d − 1)(1 + φ1 ) 1 + (f − 1)(1 + φ2 )

c e c e Sbp2 =Sy2 1 + φ0 − φ1 − φ2 − φ0 φ1 − φ0 φ2 + α1 φ21 + α2 φ22 d f d f + α3 φ1 φ3 , where α1 =

c d

−

c d2

+

c2 2d2 ,

α2 =

e f

−

e f2

+

e2 2f 2

and α3 =

(46)

ce df .

Also from (46), the bias of Sbp2 , to first order of approximation, is given by c e ∗ 2 ∗ 2 ∼ 2 b Bias(Sp ) = Sy γ α1 Cx + α2 δ04 + α3 δ03 Cx − δ21 Cx − δ22 . d f

(47)

From (46), the MSE of Sbp2 , to first order of approximation, is given by ∗ ∗ ∗ M SE(Sbp2 ) ∼ + ω12 Cx2 + ω22 δ04 − 2ω1 δ21 Cx − 2ω2 δ22 = Sy4 γ δ40 + 2ω1 ω2 δ03 Cx , where ω1 =

c d

(48)

and ω2 = fe .

From (48), the optimum values of ω1 and ω2 , are given by ω1(opt) =

∗ ∗ δ04 δ21 − δ03 δ22 ∗ − δ2 ) Cx (δ04 03

and ω2(opt) =

∗ δ22 − δ03 δ21 . ∗ 2 δ04 − δ03

Substituting the optimum values of ω1 and ω2 in (48), we can get the corresponding minimum MSE of Sbp2 as, given by "

M SE(Sbp2 )min

2

(δ ∗ − δ21 δ03 ) ∗ 2 ∼ − δ21 − 22 ∗ = Sy4 γ δ40 2 δ04 − δ03

# .

(49)

4. Comparison of Estimators Now we compare the proposed estimator Sbp2 with existing estimators. (i) By (1) and (49) M SE(Sbp2 )min 0. ∗ 2 δ04 − δ03



393

(ii) By (4) and (49) 2 M SE(Sbp2 )min < M SE(SbR1 ) if 2

∗ ∗ 2 (δ04 − 2δ22 ) + δ21 +

∗ (δ22 − δ21 δ03 ) > 0. ∗ − δ2 δ04 03

(iii) By (7) and (49) 2 M SE(Sbp2 )min < M SE(SbR2 ) if 2

2

(Cx − δ21 ) +

∗ (δ22 − δ21 δ03 ) > 0. ∗ − δ2 δ04 03

(iv) By (24) and (49) 2 M SE(Sbp2 )min < M SE(SbA1 )min if 2 2 1 + γ 83 Cx2 − 12 δ21 Cx (δ ∗ − δ21 δ03 ) ∗ 2 1− − δ40 + δ21 > 0. + 22 ∗ ∗ 2 2 1 + γ (δ40 + Cx − 2δ21 Cx ) δ04 − δ03 (v) By (36) and (49) M SE(Sbp2 )min < M SE(Sb12 ) if 2 2 (δ ∗ − δ21 δ03 ) Cx − δ21 + 22 ∗ > 0. 2 2 δ04 − δ03 (vi) By (37) and (49) M SE(Sbp2 )min < M SE(Sb22 ) if 2

2

(Cx − δ21 ) +

∗ (δ22 − δ21 δ03 ) > 0. ∗ 2 δ04 − δ03

(vii) By (38) and (49) M SE(Sbp2 )min < M SE(Sb32 ) if ∗ 2 δ04 (δ ∗ − δ21 δ03 ) ∗2 2 + δ22 + δ21 + 22 ∗ > 0. 2 4 δ04 − δ03 (viii) By (39) and (49) M SE(Sbp2 )min < M SE(Sb42 ) if 2

∗ ∗ 2 (δ04 − 2δ22 ) + δ21 +

∗ (δ22 − δ21 δ03 ) > 0. ∗ − δ2 δ04 03

(ix) By [(14), (40), (42) and (49)] 2 M SE(Sbp2 )min < [M SE(SbReg2 ), M SE(Sb52 )min , M SE(Sb72 )min ] if 2

∗ − δ21 δ03 ) (δ22 > 0. ∗ 2 δ04 − δ03


394


(x) By [(10), (12), (41), (43) and (49)] 2 M SE(Sbp2 )min < [M SE(SbY2 K )min , M SE(SbReg1 ), M SE(Sb62 )min M SE(Sb82 )min ] if !2 p ∗ δ03 ∗ > 0. δ22 p ∗ − δ21 δ04 δ04 (xi) By (44) and (49) M SE(Sbp2 )min < M SE(Sb92 ) if

Cx − δ21 2

2

+

∗ δ04 ∗ − δ22 4

2

+

δ03 Cx (δ ∗ − δ21 δ03 ) + 22 ∗ > 0. 2 2 δ04 − δ03

(xii) By (45) and (49) 2 ) if M SE(Sbp2 )min < M SE(Sb10 2

2

∗ ∗ (Cx − δ21 ) + (δ04 − 2δ22 ) + 2δ03 Cx +

∗ (δ22 − δ21 δ03 ) > 0. ∗ 2 δ04 − δ03

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. 226)] Let y be the output and x be the number of workers in a factory. N = 80, n = 25, Y = 5182.638, X = 285.125, Cy = 0.3542, Cx = 0.9485, ρyx = 0.9140, δ40 = 2.2383, δ21 = 0.5427, δ04 = 3.5360, δ22 = 2.2943, δ03 = 1.2680. Population 2: [source: (Murthy 1967, p. 228)] Let y be the output and x be the fixed capital. N = 80, n = 25, Y = 5182.638, X = 1126.463, Cy = 0.3542, Cx = 0.7507, ρyx = 0.9413, δ40 = 2.2384, δ21 = 0.4518, δ04 = 2.8306, δ22 = 2.1930, δ03 = 1.0237. Population 3: [source: Cochran (1977, p. 203)] 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 = 56.9, X = 54.2961, Cy = 0.1840, Cx = 0.1621, ρyx = 0.9237, δ40 = 1.9249, δ21 = 0.1873, δ04 = 2.5932, δ22 = 2.1149, δ03 = 0.4956. Revista Colombiana de Estadística 38 (2015) 385–397

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Population 4: [source: Sarndal, Swensson & Wretman (1992, p. 652-659)] y =P85-1985 population in thousand, x =RMT85-Revenues from the 1985 municipal taxation (in millions kronor). N = 284, n = 35, Y = 29.36, X = 245.088, Cy = 1.76, Cx = 2.43, ρyx = 0.961, δ40 = 88.92, δ21 = 7.87, δ04 = 88.88, δ22 = 78.79, δ03 = 8.77. The results based on Populations 1-4 are given in Table 1. c2 for Table 1: The percentage relative efficiency of different estimators with respect to S 0 Populations 1-4. Estimator b2 S 0 b2 S R1 b2 S R2 b2 S A1 b2 S 1 b2 S 2 b2 S 3 b2 S 4 b2 b2 , S b2 , S S 5 7 Reg2 2 2 b b b2 b2 S6 , S8 , S Reg1 , SY K 2 b S9 b2 S 10 b2 S p

Pop.1 100 104.436 111.715 136.384 130.559 111.715 214.239 104.436 131.207 214.340 139.211 35.776 226.001

Pop.2 100 181.318 110.215 124.423 119.063 110.215 246.178 181.318 119.736 268.678 179.734 58.826 351.363

Pop.3 100 320.812 103.868 120.920 102.640 103.868 444.023 320.812 103.943 639.150 411.659 223.122 806.889

Pop.4 100 434.817 158.196 331.049 125.114 158.196 273.894 434.817 338.374 461.244 350.172 288.273 463.471

We use following expression to obtain the percent relative efficiency (P RE) of different estimators with respect to S02 . PRE =

V ar(S02 ) × 100, where i = 0, R1, R2, A1, 1 − 10, Reg1, Reg2, Y K, P. M SE(Si2 )

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 MSE expressions of the Asghar et al. (2014) 2 estimator. While we note that the improved Asghar et al. (2014) estimator SbA1 performs better than several estimators. This estimator can be improved further as shown by the performance of Sbp2 . We also note that the estimators (Sb62 , Sb82 , 2 2 ) have the same MSE for all four populations. SbReg1 , SbY2 K ) and (Sb52 , Sb72 , SbReg2 Received: July 2014 — Accepted: January 2015


396


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