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An Improved Generalized Difference-Cum-Ratio-Type Estimator for thePopulation Variance in Two-PhaseSampling Using Two Auxiliary VariablesJavid Shabbira & Sat Guptab

a Department of Statistics, Quaid-i-Azam University, Islamabad,Pakistanb Department of Mathematics and Statistics, University of NorthCarolina at Greensboro, Greensboro, North Carolina, USAAccepted author version posted online: 23 Sep 2013.Publishedonline: 09 Jun 2014.

To cite this article: Javid Shabbir & Sat Gupta (2014) An Improved Generalized Difference-Cum-Ratio-Type Estimator for the Population Variance in Two-Phase Sampling Using Two AuxiliaryVariables, Communications in Statistics - Simulation and Computation, 43:10, 2540-2550, DOI:10.1080/03610918.2012.756910

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Communications in Statistics—Simulation and Computation R©, 43: 2540–2550, 2014Copyright © Taylor & Francis Group, LLCISSN: 0361-0918 print / 1532-4141 onlineDOI: 10.1080/03610918.2012.756910

An Improved GeneralizedDifference-Cum-Ratio-Type Estimator

for the Population Variance in Two-PhaseSampling Using Two Auxiliary Variables

JAVID SHABBIR1 AND SAT GUPTA2

1Department of Statistics, Quaid-i-Azam University, Islamabad, Pakistan2Department of Mathematics and Statistics, University of North Carolina atGreensboro, Greensboro, North Carolina, USA

In this paper, an improved generalized difference-cum-ratio-type estimator for the finitepopulation variance under two-phase sampling design is proposed. The expressionsfor bias and mean square error (MSE) are derived to first order of approximation.The proposed estimator is more efficient than the usual sample variance estimator,traditional ratio estimator, traditional regression estimator, chain ratio type and chainratio-product-type estimators, and Jhajj and Walia (2011) estimator. Four datasets arealso used to illustrate the performances of different estimators.

Keywords Auxiliary variables; Bias; Efficiency; MSE; Two-phase sampling.

Mathematical Subject Classification 62D05.

1. Introduction

The use of auxiliary information both at the design or estimation stages is very common toincrease the precision of the estimators of the parameters of interest by taking advantage ofthe correlation between the study and the auxiliary variables. Recently, several authors haveused or modified different types of ratio, product, and regression estimators to exploit theauxiliary information. These include Jhajj and Srivastava (1980), Tripathy (1980), Singhand Singh (2003), Jhajj et al. (2005), Kadilar and Cingi (2006), Koyuncu and Kadilar (2010),Shabbir and Gupta (2010), Jhajj and Walia (2011), and Choudhury and Singh (2012). Inthis paper, we propose a new generalized difference-cum-ratio-type estimator for finitepopulation variance and compare its performance with the usual sample variance estimator,ratio estimator, regression estimator, chain ratio-type estimator, chain ratio-product-typeestimator, and Jhajj and Walia (2011) estimator.

Consider a finite population of N identifiable units. Let yi and (xi, zi) be the values ofstudy variable (y) and two auxiliary variables (x, z), respectively. Let y and (x, z) be thesample means corresponding to the population means (Y ) and (X, Z), respectively. Also,let s2

y and (s2x , s2

z ) be the sample variances corresponding to the population variances S2y and

Received July 9, 2012; Accepted December 5, 2012Address correspondence to Javid Shabbir, Department of Statistics, Quaid-i-Azam University,

Islamabad 45320, Pakistan; E-mail: js@qau.edu.pk

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Estimator for the Population Variance 2541

(S2x , S2

z ), respectively. The usual ratio, product, and regression estimators require advanceknowledge of population parameters for the auxiliary variables to estimate the populationvariance S2

y . Chand (1975) and Kiregyera (1980, 1984) have discussed a situation wheninformation on x is unknown but another auxiliary variable z is easily available. When pop-ulation mean and population variance of one auxiliary variable z are known in advance andthe population mean and population variance of the other auxiliary variable x are unknown,then we seek to estimate S2

x through a two-phase sampling design. Using a simple randomsample without replacement (SRSWOR) sampling scheme, in the first phase, a preliminarylarge sample of size n′ is selected from a target population and information on the studyvariable y and the auxiliary variables x and z is observed. In the second phase, a subsampleof size n is drawn either from first-phase sample n′ units or independently from the targetpopulation and information on both the study variable y and the auxiliary variables x and z istaken.

We use the following standard terminology:

y = 1

n

n∑i=1

yi, x = 1

n

n∑i=1

xi, z = 1

n

n∑i=1

zi, x ′ = 1

n′

n′∑i=1

xi, z′ = 1

n′

n′∑i=1

zi,

s2y = 1

n − 1

n∑i=1

(yi − x)2, s2x = 1

n − 1

n∑i=1

(xi − x)2, s2z = 1

n − 1

n∑i=1

(zi − z)2,

s ′2y = 1

n′ − 1

n′∑i=1

(yi − y)2, s ′2x = 1

n′ − 1

n′∑i=1

(xi − x)2, s ′2z = 1

n′ − 1

n′∑i=1

(zi − z)2,

Y = 1

N

N∑i=1

yi, X = 1

N

N∑i=1

xi, Z = 1

N

N∑i=1

zi, S2y = 1

N − 1

N∑i=1

(yi − Y )2,

S2x = 1

N − 1

N∑i=1

(xi − X)2, S2z = 1

N − 1

N∑i=1

(zi − Z)2.

Also, λrst = μrst

μr/2200μ

s/2020μ

t/2002

, where μrst = 1N−1

∑Ni=1 (yi − Y )r (xi − X)s(zi − Z)t .

We also define the following error terms:

Let ψ0 = s2y

S2y−1, ψ ′

0 = s ′2y

S2y−1, ψ1 = s2

x

S2x−1, ψ ′

1 = s ′2x

S2x−1, ψ2 = s2

z

S2z−1, ψ ′

2 = s ′2z

S2z−1

To first order of approximation, we have

E(ψi) = E(ψ ′i ) = 0 (i = 0, 1, 2),

E(ψ2

0

) = η(λ400 − 1), E(ψ ′2

0

) = E(ψ0ψ′0) = η′(λ400 − 1), E

(ψ2

1

) = η(λ040 − 1),

E(ψ ′2

1

) = E(ψ1ψ′1) = η′(λ040 − 1), E

(ψ2

2

) = η(λ004 − 1),

E(ψ ′2

2

) = E(ψ2ψ′2) = η′(λ004 − 1),

E(ψ0ψ1) = η(λ220 − 1), E(ψ0ψ′1) = E(ψ ′

0ψ′1) = E(ψ ′

0ψ1) = η′(λ220 − 1),

E(ψ0ψ′2) = E(ψ ′

0ψ′2) = E(ψ ′

0ψ2) = η′(λ202 − 1),

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2542 Shabbir and Gupta

E(ψ1ψ′2) = E(ψ ′

1ψ′2) = E(ψ ′

1ψ2) = η′(λ022 − 1), η =(

1

n− 1

N

), η′ =

(1

n′ − 1

N

),

η′′ =(

1

n− 1

n′

).

2. Existing Estimators

2.1. When Using Single Auxiliary Variable

First note that the MSEs of the usual sample variance (S2y ), ratio estimator (S2

R), regression

estimator (S2Reg), and Jhajj and Walia (2011) estimator (S2

JW) are as follows:

The MSE of S2y is given as

MSE(S2

y

) = ηS4y (λ400 − 1) . (2.1)

The usual ratio estimator (S2R) is given by

S2R = s2

y

(s ′2x

s2x

). (2.2)

The bias and MSE, respectively, of S2R , to first order of approximation, are given by

B(S2

R

) ∼= S4yη

′′ [(λ400 − 1) − (λ220 − 1)] (2.3)

and

MSE(S2

R

) ∼= S4y [η (λ400 − 1) + η′′ (λ040 − 1) − 2η′′ (λ220 − 1)]. (2.4)

The regression estimator (S2Reg) and its MSE are given by

S2Reg = s2

y + b(s2y ,s2

x )(s ′2x − s2

x

), (2.5)

where b(s2y ,s2

x ) is the sample regression coefficient whose population regression coefficient

is β = S2y (λ220−1)

S2x (λ040−1) and

MSE(S2

Reg

) = S4y

[η (λ400 − 1) − η′′ (λ220 − 1)2

(λ040 − 1)

]. (2.6)

Jhajj and Walia (2011) estimator (S2JW) is given by

S2JW = s2

y + θ(s ′2y − s2

y

) [s ′2x

s2x + θ

(s ′2x − s2

x

)]α1

, (2.7)

where θ and α1 are unknown constants.The bias of S2

JW, to first degree of approximation, is given by

B(S2

JW

) ∼= S4yη

′′ (1 − θ )2 α1

[1

2(α1 + 1) (λ040 − 1) − (λ220 − 1)

]. (2.8)

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Estimator for the Population Variance 2543

The MSE of S2JW, to first degree of approximation for α1(opt) = (λ220 − 1) / (λ020 − 1)

is given by

MSE(S2

JW

)min

∼= S4y

[η (λ400 − 1) + (θ2 − 2θ )η′′ (λ400 − 1) − η′′ (1 − θ )2 (λ220 − 1)2

(λ040 − 1)

].

(2.9)

For optimum value of θ = 1, Eq. (2.9) becomes

MSE(S2

JW

)min

∼= S4yη

′ (λ400 − 1) . (2.10)

2.2. When Using Two Auxiliary Variables

The usual chain ratio-type estimator when variance of the second auxiliary variable z isknown, is given by

S2R(t) = s2

y

(s ′2x

s2x

)(S2

z

s ′2z

). (2.11)

The bias and MSE, respectively, of S2R(t), to first order of approximation, are given by

B(S2

R(t)

) ∼= S4y [η′φ1 + η′′φ2] (2.12)

and

MSE(S2

R(t)

) ∼= S4y [η (λ400 − 1) + η′φ3 + η′′φ4], (2.13)

where φ1 = (λ004−1)−(λ202−1), φ2 = (λ040−1)−(λ220−1), φ3 = (λ004−1)−2(λ202−1)and φ4 = (λ040 − 1) − 2(λ220 − 1).

The usual regression estimator for population variance S2y when S2

z is known, is givenby

S2Reg(t) = s2

y + b1(s2y ,s2

x )(s ′2x − s2

x

) + b2(s2y,s

2z )

(S2

z − s ′2z

), (2.14)

where b1(s2y ,s2

x ) and b2(s2y ,s2

x ) are the sample regression coefficients, with corresponding pop-

ulation regression coefficients being β1 = S2y (λ220−1)

S2x (λ040−1) and β2 = S2

y (λ202−1)S2

z (λ004−1) , respectively.

The MSE of S2Reg(t) is given by

MSE(S2

Reg(t)

) = S4y

[η (λ400 − 1) − η′ (λ202 − 1)2

(λ004 − 1)− η′′ (λ220 − 1)2

(λ040 − 1)

]. (2.15)

The expression given in Eq. (2.15) is equal to the following MSE of difference typeand a general class of chain ratio-type estimators.

(i) S2D(t) = s2

y+t1(s ′2x − s2

x )+t2(S2z − s ′2

z ), where t1 and t2 are constants whose optimumvalues are given by: t1(opt) = β1 and t2(opt) = β2.

(ii) S2R(t) = s2

y ( s ′2x

s2x

)g1 ( S2z

s ′2z

)g2 , where g1 and g2 are constants with optimum values

g1(opt) = (λ220 − 1)

(λ040 − 1)and g2(opt) = (λ202 − 1)

(λ004 − 1).

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2544 Shabbir and Gupta

Note: The minimum MSEs of S2D(t) and S2

R(t) are exactly equal to the MSEs of the

regression estimator S2Reg(t).

So, we can write MSE(S2D(t))min = MSE(S2

R(t))min = MSE(S2Reg(t)).

Another estimator for population variance S2y when S2

z is known can be developed onthe lines of Choudhury and Singh (2012). This class is given by

S2RP(t) = s2

y

[k

(s ′2x

s2x

)(S2

z

s ′2z

)+ (1 − k)

(s2x

s ′2x

) (s ′2z

S2z

)], (2.16)

where k is a constant.The bias of S2

RP(t), to first degree of approximation, is given by

B(S2

RP(t)

) ∼= S2y [k{η (λ040 − 1) + η′ (λ004 − 1)} + (1 − 2k){η′ (λ202 − 1)

+ η′′ (λ220 − 1)}]. (2.17)

The minimum MSE of S2RP , to first degree of approximation at optimum value k, i.e.,

kopt = 12 [1 + { η′(λ202−1)+η′′(λ220−1)

η′(λ004−1)+η′′(λ040−1) }], is given by

MSE(S2

RP(t)

)min

∼= S4y

[η (λ400 − 1) − {η′ (λ202 − 1) + η′′ (λ220 − 1)}2

η′ (λ004 − 1) + η′′ (λ040 − 1)

]. (2.18)

3. Proposed Estimator

We propose a generalized difference-cum-ratio-type estimator for the population varianceS2

y under two-phase sampling design using two auxiliary variables. This estimator utilizesthe information on variances of both of the auxiliary variables x and z. This estimator isgiven by

S2P = [

s2y + θ

(s ′2y − s2

y

)] [s ′2x

s2x + θ

(s ′2x − s2

x

)]α1

[s ′2z

s2z + θ

(s ′2z − s2

z

)]α2

×[

S2z

S2z + θ

(s ′2z − S2

z

)]α3

, (3.1)

where θ and αi (i = 1, 2, 3) are unknown constants.For α2 = α3 = 0, Eq. (3.1) becomes the Jhajj and Walia (2011) estimator.Expanding Eq. (3.1) in terms of ψi and ψ ′

i , to first-degree approximation, we have

S2P

∼= S2y [1 + A] , (3.2)

where

A = ψ0 + θ (ψ ′0 − ψ0) + (1 − θ )α1(ψ ′

1 − ψ1) + (1 − θ )α2(ψ ′2 − ψ2) − θα3ψ

′2

− α1ψ1(ψ ′1 − ψ1) + α1θ

2(ψ ′1 − ψ1)2 + 2α1θψ1(ψ ′

1 − ψ1) − α1θψ ′1(ψ ′

1 − ψ1)

+ 1

2α1(α1 + 1)(1 − θ )2(ψ ′

1 − ψ1)2 − α2ψ2(ψ ′2 − ψ2) + α2θ

2(ψ ′2 − ψ2)2

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Estimator for the Population Variance 2545

+ 2α2θψ2(ψ ′2 − ψ2) − α2θψ ′

2(ψ ′2 − ψ2) + 1

2α2(α2 + 1)(1 − θ )2(ψ ′

2 − ψ2)2

+ (1 − θ )α1ψ0(ψ ′1 − ψ1) + (1 − θ )α2ψ0(ψ ′

2 − ψ2) − θα3ψ0ψ′2

+ θ (1 − θ )α1(ψ ′0 − ψ0)(ψ ′

1 − ψ1) + θ (1 − θ )α2(ψ ′0 − ψ0)(ψ ′

2 − ψ2)

− θ2α3ψ′2(ψ ′

0 − ψ0) − θ (1 − θ )α1α3ψ′2(ψ ′

1 − ψ1)

+ (1 − θ )2α1α2(ψ ′1 − ψ1)(ψ ′

2 − ψ2).

Using Eq. (3.2), the bias of S2P , to first degree of approximation, is given by

B(S2

P

) ∼= S2y [η′′A1 − η′A2], (3.3)

where

A1 = (1 − θ )2 α1 (λ040 − 1) + (1 − θ)2 α2 (λ004 − 1) + 1

2α1 (α1 + 1) (1 − θ)2 (λ040 − 1)

+ 1

2α2 (α2 + 1) (1 − θ)2 (λ004 − 1) + (1 − θ )2 α1α2 (λ022 − 1)

− (1 − θ )2 α1 (λ220 − 1) − (1 − θ )2 α2 (λ202 − 1) and A2 = θα3 (λ202 − 1) .

Using Eq. (3.2), the MSE of S2P , to first degree of approximation, is given by

MSE(S2

P

) ∼= S4yE[ψ0 + θ (ψ ′

0 − ψ0) + A3]2, (3.4)

where A3 = (1 − θ ) α1(ψ ′

1 − ψ1) + (1 − θ) α2

(ψ ′

2 − ψ2) − θα3ψ

′2.

From Eq. (3.4), the MSE of S2P , to first degree of approximation, is given by

MSE(S2

P

) ∼= S4y [η(λ400 − 1) + (θ2 − 2θ )η′′ (λ400 − 1) + η′′ (1 − θ )2 A4 + η′A5], (3.5)

where

A4 = α21 (λ040 − 1) + α2

2 (λ004 − 1) − 2α1 (λ220 − 1) − 2α2 (λ202 − 1) + 2α1α2 (λ022 − 1)

and

A5 = α23θ

2 (λ004 − 1) − 2α3θ (λ202 − 1) .

Using Eq. (3.5), the optimum values of αi (i = 1, 2, 3) are given by

α1opt = (λ004 − 1) (λ220 − 1) − (λ022 − 1) (λ202 − 1)

(λ040 − 1) (λ004 − 1) − (λ022 − 1)2 ,

α2opt = (λ040 − 1) (λ202 − 1) − (λ022 − 1) (λ220 − 1)

(λ040 − 1) (λ004 − 1) − (λ022 − 1)2 , and α3opt = (λ202 − 1)

θ (λ004 − 1).

Substituting the optimum values of αi (i = 1, 2, 3) in Eq. (3.5), we get the minimumMSE of S2

P as

MSE(S2

P

)min

∼= S4y [η (λ400 − 1) + (θ2 − 2θ )η′′ (λ400 − 1) − η′′ (1 − θ )2 A6 − η′A7],

(3.6)

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2546 Shabbir and Gupta

where

A6 = (λ220 − 1)2 (λ004 − 1) + (λ202 − 1) (λ040 − 1) − 2 (λ220 − 1) (λ202 − 1) (λ022 − 1)

(λ004 − 1) (λ040 − 1) − (λ022 − 1)2 ,

and

A7 = (λ202 − 1)2

(λ004 − 1).

Although Jhajj and Walia (2011) have presented results for various values of θ , theirnumerical results clearly show that the optimal value of θ is 1. This is obvious even fromexpression (3.5), when we optimize the MSE relative to θ.

For θ = 1, Eq. (3.6) becomes

MSE(S2

P (θ=1)

)min

∼= S4y

[η (λ400 − 1) − η′′ (λ400 − 1) − η′ (λ202 − 1)2

(λ004 − 1)

]. (3.7)

4. Comparison

4.1. When Using Single Auxiliary Variable

We compare the proposed estimator S2P with the usual estimator S2

y , ratio estimator (S2R),

regression estimator (S2Reg), and Jhajj and Walia (2011) estimator (S2

JW).

Condition (i): By Eqs. (2.1) and (3.7),

MSE(S2

P (θ=1)

)min < MSE(S2

y ) if

η′′ (λ400 − 1) + η′ (λ202 − 1)2

(λ004 − 1)> 0.

Condition (ii): By Eqs. (2.4) and (3.7),

MSE(S2

P (θ=1)

)min < MSE

(S2

R

)if

η′′MSE(S∗2R )

ηS4y

+ η′ (λ202 − 1)2

(λ004 − 1)> 0, where the ratio estimator S∗2

R = s2y

(S2

x

s2x

)has

MSE(S∗2

R

) ∼= ηS4y [(λ400 − 1) + (λ040 − 1) − 2 (λ220 − 1)].

Condition (iii): By Eqs. (2.6) and (3.7),

MSE(S2

P (θ=1)

)min < MSE

(S2

Reg

)if

η′′ (λ400 − 1)(1 − ρ2

c

) + η′ (λ202 − 1)2

(λ004 − 1)> 0,

where ρc = (λ202 − 1)√(λ400 − 1)

√(λ040 − 1)

.

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Estimator for the Population Variance 2547

Condition (iv): By Eqs. (2.10) and (3.7),

MSE(S2

P (θ=1)

)min < MSE

(S2

JW

)min if

η′ (λ202 − 1)2

(λ004 − 1)> 0.

Note: All of the conditions (i)–(iv) will always hold true.

4.2. When Using Two Auxiliary Variables

We compare the proposed estimator S2P with, chain ratio estimator (S2

R(t)), regression

estimator (S2Reg(t)), and chain ratio-product estimator (S2

RP(t)).

Condition (v): By Eqs. (2.13) and (3.7),

MSE(S2

P (θ=1)

)min < MSE

(S2

R(t)

)if

η′′MSE(S∗2

R

)ηS4

y

+ η′(√

(λ004 − 1) − (λ202 − 1)√(λ004 − 1)

)2

> 0.

Condition (vi): By Eqs. (2.15) and (3.7),

MSE(S2

P (θ=1)

)min < MSE

(S2

Reg(t)

)if

η′′ (λ400 − 1)(1 − ρ2

c

)> 0.

Condition (vii): By Eqs. (2.18) and (3.7),

MSE(S2

P (θ=1)

)min < MSE

(S2

RP (t)

)min if

η′′ (λ400 − 1) + η′ (λ202 − 1)2

(λ004 − 1)− {η′ (λ202 − 1) + η′′ (λ220 − 1)}2

η′ (λ004 − 1) + η′′ (λ040 − 1)> 0.

Table 1The percentage relative efficiency of different estimators with respect to S2

y for Data 1

Using one auxiliary variable (x) Using two auxiliary variables (x, z)

θ S2R S2

Reg S2JW S2

R(t) S2Reg(t) S2

RP(t) S2P

0.0 150.45 156.25 156.25 221.02 237.20 236.09 246.500.5 150.45 156.25 173.40 221.02 237.20 236.09 282.261.0 150.45 156.25 180.00 221.02 237.20 236.09 296.611.5 150.45 156.25 173.41 221.02 237.20 236.09 282.262.0 150.45 156.25 156.25 221.02 237.20 236.09 246.50

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Table 2The percentage relative efficiency of different estimators with respect to S2

y for Data 2

Using one auxiliary variable (x) Using two auxiliary variables (x, z)

θ S2R S2

Reg S2JW S2

R(t) S2Reg(t) S2

RP(t) S2P

0.0 120.51 130.69 130.69 171.86 193.44 187.17 303.520.5 120.51 130.69 206.78 171.86 193.44 187.17 530.441.0 120.51 130.69 256.58 171.86 193.44 187.17 706.501.5 120.51 130.69 206.78 171.86 193.44 187.17 530.442.0 120.51 130.69 130.69 171.86 193.44 187.17 303.52

Note: Conditions (v) and (vi) will always hold true. Condition (vii) is likely to holdtrue.

5. Numerical Examples

We use the following four datasets for numerical comparison of the efficiencies of differentestimators:

Data 1: [Source: Murthy (1967, p. 228)]Let y = Factory output, x = Fixed capital, and z = Number of workers for a population ofsize 40.For this population, we haveN = 40, n′ = 15, n = 10, λ220 = 2.61055, λ202 = 1.88058, λ022 = 2.00447, λ400 =3.69323, λ040 = 2.18902, λ004 = 1.73235, S2

y = 23154.86.

Data 2: [Source: Sukhatme and Sukhatme (1970, p. 185]Let y = Area under wheat in acres during 1937, x = Area under wheat in acres during1936, and z = Area under wheat in acres during 1931.For this population, we haveN = 34, n′ = 15, n = 8, λ220 = 2.43316, λ202 = 2.60782, λ022 = 1.67304, λ400 =3.44263, λ040 = 3.18507, λ004 = 2.66185, S2

y = 23154.86.

Data 3: [Source: Singh and Mangat (1996)]Let y = Irrigated area, x = Number of tubewells, and z = Number of tractors.

Table 3The percentage relative efficiency of different estimators with respect to S2

y for Data 3

Using one auxiliary variable (x) Using two auxiliary variables (x, z)

θ S2R S2

Reg S2JW S2

R(t) S2Reg(t) S2

RP(t) S2P

0.0 220.04 220.15 220.15 621.00 641.56 631.34 681.580.5 220.04 220.15 260.15 621.00 641.56 631.34 1194.151.0 220.04 220.15 276.92 621.00 641.56 631.34 1593.651.5 220.04 220.15 260.15 621.00 641.56 631.34 1194.152.0 220.04 220.15 220.15 621.00 641.56 631.34 681.58

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Estimator for the Population Variance 2549

Table 4The percentage relative efficiency of different estimators with respect to S2

y for Data 4

Using one auxiliary variable (x) Using two auxiliary variables (x, z)

θ S2R S2

Reg S2JW S2

R(t) S2Reg(t) S2

RP(t) S2P

0.0 162.57 164.00 164.00 278.03 282.29 281.20 527.240.5 162.57 164.00 256.74 278.03 282.29 281.20 1076.831.0 162.57 164.00 316.36 278.03 282.29 281.20 1650.241.5 162.57 164.00 256.74 278.03 282.29 281.20 1076.832.0 162.57 164.00 164.00 278.03 282.29 281.20 527.24

For this population, we haveN = 69, n′ = 30, n = 15, λ220 = 8.09156, λ202 = 7.10378, λ022 = 6.70659, λ400 =9.47695, λ040 = 7.94482, λ004 = 6.31931, S2

y = 85943.8.

Data 4: [Source: Ahmed (1995)]Let y = Number of literate persons, x = Number of households, and z = Total populationin the village.For this population, we haveN = 340, n′ = 120, n = 50, λ220 = 7.31398, λ202 = 9.12905, λ022 = 7.13648, λ400 =10.90334, λ040 = 8.05448, λ004 = 9.25523, S2

y = 71379.47.

The results based on Data 1–4 are given in Tables 1–4.We use the following expression to obtain the percent relative efficiency (PRE) of

different estimators with respect to S2y .

PRE = MSE(S2y )

MSE(Sj )× 100, wherej = R, Reg, JW, R(t), Reg(t), RP(t), P .

In Tables 1–4, we observed that the efficiency of the proposed estimator (S2P ) is

considerably higher as compared to the usual variance estimator (S2y ), ratio estimator (S2

R),

regression estimator (S2Reg), Jhajj and Walia estimator (S2

JW), chain ratio-type estimator

(S2R(t)), another regression estimator (S2

Reg(t)), and chain ratio-product-type estimator (S2RP (t))

for different values of θ under two-phase sampling design. The maximum gain in efficiencyis observed for θ = 1 in each dataset.

Acknowledgments

The authors are thankful to the learned referees for their valuable suggestions that helpedimprove the manuscript.

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