Vishwakarma 2015

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Commun. Math. Stat.

DOI 10.1007/s40304-015-0069-7

An Efficient Class of Estimators for the Mean of a Finite

Population in Two-Phase Sampling UsingMulti-Auxiliary Variates

Gajendra K. Vishwakarma1· Manish Kumar1

Received: 9 June 2014 / Revised: 9 July 2015 / Accepted: 14 September 2015

© School of Mathematical Sciences, University of Science and Technology of China and Springer-VerlagBerlin Heidelberg 2015

Abstract This paper presents an efficient class of estimators for estimating the pop-

ulation mean of the variate under study in two-phase sampling using information

on several auxiliary variates. The expressions for bias and mean square error (MSE)

of the proposed class have been obtained using Taylor series method. In addition, the

minimum attainable MSE of the proposed class is obtained to the first order of approx-

imation. The proposed class encompasses a wide range of estimators of the sampling

literature. Efficiency comparison has been made for demonstrating the performanceof the proposed class. An attempt has been made to find optimum sample sizes under

a known fixed cost function. Numerical illustrations are given in support of theoretical

findings.

Keywords Auxiliary variate · Study variate · Two-phase sampling · Bias · MSE

Mathematics Subject Classification 62D05

1 Introduction

Information on auxiliary variates are widely utilized for obtaining precise estimator

of a population parameter such as the mean, the variance, or the distribution function

of a study variate. A great variety of techniques (i.e., ratio, product, regression, ratio-

B Gajendra K. [email protected]

Manish Kumar

[email protected]

1 Department of Applied Mathematics, Indian School of Mines, Dhanbad, Jharkhand 826004, India

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G. K. Vishwakarma, M. Kumar

cum-product, dual to ratio, dual to ratio-cum-product, etc.) have been developed in

sampling literature for the estimation of the population parameters.

The classical ratio method of estimation utilizes single auxiliary variate and gives

more efficient estimator than the usual unbiased estimator, provided that the variate

under study is highly positively correlated with the auxiliary variate. It is also quiteoften observed that in large-scale sample surveys, the information on several auxiliary

variates can be fruitfully utilized at the design stage, at the estimation stage, or at

both stages to obtain improved designs and for achieving precise estimators of the

parameters under consideration.

Over the years, the ratio method of estimation has drawn attention of social

researchers and scientists due to its intuitive appeal, computational simplicity and

applicability to a general design. Moving along this direction, Olkin [8] made the

first attempt for developing multivariate ratio estimator of the population mean of the

study variate utilizing information on multi-auxiliary variates at the estimation stage.Following the approach adopted by Olkin [8], several authors developed multivari-

ate product, multivariate difference and multivariate regression estimators (see Singh

[15], Raj [9], Mukerjee et al. [7]).

As mentioned earlier, many authors dealt with the problem of estimating the mean

using auxiliary information. Hence, a wide range of estimators have been developed up

to now, adopting different approaches and techniques. Some of them, in the optimum

cases, are equivalent to the linear regression estimator, while others are not. Some

recent contributions towards this have been made by Diana and Tommasi [5], Diana

and Perri [4], Diana et al. [3], Vishwakarma and Gangele [21], Vishwakarma andKumar [22] and Vishwakarma et al. [23].

Utilizing prior information on parameters of auxiliary variates, Srivastava [17],

Srivastava and Jhajj [18] and Diana and Perri [4] have suggested classes of estimators

for estimating population mean Y of a study variate Y . However, in many practical

situations, the prior information on parameters of auxiliary variates are not available.

In that case, two-phase sampling technique is adopted. Taking into consideration the

two-phase sampling scheme, Dash and Mishra [2] suggested a class of estimators for

Y in the presence of two auxiliary variates.

Srivastava [17] suggested a class of estimators for ¯Y using information on the

known means of p auxiliary variates X 1, X 2, . . . , X p in simple random sampling

without replacement (SRSWOR) as

¯ yh = ¯ yh (u1, u2, . . . , u p) = ¯ yh (u), (1.1)

where ui = ¯ x i / X i , (i = 1, 2, . . . , p) , h(·) is a parametric function such that

h(e) = 1, and the function h satisfies certain regularity conditions. Also, e denotes

the column vector of p unit elements, ¯ y denotes the sample mean of Y , ¯ x i and ¯ X idenote, respectively, the sample mean and population mean of auxiliary variate X i (i

=1, 2, . . . , p).

Srivastava and Jhajj [18] defined a class of estimators for Y using information on

the mean and variance of single auxiliary variate X as

¯ yt = ¯ yt (u,v), (1.2)

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An Efficient Class of Estimators for the Mean of a Finite...

where u = ¯ x / ¯ X , v = s2 x /S 2 x , t (·, ·) is a parametric function such that t (1, 1) = 1,

and the function t is continuous and bounded, with continuous and bounded first and

second partial derivatives in R.

Diana and Perri [4] developed a general class of estimators for Y using information

on means and variances of multi-auxiliary variates in SRSWOR as

¯ yg = g( ¯ y, x, s) = g( ¯ y, ¯ x 1, ¯ x 2, . . . , ¯ x p, s21 , s2

2 , . . . , s2 p), (1.3)

where g is a parametric function such that g( Y , X, S) = Y and satisfying certain

regularity conditions. Also, ¯ x i and s2i denote, respectively, the sample mean and sample

variance of X i (i = 1, 2, . . . , p).

Dash and Mishra [2] considered an improved class of estimators for Y using two

auxiliary variates X 1 and X 2 in two-phase sampling as

ˆY mc = h( ¯ y, ¯ x 1, ¯ x 2, ¯ x 1, ¯ x 2), (1.4)

where h is a function such that h ( Y , ¯ X 1, ¯ X 2, ¯ X 1, ¯ X 2) = Y and satisfies certain regu-

larity conditions as given in Srivastava [17].

In this paper, an attempt is made to propose a general class of estimators for the finite

population mean of the study variate Y in two-phase sampling using information on

multi-auxiliary variates. The asymptotic expression for the mean square error (MSE)

of the proposed class is obtained along with its minimum attainable value. Some of the

estimators of the sampling literature are identified as members of the proposed class.

Efficiency comparison is made for judging the merit of the proposed class over the

usual unbiased estimator. In addition, the optimum sample sizes of the first-phase and

second-phase samples along with the optimum MSEs have been obtained for the pro-

posed class and its members. Lastly, an empirical study is carried out for demonstrating

the magnitude of possible gain in efficiency, which is followed by Sect. 7.

2 Proposed Class of Estimators

Let vi = ¯ x i / ¯ x i , (i = 1, 2, . . . , p), where ¯ x i and ¯ x i denote, respectively, the sample

means of auxiliary variate X i based on the first-phase and second- phase samples. Also,

let v denote the column vector containing elements {v1, v2, . . . , v p}, and e denote the

column vector consisting of p unit elements.

Considering the approach adopted by Srivastava [17], we define a class of estimators

for the population mean Y under two-phase sampling as

¯ ym = ¯ ym (v), (2.1)

where the vector v assumes values in a bounded closed convex subset, R, of p-

dimensional real space containing the unit vector e. Also, the function m(v) of the

vector v satisfies the following conditions:

(i) It is continuous and bounded in R;

1 3




(ii) Its first and second partial derivatives exist and are continuous and bounded in

R;

(iii) m(e) = 1.

Now expanding m (v) at the point v =

e in a second-order Taylor’s series, we have

m(v) = m(e) + (v − e)m(e) + 1

2(v − e)m(v∗)(v − e)

or m(v) = 1 + δm(e) + 1

2δ

m(v∗)δ, (2.2)

where v∗ = e + θδ , (0 < θ < 1) and θ may depend on v , m(e) denotes the ( p × 1)

column vector of first partial derivatives of m (v) at the point v = e , m(v∗) denotes

the ( p

× p) matrix of second partial derivatives of m (v) at the point v

= v∗ , and δ

denotes the (1 × p) row vector, which is given by

δ =

δ1 δ2 . . . δ p

; δi = vi − 1 , (i = 1, 2, . . . , p).

Substituting (2.2) in (2.1), we have

¯ ym = ¯ y

1 + δm(e) + 1

2δm(v∗)δ

. (2.3)

To obtain the bias and MSE of the proposed class ¯ ym , we consider

0 = ¯ yY

− 1 , i = ¯ x i¯ X i

− 1 , i =

¯ x i¯ X i

− 1 , (i = 1, 2, . . . , p).

Also, δi = vi − 1 = ¯ x i

¯ x i− 1 = (i −

i )(1 + i )−1.

Then, we have

E (0

) =

E (i) =

E (i) =

0,

E (δ) = 0, E (20 ) = f 1C 20 ,

E (0δ) = ( f 1 − f 2)b = f 3b,

E (δδ ) = ( f 1 − f 2)A = f 3A,

(2.4)

where f 1 =

1n − 1

N

, f 2 =

1n − 1

N

, f 3 = f 1 − f 2 =

1n − 1

n

.

C 0 denotes the coefficient of variation of study variate Y , 0 denotes the (1 × p) row

vector consisting of all zero elements, and b denotes the (1 × p) row vector, which

is given by

b = b1 b2 . . . b p

; bi = ρ0i C 0C i , (i = 1, 2, . . . , p).

Also, A denotes the ( p × p) positive definite matrix and is given by

A = [ai j ] = [ρi j C i C j ] , (i, j = 1, 2, . . . , p),

1 3




where C i denotes the coefficient of variation of X i , ρ0i is the correlation coefficient

between Y and X i , and ρi j is the correlation coefficient between X i and X j (i = j ).

Hence, the proposed class ¯ ym at (2.3) takes the following form:

¯ ym = Y (1 + 0)

1 + δm(e) + 1

2δm(v∗)δ

or ¯ ym − Y = Y

0 + δ

m(e) + 0δm(e) + 1

2δ

m(v∗)δ + 1

20δ

m(v∗)δ

.

(2.5)

Taking expectation on both sides of (2.5) and using results in (2.4), we get the bias of

¯ ym as

Bias(

¯ ym )

= O(n−1). (2.6)

Again from (2.5), on neglecting higher order terms, we have

¯ ym − Y = Y

0 + δm(e)

. (2.7)

Squaring both sides of (2.7), taking expectation and using the results in (2.4), we

obtain the MSE of ¯ ym to the terms of order O(n−1) as

MSE( ¯ ym ) = Y 2 f 1C 20 + f 3 2bm(e) + m(e) Am(e) . (2.8)

The MSE of ¯ ym at (2.8) is minimized for

m(e) = −A−1b (2.9)

and the minimum attainable MSE is given by

MSE( ¯ ym )min = Y 2

f 1C 20 − f 3bA−1b = Y 2C 20

f 1 − f 3 R2

, (2.10)

where R is the multiple correlation coefficient of Y on { X 1, X 2, . . . , X p}. Also, for

the minimum MSE obtained in (2.10), the asymptotic optimum estimator (AOE) in

the proposed class is denoted by ¯ ym(opt) .

Remark 2.1 If we consider a wider class of estimators for Y , say, ¯ y M = M ( ¯ y, v),

where the function M satisfies M ( Y , e) = Y and M 0( Y , e) = 1, M 0 denoting the first

partial derivative of M with respect to ¯ y, then the minimum MSE of ¯ y M is equal to

(2.10) and is not reduced further.

3 Members of the Proposed Class

In the previous section, we defined a class of estimators ¯ ym for estimating the unknown

mean Y in two-phase sampling, under the assumption that the population means¯ X i , (i = 1, 2, . . . , p), of the p auxiliary variates are unknown.

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Table 1 Members of the class of estimators ¯ ym

Authors Estimators

Sukhatme [20]

¯ y Rd

= ¯ y ¯ x

¯ x

¯ yPd = ¯ y ¯ x

¯ x

Srivastava [16] ¯ yds = ¯ y ¯ x

¯ x

α

Singh and Vishwakarma [13] ˆY Md = ¯ y ¯ x + θ x

¯ x + θ x

Kumar and Bahl [6] ¯ y∗ D R

= ¯ y ¯ x ∗

¯ x = ¯ y

(1 + g) − g

¯ x

¯ x

;¯ x ∗ = (1 + g) ¯ x − g ¯ x ; g = n

n−n

Singh and Vishwakarma [14] ˆY Re Md = ¯ ye x p ¯ x −¯ x

¯ x + ¯ x

ˆY P eMd = ¯ ye x p ¯ x − ¯ x

¯ x + ¯ x

Singh and Ruiz Espejo [12] ¯ y(d ) R P

= ¯ y

k ¯ x ¯ x

+ (1 − k ) ¯ x ¯ x

Singh and Choudhury [11] ¯ y∗Pd R

= ¯ y

α ¯ x

¯ x + (1 − α) ¯ x ∗

¯ x

= ¯ y

α ¯ x ¯ x + (1 − α)

(1 + g) − g

¯ x ¯ x

;¯ x ∗ = (1 + g) ¯ x − g ¯ x ; g = n

n−n

In sampling literature, several estimators have been suggested in two-phase sam-

pling by following different approaches and techniques. Also, many of them can be

regarded as particular members of the proposed class ¯ ym . Any estimator, which is a

member of ¯ ym , is optimum if it attains the minimum MSE as given in (2.10).

In Table 1, some of the members of the proposed class have been listed. The

expressions for the MSE of these members are given to the terms of order O(n−1). It

has also been verified, with the help of an empirical study, that some of these members

do not attain the minimum variance bound (MVB) as that in (2.10) and hence are not

optimum.In Table 1, α, θ and k denote the scalars, which are suitably determined so as to

minimize the MSEs of the concerned estimators. Also, the expressions for the MSEs

of various estimators to the terms of order O(n−1) are given by

MSE( ¯ y Rd ) = Y 2

f 1C 20 + f 3

C 21 − 2ρ01C 0C 1

, (3.1)

MSE( ¯ yPd ) = Y 2

f 1C 20 + f 3

C 21 + 2ρ01C 0C 1

, (3.2)

MSE( ¯ yds ) = Y 2

f 1C 20 + f 3

α2C 21 − 2αρ01C 0C 1

, (3.3)

MSE( Y Md ) = Y 2

f 1C 20 + f 3

F 2C 21 + 2 F ρ01C 0C 1

; F = 1 − θ

1 + θ , (3.4)

MSE( ¯ y∗ D R ) = Y 2

f 1C 20 + f 3

g2C 21 − 2gρ01C 0C 1

, (3.5)

1 3




MSE( Y Re Md ) = Y 2

f 1C 20 + f 3

1

4C 21 − ρ01C 0C 1

, (3.6)

MSE( Y PeMd ) = Y 2

f 1C 20 + f 3

1

4C 21 + ρ01C 0C 1

, (3.7)

MSE( ¯ y(d ) R P ) = Y 2

f 1C 20 + f 3

(1 − 2k )2C 21 + 2(1 − 2k )ρ01C 0C 1

, (3.8)

MSE( ¯ y∗Pd R ) = Y 2

f 1C 20 + f 3

ω2C 21 + 2ωρ01C 0C 1

; ω = (1 + g)α − g,

(3.9)

where the term C 1 denotes the coefficient of variation of X , and ρ01 denotes the

correlation coefficient between Y and X .

Furthermore, the minimum attainable MSEs of the estimators

¯ yds , ˆY Md ,

¯ y

(d ) R P and

¯ y∗Pd R are given, respectively, by

MSE( ¯ yds )mi n = Y 2C 20

f 1 − f 3ρ2

01

, (3.10)

MSE( Y Md )mi n = Y 2C 20

f 1 − f 3ρ2

01

, (3.11)

MSE( ¯ y(d ) R P )mi n = Y 2C 20

f 1 − f 3ρ2

01

, (3.12)

MSE( ¯ y∗Pd R)mi n = Y 2C 20

f 1 − f 3ρ2

01

. (3.13)

4 Efficiency Comparison

The variance of the sample mean ¯ y in SRSWOR is

V ( ¯ y) = f 1 Y 2C 20 . (4.1)

For making efficiency comparison of the proposed class ¯ ym with the sample mean ¯ y,

we have from (2.8) and (4.1),

(i) MSE( ¯ ym ) = V ( ¯ y) if

m(e) = −2A−1b .

(ii) MSE( ¯ ym ) < V ( ¯ y) if

m(e) = −A−1b .

5 Optimum Values of n and n

In this section, we shall obtain the optimum sample sizes of the first-phase and second-

phase samples such that the MSE of the proposed class ¯ ym is minimized for a specified

cost.

1 3




Let c1 and c2 be the costs per unit associated with the first-phase and second-phase

samples, respectively. The total cost of the survey is, therefore, given by

c

= c1n

+ c2n . (5.1)

The optimum values of n and n that minimize the MSE of ¯ ym for fixed cost c ≤ c0,

are obtained by using the function φ as

φ = MSE( ¯ ym ) + λ(c1n + c2n − c0), (5.2)

where λ is the Lagrange’s multiplier.

Now from (2.8), we have

MSE( ¯ ym ) = f 1 Y 2C 20 + f 3 Y 2

2bm(e) + m(e)

Am(e)

= f 1 S 20 − f 3 Y 2−2bm(e) −

m(e)

Am(e)

,

or MSE( ¯ ym ) =

1

n− 1

N

S 20 −

1

n− 1

n

ξ, (5.3)

where S 0 = Y C 0, and ξ = Y 2−2bm(e) −

m(e)

Am(e)

.

Substituting (5.3) in (5.2), we have

φ =

1

n− 1

N

S 20 −

1

n− 1

n

ξ + λ(c1n

+ c2n − c0). (5.4)

Differentiating (5.4) with respect to n

and n, equating the results to zero, and using

(5.1), we obtain the optimum values of n

and n as

n

=

c√

A

c2 + c1√ A, (5.5)

n = n√

A= c

c2 + c1

√ A

, (5.6)

where A = c2ξ

c1(S

20 −ξ )

.

Hence, substituting the values of n and n from (5.5) and (5.6) in (5.3), the optimum

MSE of the proposed class ¯ ym is obtained as

MSE( ¯ ym )opt = (c

2 + c

1

√ A)

c

1 − c

N (c2 + c1√

A)

S 20 −

1 − 1√

A

ξ

.

(5.7)

Remark 5.1 On considering the minimum MSE of ¯ ym at (2.10), the optimum values

of n and n for fixed cost c ≤ c0 are obtained on replacing A in (5.5) and (5.6) by A∗,

1 3




Table 2 Optimum n and n along with the optimum MSEs of various estimators for fixed cost c ≤ c0

Estimators n n Optimum MSEs

¯ y c

c2

c2c

− 1 N S 2

0

¯ y Rd c√

A1

c2 + c1√

A1

cc2+c1

√ A1

(c2 + c1√

A1)c

1 − c

N (c2 + c1√

A1)

S 2

0 −

1 − 1√

A1

ξ 1

¯ yP d c√

A2c2 + c1

√ A2

cc2 + c1

√ A2

(c2 + c1√

A2)c

1 − c

N (c2 + c1√

A2)

S 2

0 −

1 − 1√

A2

ξ 2

¯ ydsc√

A3

c2 + c1√

A3

cc2 + c1

√ A3

(c2 + c1√

A3)c

1 − c

N (c2 + c1√

A3)

S 2

0 −

1 − 1√

A3

ξ 3

ˆY Md

c√

A4c2 + c1

√ A4

cc2 + c1

√ A4

(c2 + c1√

A4)c

1 − c

N (c2 + c1√

A4)

S 2

0 −

1 − 1√

A4

ξ 4

¯ y∗ D R

c√

A5

c2 + c1

√ A5

c

c2 + c1

√ A5

(c2 + c1

√ A5)

c

1 − c

N (c2 + c1

√ A5)

S 2

0 −

1 − 1√

A5

ξ 5

ˆY

Re Md

c√

A6

c2 + c1√ A6

c

c2 + c1√ A6

(c2

+c1

√ A6)

c1

− c

N (c2 + c1√ A6) S 2

0 − 1 −

1

√ A6 ξ

6

ˆY P eMd c√

A7

c2 + c1√

A7

cc2 + c1

√ A7

(c2 + c1√

A7)c

1 − c

N (c2 + c1√

A7)

S 2

0 −

1 − 1√

A7

ξ 7

¯ y(d ) R P

c√

A8

c2 + c1√

A8

cc2 + c1

√ A8

(c2 + c1√

A8)c

1 − c

N (c2 + c1√

A8)

S 2

0 −

1 − 1√

A8

ξ 8

¯ y∗P d R

c√

A9

c2 + c1√

A9

cc2 + c1

√ A9

(c2 + c1√

A9)c

1 − c

N (c2 + c1√

A9)

S 2

0 −

1 − 1√

A9

ξ 9

where

A∗ = c2ξ

∗c1(S 20 − ξ ∗) ; ξ ∗ = S 20 R2,

and consequently, the optimum MSE in this situation is obtained as

MSE( ¯ ym )∗opt =

(c2 + c1

√ A∗)

c

1 − c

N (c2 + c1

√ A∗)

S 20 −

1 − 1√

A∗

ξ ∗

.

The optimum values of n and n along with the optimum MSEs of various estimators

for fixed cost c ≤ c0 are listed in Table 2.The notations used in Table 2 are as follows:

A1 = c2ξ 1

c1(S 20 − ξ 1); ξ 1 = Y 2

−C 21 + 2ρ01C 0C 1

,

A2 = c2ξ 2

c1(S 20 − ξ 2); ξ 2 = Y 2

−C 21 − 2ρ01C 0C 1

,

A3

=

c2ξ 3

c1(S 20 − ξ 3) ;

ξ 3

= Y 2 −

α2C 21

+ 2αρ01C 0C 1 ,

A4 = c2ξ 4

c1(S 20 − ξ 4); ξ 4 = Y 2

−F 2C 21 − 2F ρ01C 0C 1

,

A5 = c2ξ 5

c1(S 20 − ξ 5); ξ 5 = Y 2

−g2C 21 + 2gρ01C 0C 1

,

1 3




A6 = c2ξ 6

c1(S 20 − ξ 6); ξ 6 = Y 2

−1

4C 21 + ρ01C 0C 1

,

A7 = c2ξ 7

c1(S

2

0 − ξ 7)

; ξ 7 = Y 2

−1

4C 21 − ρ01C 0C 1

,

A8 = c2ξ 8

c1(S 20 − ξ 8); ξ 8 = Y 2

−(1 − 2k )2C 21 − 2(1 − 2k )ρ01C 0C 1

,

A9 = c2ξ 9

c1(S 20 − ξ 9); ξ 9 = Y 2

−ω2C 21 − 2ωρ01C 0C 1

.

6 Empirical Study

To examine the merits of the proposed class ¯ ym , three natural population datasetshave been considered. The description of the populations and the values of various

parameters are given as follows:

Population I—Source: Cochran [1]

Y : Number of ‘placebo’ children,

X 1: Number of paralytic polio cases in the ‘placebo’ group,

X 2: Number of paralytic polio cases in the ‘not inoculated’ group,

N = 34, n = 15, n = 10, Y = 4.92, ¯ X 1 = 2.59, ¯ X 2 = 2.91, ρ01 = 0.7326,

ρ02 = 0.6430, ρ12 = 0.6837, C 20 = 1.0248, C 21 = 1.5175, C 22 = 1.1492 .

Population II—Source: Srivnstava et al. [19]

Y : The measurement of weight of children,

X 1: Mid arm circumference of children,

X 2: Skull circumference of children,

N = 55, n = 30, n = 18, Y = 17.08, ¯ X 1 = 16.92, ¯ X 2 = 50.44, ρ01 = 0.54,

ρ02 = 0.51, ρ12 = −0.08, C 20 = 0.0161, C 21 = 0.0049, C 22 = 0.0007.

Population III—Source: Sahoo and Swain [10]

Y : Yield of rice per plant,

X 1: Number of tillers, X 2: Percentage of sterility,

N = 50, n = 30, n = 15, Y = 12.842, ¯ X 1 = 9.04, ¯ X 2 = 18.77, ρ01 =0.7133,

ρ02 = −0.2509, ρ12 = 0.0224, C 0 = 0.3957, C 1 = 0.2627, C 2 = 0.0970.

The MSEs along with the percent relative efficiencies (PREs) of various estimators of

Y have been computed and findings are presented in Table 3. The PREs are obtained

for various suggested estimators of Y with respect to the usual unbiased estimator ¯ yusing the formula:

PRE(φ, ¯ y) = V ( ¯ y)

MSE(φ)× 100,

where φ = ¯ y, ¯ y Rd , ¯ yPd , ¯ yds , ˆY Md , ¯ y∗ D R, ˆY Re Md ,

ˆY PeMd , ¯ y(d ) R P , ¯ y∗

Pd R, ¯ ym(opt) .

1 3




Table 3 MSEs and PREs of various estimators of Y

Estimator Auxiliary Population I Population II Population III

variates used MSE PRE MSE PRE MSE PRE

¯ y – 1.75106 100.00 0.175537 100.00 1.20505 100.00

¯ y Rd X 1 1.50119 116.65 0.145116 120.96 0.769203 156.66

X 2 1.55225 112.81 0.157876 111.19 * *

¯ yP d X 2 * * * * 1.15089 104.71

¯ yds X 1 1.30727 133.95 0.145102 120.98 0.767102 157.09

X 2 1.40919 124.26 0.148389 118.29 * *

ˆY Md X 1 1.30727 133.95 0.145102 120.98 0.767102 157.09

X 2 1.40919 124.26 0.148389 118.29 1.15086 104.71

¯ y∗

D R

X 1 3.7002 47.32 0.15373 114.19 0.769203 156.66

X 2 3.20798 54.58 0.152449 115.14 * *

ˆY Re Md X 1 1.32002 132.66 0.152385 115.19 0.892283 135.05

X 2 1.41984 123.33 0.165572 106.02 * *

ˆY P eMd X 2 * * * * 1.16504 103.43

¯ y(d ) R P

X 1 1.30727 133.95 0.145102 120.98 0.767102 157.09

X 2 1.40919 124.26 0.148389 118.29 1.15086 104.71

¯ y∗P d R

X 1 1.30727 133.95 0.145102 120.98 0.767102 157.09

X 2 1.40919 124.26 0.148389 118.29 1.15086 104.71

¯ ym(opt ) X 1 1.30727 133.95 0.145102 120.98 0.767102 157.09

X 2 1.40919 124.26 0.148389 118.29 1.15086 104.71

X 1 and X 2 1.27591 137.24 0.112954 155.41 0.705765 170.74

Bold values indicate the maximum PRE

* Data are not applicable

From Table 3, it is observed that:

(i) Among the members of proposed class

¯ ym , the performances of

¯ yds , ˆY Md ,

¯ y

(d ) R P ,

and ¯ y∗Pd R are same, and are equal to thatof the AOE ¯ ym(opt). Hence, these membersare better than the other members and also the usual unbiased estimator ¯ y as well.

(ii) In all the three populations, the proposed class ¯ ym(opt ), based on two auxiliary

variates, exhibits better performances over the usual unbiased estimator ¯ y and the

other estimators, which are based on single auxiliary variates.

7 Discussion and Conclusion

In this paper, we have proposed a generalized class of estimators for the mean of

a study variate by utilizing information on the means of several auxiliary variates

under the two-phase sampling scheme. The proposed class can also be viewed as a

generalization over the class of estimators suggested by Dash and Mishra [ 2].

For specific choices of the parametric function m(v) in (2.1), several estimators

could be developed. Some of the available estimators are listed in Table 1. Among

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the estimators in Table 1, it has been established theoretically as well as empirically

that the estimators suggested by Srivastava [16], Singh and Vishwakarma [13], Singh

and Ruiz Espejo [12] and Singh and Choudhury [11] attain the MVB and hence are

optimum, while the others are not.

It follows from Table 3 that the performance of ¯ ym(opt ), based on two auxiliaryvariates, is better as compared to the other estimators based on single auxiliary variates.

Thus, the more the auxiliary variates used, the more efficient are the estimators. The

efficiency of any estimator also depends on the correlation between the study and

auxiliary variates. The more highly (i.e., positively high or negatively high) correlated

the study and auxiliary variates, the more precise are the estimators.

Hence, in the present work, an attempt has been made to illustrate how the problem

of estimating the mean of a study variate can be treated in a unified framework by

defining a class of estimators based on the means of several auxiliary variates. This

work could also be extended further through the utilization of information on thevariances of multi-auxiliary variates.

Moreover, the optimum sample sizes of the first-phase and second- phase samples

have been derived for various suggested estimators of Y under a specified cost function.

In addition, the expressions for the optimum MSEs have been obtained for the specified

cost.

Acknowledgments The authors are thankful to the editor-in-chief Prof. Zhi-Ming Ma and the anonymous

reviewer for their valuable suggestions that led to the improvement of the original article in the present

form.

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