A General Class of Chain-Type Estimators in the Presence of Non-Response Under Double Sampling Scheme

(1)

Journal of Modern Applied Statistical

Methods

Volume 9 | Issue 2

Article 19

11-1-2010

A General Class of Chain-Type Estimators in the

Presence of Non-Response Under Double

Sampling Scheme

Sunil Kumar

University of Jammu, (J & K), India, [email protected]

Housila P. Singh

Vikram University, Ujjain, India

Sandeep Bhougal

Shri Mata Vaishno Devi University, Kakryal, Jammu (J & K), India, [email protected]

Follow this and additional works at:

http://digitalcommons.wayne.edu/jmasm

Part of the

Applied Statistics Commons

,

Social and Behavioral Sciences Commons

, and the

Statistical Theory Commons

This Regular Article is brought to you for free and open access by the Open Access Journals at DigitalCommons@WayneState. It has been accepted for inclusion in Journal of Modern Applied Statistical Methods by an authorized administrator of DigitalCommons@WayneState.

Recommended Citation

Kumar, Sunil; Singh, Housila P.; and Bhougal, Sandeep (2010) "A General Class of Chain-Type Estimators in the Presence of Non-Response Under Double Sampling Scheme," Journal of Modern Applied Statistical Methods: Vol. 9: Iss. 2, Article 19.

(2)

512 A General Class of Chain-Type Estimators in the Presence of Non-Response

Under Double Sampling Scheme

Sunil Kumar

Housila P. Singh

Sandeep Bhougal

University of Jammu, (J & K), India

Vikram University, Ujjain, India

Shri Mata Vaishno Devi University, Kakryal, Jammu (J & K), India

General class chain ratio type estimators for estimating the population mean of a study variable are examined in the presence of non-response under a double sampling scheme using a factor-type estimator (FTE). Properties of the suggested estimators are studied and compared to those of existing estimators. An empirical study is carried out to demonstrate the performance of the suggested estimators; empirical results support the theoretical study.

Key words: Double sampling, factor-type estimator, chain ratio estimator, non-response.

Introduction

Over the last five decades one of the major developments in sample surveys is the use of an auxiliary variable x , correlated with the study

variable y in order to obtain estimates of the

population total or mean of the study variable. Various estimation procedures in sample surveys require advance knowledge of some auxiliary

variable x , which is then used to increase the _i

precision of estimates. When the population mean X is not known, it can be estimated from a preliminary large sample on which only the auxiliary characteristic x is observed. The value of X in the estimator is then replaced by its estimate, and a smaller second-phase sample of

the variable of interest (study variable) y is

taken. This technique, known as double sampling or two-phase sampling, is particularly appropriate if the x values are easily accessible i

and are much less expensive to collect than the

Sunil Kumar is in the Department of Statistics in University of Jammu College. Housila P. Singh is a Professor in the School of Studies in Statistics. Email: [email protected]. Sandeep Bhougal is in the School of Mathematics.

Email: [email protected].

i

y values (Sitter, 1997; Hidiroglou & Sarndal,

1998). Neyman (1938) was the first to describe the concept of double sampling in connection with collecting information on strata sizes is a stratified sampling (Singh & Espejo, 2007).

In some practical situations it is observed that, when conducting a sample survey, complete information for all the units selected in the sample is not obtained due to the occurrence of non-response. Hansen and Hurwitz (1946) considered the problem of non-response while estimating the population mean by taking a sub-sample from the non-response group with the help of an unbiased estimator; they suggested combining the information available from response and non-response groups. Further, rectification in the estimation procedure for the population mean in the presence of non-response using auxiliary variable was proposed by Cochran(1977), Rao (1986, 1987), Khare and Srivastava (1993, 1995, 1997), Okator and Lee (2000), Tabasum and Khan (2004, 2006), and Singh and Kumar (2008a, 2008b, 2008c, 2009a, 2009b) using the Hansen and Hurwitz (1946) technique. This article develops a one parameter family of chain ratio type estimators with two auxiliary variables in the presence of non-response. The proposed family is based on factor type estimators (FTE) developed by Singh and Shukla (1987) and Singh, et al. (1994) and empirical studies support the results.

(3)

KUMAR, SINGH & BHOUGAL

513

The Proposed Strategy

Consider a finite population

(

U ,U ,...,UN

)

U = 1 2 of size N . Let y be the

study variable, x be the main auxiliary variable ₁

with an unknown mean that is highly correlated

with main character y , and x be an additional 2

auxiliary variable with known mean that is less correlated with y than is x . A large first phase ₁

sample of size n′ from the finite population U is selected by simple random sampling without replacement (SRSWOR). A smaller second

phase sample of size n is selected from n′ by

SRSWOR. Non-response occurs in the second

phase sample of size n in which n units 1

respond and n units do not. From the ₂ n non-₂

respondents, by SRSWOR a sample of size 1

2 >

=n k;k

r units is selected where k is the

inverse sampling rate at the second phase sample of size n with all r units responding. Thus,

(

n1+r

)

are the responding units on the study

variable y , consequently the estimator for the

population mean Y of the study variable y

using a sub-sampling scheme envisaged by Hansen and Hurwitz (1946) is defined as

r * _w _y _w _y y = ₁ ₁+ ₂ ₂ , (1) where n n w₁ = ₁ , w₂ =n₂ n,



= = 1 1 1 1 n i i n y y and



= = r i i r y r y 1 2 .

It is known that the estimator _{y is an unbiased}*

estimator of the population mean Y of the study

variable y and has a variance as given by

( )

2 2 2 1 y * y( ) * _θ _S _θ _S y Var = + , (2) where       − = N n θ₁ 1 1 ,

(

)

n k W θ* ₌ 2 −1 _, _W _N _N 2 2 = , and 2 y S and 2 2 ) ( y

S are the population mean

square of the variable y for the entire

population and for the non-responding group of the population. Similarly, for estimating the

population mean X of the auxiliary variable _i

(

i 1,2

)

;

xi = , the unbiased estimator x is given i*

by ) r ( i ) ( i * i wx w x x ₌ ₁ ₁ ₊ ₂ ₂ , (3)

where x_i₍₁₎ and x_i₍₂_r₎ are the sample means of the auxiliary variable xi;

(

i=1,2

)

based on n 1

and r units respectively.

The variance of * i x is given by

( )

2 2 2 1 x * x( ) * i θ S i θ S i x Var ₌ ₊ , (4) where 2 i x S and 2 2) ( xi

S are the population mean

square of xi;

(

i=1,2

)

for the entire population and the non-responding group of the population. The Proposed Class of Strategy

Using an unknown constant t>0 and

two auxiliary variables x and ₁ x , a general 2

class of chain ratio type of strategy

[

D,y*(t)

]

F is

defined for estimating the population mean Y of

the study variable y in the presence of

non-response as follows:

( )

_{{ }}

{ }

( )

_{( )}

      = t λ φ t λ φ , g y ) t ( y* * F 2 1 310 , (5) where

( )

{ }

λ t λ

( )

t

{

1 λ

( )

t

} ( )

g2 0,1;i 1,2 φ i = i + − i = ,

( )

t _A _θθ_BB _C λ + + = 1 ,

( )

_A _θ_B _C C t λ + + = 2 ,

( )(

−1 −2

)

= t t A , B=

( )(

t−1 t−4

)

,

(

−2

)( )(

−3 −4

)

= t t t C , θ =n N,       − ′ = N n θ₂ 1 1 ,       ′ − = n n θ₃ 1 1 ,

(4)

514 ( )

* β α * _x X x X β , α g _            = 2 2 1 1 1 ,

( )

' β α ' _x X x X β , α g _            = 2 2 1 1 2 ,

( )

* β ' α * ' x x x x β , α g _            = 2 2 1 1 3 ,



= = 1 1 1 1 N i i N x X ,



= = 2 1 2 2 N i i N x X ,



= = ' n i ' i ' _x _n x 1 1 1 1 and



= = ' n i ' i ' _x _n x 2 1 2 2 .

In order to identify some of the members of the proposed strategy and compare their efficiencies, certain classical strategies are put forth:

(i)

[ ]

D,y ; y* y*g₃

( )

1,0

R *

R = (6)

by Khare and Srivastava (1993), Okafor and Lee (2000) and Tabasum and Khan (2004) (ii)

[ ]

D,y ; y* y*g₃

(

1,0

)

P *

P = − (7)

by Khare and Srivastava (1993)

(iii)

[ ]

D,y_C* ; y_C* = y*g₃

( ) ( )

1,0 g₂ 0,1 . (8) Some Strategies of the Class

For t=1 and 4 respectively, (i)

[

( )

] [ ]

* C * F D,y y , D 1 = , (9) (ii)

[

( )

] [ ]

* R * F D,y y , D 4 = . (10)

Further, for t=2 and 3,

( )

[

D,y 2

]

; y*

( )

2 = y*g₃

( ) (

1,0 g₂ 0,−1

)

F * F , (11)

( )

3

( ) (

₍

)

₎

2

1

3

3 1 0

0 1

* * * F F

h

D, y

; y

y g

,

hg

,

+











₌

_

_





₋







(12) where h=n

(

N−n

)

−1, and *

( )

2 F y is a chain type

estimator in D in which X is estimated ₁

through the product estimator utilizing X ₂

where non-response on auxiliary variable x and ₁

( )

3

* F

y is a chain type estimator in D in which

1

X is estimated utilizing a dual to ratio

estimator with non-response on auxiliary variable x . 1

Properties of the Proposed Strategy

To obtain the bias and mean square error (MSE) of the proposed general class of strategy

[

D,y*_F(t)

]

, under the large sample

approximation,

(

1

0

)

*

y

₌

Y

_{+ ε}

,

_x

₁*

₌

_X

₁

(

1 _{+ ε}

₁

)

_,

(

)

2 2

1

2 *

x

₌

X

_{+ ε}

,

_x₁' ₌_X₁

(

1_{+ ε}₁'

)

_, and ₂' ₂

(

1 '₂

)

x =X + ε , such that

( ) ( ) ( )

_ε₀ =_E _ε₁ =_E _ε₂ =_E

( ) ( )

_ε₁' =_E_ε'₂ =0 E and

( )

2 2 2 1 2 0 θ Sy θ*Sy( ) ε E ₌ ₊ ,

( )

2 2 2 1 2 1 1 x1( ) * x θ S S θ ε E ₌ ₊ ,

( )

2 2 2 1 2 2 2 x2( ) * x θ S S θ ε E ₌ ₊ ,

( )

2 2 1 1 2 x ' _θ _S ε E = ,

( )

2 2 2 2 2 x ' _θ _S ε E = , E

( )

ε₀ε₁ θ₁S_yx θ*S_yx₍₂₎ 1 1 + = ,

( )

2 1 ' 1 0

ε

θ

S

yx

ε

E

₌

,

( )

₁ ₁' ₂ 2 1 x

S

θ

ε

E

₌

,

( )

2 2 ' 2 0

ε

θ

S

yx

ε

E

₌

,

( )

2 1 2 2 1ε' θ Sxx ε E ₌ ,

( )

1 2 2 x1x2 ' '_ε _θ _S ε E ₌ , where

(5)

KUMAR, SINGH & BHOUGAL

515 (

)



=

(

)(

)

− − − = N i i i yx y Y x X N S 1 1 1 1 1 1 ,

(

)



=

(

)(

)

− − − = N i i i yx y Y x X N S 1 2 2 1 1 2 ,

(

)



=

(

)(

)

− − − = 2 1 1 2 1 1 2 2 2 ₁ 1 N i ) ( i i ) ( yx _N y Y x X S ,

(

)



=

(

)(

)

− − − = N i i i x x _N x X x X S 1 2 2 1 1 1 1 2 1 ,



= = 2 1 2 1 2 1 N i i ) ( x N X ,



= = N i i N y Y 1 ,



= = 2 1 2 2 N i i N y Y , 1

N and N₂

(

=N−N₁

)

are the sizes of the responding and non-responding units from the finite population N .

Expressing the proposed estimator

( )

t y* F in terms of sε' ,

( )

(

)

(

₍

)

₎

( )

{

( )

}

(

)

( )

{

( )

}

(

)

1 1 1 1 2 0 1 1 ₂ ₂ ₂

1

1 ₁

₁

* F ' ' '

y t

t

Y

.

t

− −

=





+ ε

_

λ

+ − λ

− ε

_

+ ε

_

_

+ ε

_

_λ

_{+ − λ}

_{− ε}

_

(13)

It is assumed that _λ₂

( )

_t _ε₂' <1_{, because}

( )

t _A _θC_B _C λ

+ + =

2 , for any choice of t ,

( )

1 2 t < λ . Thus, if _ε'₂ <1_,

( )

₁ 2 2 t ε' < λ is a

valid assumption, expanding the right hand side of (13) and neglecting the terms involving powers of sε' greater than two results in

( )

(

2

)

2 0 1 1 1 1 1 0 1 0 1 2 1 2 1 2 0 2 2 2 1 * F ' ' ' ' ' ' ' ' ' y t Y , t =  + ε − ε + ε + ε − ε ε − ε ε + ε ε      +λ ε − ε ε + ε ε + ε ε − λ ε    

( )

{

}

( )

(

2

)

2 0 1 1 1 1 1 0 1 0 1 2 1 2 1 2 0 2 2 2 * F ' ' ' ' ' ' ' ' ' y t Y Y t − = _{ε − ε + ε + ε − ε ε − ε ε + ε ε}      +λ ε − ε ε + ε ε + ε ε − λ ε     , (14) where λ

( )

t =λ₁

( )

t −λ₂

( )

t .

Taking expectations of both sides of (14), results in the bias of y*

( )

t

F to the first degree of approximation, as

( )

(

)

(

_{( )}

₍

1

)

_{( )}

1

(

₎

1

)

1 2 2 2 2 3 2 2 2 2 2

1

*

1

yx x yx ( ) x ( ) * F yx x

K

S

K

S

B y t

,

t

K

S



_θ

₋

_{+ θ}

₋





= 



−λ

θ λ

−







(15) where 1 1 1 1 1 2 x y yx x yx yx _S S ρ S S K = = , 2 2 2 2 2 2 x y yx x yx yx _S S ρ S S K = = , 1 1 1 x y yx yx _S _S S ρ = and 2 2 2 x y yx yx _S _S S ρ = .

Squaring both sides of (14) and neglecting terms of sε involving power greater than two, '

( )

{

( )

}

2 2 1 1 0 2 2 ε t λ ε ε ε Y Y t y*_F  = − + ′+ ′      ₋

( )

(

)

( )

2 2 2 2 2 0 1 1 2 2 ₂ 0 1 0 1 0 2 1 1 1 2 1 2

2

* F

t

y t

Y

t

.

t

′



_{ε + ε + ε + λ}

_ε





′

−

=



− ε ε + ε ε + λ

ε ε





_{− ε ε − λ}

_′

_{ε ε + λ}

_′

_{ε ε}

_{′ ′}







(16) Taking expectations of both sides of (16), gives

the mean square error of y*

( )

t

F to the first

(6)

516 ( )

(

)

(

)

( )

(

( )

)

(

)

{

}

1 1 2 2 1 1 2 2 1 3 2 2 2 2 2 2 2

1 2

2 1 2

* F y yx x yx x * y( ) yx ( ) x ( )

MSE y t

S

K

S

t

K

S

K

S

=



_θ

_{+ θ}

₋









_+λ

_{θ λ}

₊











+θ

+ −









, (17) where 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 yx ( ) yx ( ) yx ( ) yx ( ) x ( ) x ( ) yx ( ) yx ( ) y( ) x ( ) S S K , S S S S S = = ρ ρ = . Corollary Letting λ

( )

t =−1,λ2

( )

t =1 for

t

=

1

in

(15) and (17), the bias and MSE of *

C y , respectively given by

( )

(

)

(

)

(

)

1 1 1 1 2 2 2 3 2 2 2 2 2

1

yx x * * * F C yx ( ) x ( ) yx x

K

S

B y

y

K

S

K

S



_θ

₋











=

= +θ

_

−

_



_+θ

₋







(18) and

( )

(

)

(

)

(

)

(

)

{

}

1 1 2 2 1 1 2 2 1 3 2 2 2 2 2 2 2

1 1 2

* * F C y yx x yx x * y( ) yx ( ) x ( )

MSE y

y

S

K

S

K

S

K

S

=



_θ

_{+ θ}

₋











= +θ

_

−

_





+θ

+ −









. (19)

To obtain the bias and MSE of y*_R, assume that

( )

t =λ₂

( )

t =0 λ for t=4, in (15) and (17),

( )

(

)

(

₍

1

)

1

₎

1 1 2 3 2 2 2

1

4

1

yx x * * F R _* yx ( ) x ( )

K

S

B y

y

K

S



_θ

₋







=

_

_

+θ

−





(20) and

( )

(

)

(

)

(

)

{

}

1 1 1 1 2 2 1 3 2 2 2 2 2

4 1 2

1 2

* * F R y yx x * y( ) yx ( ) x ( )

MSE y

y

S

K

S

K

S

=



_θ

_{+ θ}

₋







= 

_+θ

_{+ −}







. (21) The MSE

( )

y*

( )

t F is minimized, when

( )

t Kyx2 λ =− . (22) Thus, substituting (22) in (17), results in the optimum mean square error of y*

( )

t

F , as

( )

(

)

(

)

(

)

{

}

2 2 1 1 1 1 2 2 2 2 1 2 3 2 2 2 2 2

1 2

* F _opt y yx x yx x * y( ) yx ( ) x ( )

MSE y t

S

K S

K

S

K

S

=



_θ

_{− θ}

_{+ θ}

₋









_+θ

_{+ −}







. (23) Efficiency Comparisons From (2), (19), (21) and (23),

( )

(

( )

)

(

)

(

)

1 1 2 2 1 1 2 2 2 3 2 2 2 2

1 2

* * F _opt yx x yx x * yx ( ) x ( )

Var y

MSE y t

K

S

K S

K

S

−



_θ

₋

_{− θ}







=

_

_

+θ

−





, (24)

( )

(

1 2

)

2 2 0 2 − 2 2 > = − opt yx x * F * C MSE y t θ K S y MSE when 1 2 < yx K , (25)

( )

2 2 2 * * 2 2 x yx opt F k MSE y t K S y MSE − =θ (26) It is explicit from the equations (24)-(26) that

the proposed class of estimator y*

( )

t

F is more

efficient than:

(i) The usual unbiased estimator _y*_;

(ii) The estimator *

C

y when 1

2 <

yx

K ; and

(iii) The estimator *

R

y , the ratio type estimator

(7)

KUMAR, SINGH & BHOUGAL

517

Tabasum and Khan (2004) and Okafor and Lee (2000).

Thus, it may be concluded that the general chain ratio type class of proposed strategy

[

D,y*

( )

t

]

F is

more efficient than the usual unbiased estimator

*

y , the estimator *

C

y and the ratio type

estimator *

R y . Empirical Study

To examine the effectiveness of the suggested class of chain ratio types, data sets studied by Khare and Sinha (2007) are considered. The data, from the Department of Paediatrics, Banaras Hindu University during 1983-1984, is the physical growth of an upper socio economic group of 95 school age children of Varanasi under ICMR study. The first 25% (i.e., 24 children) have been considered as non-responding units. The descriptions of the variates are given below:

Population I:

y: Height (in cm.) of the children,

1

x : Skull circumference (cm) of the

children,

2

x : Chest circumference (cm) of the

children. For this population:

1 2 1 2 2 2 2

115 9526

51 1726

55 8611

35 6041

2 3662

10 7155

y x x

Y

.

,

X

.

,

X

.

,

S

.

,

S

.

,

S

.

,

=

1 2 1 2 1 2 2 2 2 2 2 2

26 0532

1 6079

9 1060

0 3740

0 620

0 571

y( ) x ( ) x ( ) yx yx yx ( )

S

.

,

S

.

,

S

.

,

.

,

.

,

.

,

=

ρ =

ρ

=

2 1 2 1 2 2 2

0 401

0 2970

0 570

95

35

45

yx ( ) x x x x ( )

.

,

.

,

.

,

N

,

n

,

n

ρ

=

ρ

=

ρ

=

′ =

Population II:

y: Weight (kg) of the children,

1

x : Chest circumference (cm) of the

children,

2

x : Mid-arm circumference (cm) of the

children. For this population,

1 2 1 2 2 2 2

19 4968

55 8611

16 7968

9 2662

10 7155

2 1115

y x x

Y

.

,

X

.

,

X

.

,

S

.

,

S

.

,

S

.

,

=

1 2 1 2 1 2 2 2 2 2 2 2

5 5424

9 1060

1 4323

0 846

0 797

0 729

y( ) x ( ) x ( ) yx yx yx ( )

S

.

,

S

.

,

S

.

,

.

,

.

,

.

,

=

ρ =

ρ

=

(8)

518

2 1 2 1 2 2 2

0 757

0 725

0 641

95

35

45

yx ( ) x x x x ( )

.

,

.

,

.

,

N

,

n

,

n

ρ

=

ρ

=

ρ

=

′ =

The percent relative efficiencies (PREs) of the

estimators *

R

y and *

C

y have been computed

along with the proposed estimator y*

( )

t

F at its

optimum with respect to the usual unbiased

estimator _y*_{for two data sets for different}

values of k ; results are displayed in Table 1.

References

Cochran, W. G. (1977). Sampling

techniques, (3rd_Ed._{). New York: John Wiley &}

Sons.

Hansen, M. H., & Hurwitz, W. N. (1946). The problem of non-response in sample surveys. Journal of the American Statistical

Association, 41, 517-529.

Results and Conclusion

Table 1 shows that the percent relative efficiency (PRE) of the proposed estimator

( )

t y*

F at its optimum with respect to y* is at its

maximum over the ratio estimator *

R

y and the

estimator *

C

y in both the populations. In

population I, the PREs of all the estimators decreases with the increase in the value of k while in population II, the PREs of all the estimators increases with the increase in the value of k . Further, it is envisaged that the estimator

{ }

y*_F

( )

t _opt is the best estimator among

* y , *

R

y and *

C

y in both the populations.

Hidiroglou, M. A., & Särndal, C. E. (1998). Use of auxiliary information for two phase sampling. Survey Methodology, 873-878.

Khare, B. B., & Sinha, R. R. (2007). Estimation of the ratio of the two population means using multi auxiliary characters in the presence of non-response. Statistical Technology In Life Testing, Reliability, Sampling Theory and

Quality Control, 163-171.

Table 1: Percent Relative Efficiencies (PREs) of the Different Estimators with Respect to

*

y _{for Different Values of k}

Estimator Population - I Population - II

( )

1k

( )

15

( )

14

( )

13

( )

12

( )

15

( )

14

( )

13

( )

12 * y 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 * R y 124.27 121.21 117.27 112.01 129.65 129.71 129.79 129.92 * C y 144.83 144.27 143.52 142.45 168.52 175.91 186.72 204.05

( )

{ }

opt * F t y 145.16 144.66 143.97 142.99 178.81 188.82 203.89 229.16

(9)

KUMAR, SINGH & BHOUGAL

519

Khare, B. B., & Srivastava, S. (1993). Estimation of population mean using auxiliary character in presence of non-response. National

Academy of Science and Letters India, 16, 111-114.

Khare, B. B., & Srivastava, S. (1995). Study of conventional and alternative two-phase sampling ratio, product and regression estimators in presence of non-response.

Proceedings of the Indian National Science Academy, 65, 195-203.

Khare, B. B., & Srivastava, S. (1997). Transformed ratio type estimators for the population mean in the presence of non response. Communications in Statistical Theory

and Methods, 26, 1779-1791.

Neyman, J. (1938). Contribution to the theory of sampling human populations. Journal

of the American Statistical Association, 33, 101-116.

Okafor, F. C., & Lee, H. (2000). Double sampling for ratio and regression estimation with sub-sampling the non-respondents. Survey

Methodology, 26(2), 183-188.

Rao, P. S. R. S. (1986). Ratio estimation with sub sampling the non- respondents. Survey

Methodology, 12, 217-230.

Rao, P. S. R. S. (1987). Ratio and regression estimates with sub sampling the non -respondents. Paper presented at a special

contributed session of the International Statistical Association Meeting, Sept., 2-16,

1987, Tokyo, Japan.

Singh, H. P., & Kumar, S. (2008 a). Estimation of mean in presence of non-response using two phase sampling scheme. Statistics

Papers, DOI 10.1007/s00362-008-040-5.

Singh, H. P., & Kumar, S. (2008 b). A regression approach to the estimation of finite population mean in presence of non-response.

Australian and New Zealand Journal of Statistics, 50(4), 395-408.

Singh, H. P., & Kumar, S. (2008 c). A general family of estimators of finite population ratio, product and mean using two phase sampling scheme in the presence of non-response. Journal of Statistical Theory and

Practice, 2(4), 677-692.

Singh, H. P., & Kumar, S. (2009 a). A general class of estimators of the population mean in survey sampling using auxiliary information with sub sampling the non-respondents. Korean Journal of Applied

Statistics, 22(2), 387-402.

Singh, H. P., & Kumar, S. (2009 b). A General Procedure of Estimating the Population Mean in the Presence of Non-Response under Double Sampling using Auxiliary Information.

SORT, 33(1), 71-84.

Singh, H. P., & Espejo, R. M. (2007). Double sampling ratio-product estimator of a finite population mean in sample survey.

Journal of Applied Statistics, 34(1), 71-85. Singh, V. K., & Shukla, D. (1987). One parameter family of factor type ratio estimator.

Metron, 45(1, 2), 273-283.

Singh, V. K., Singh, G. N., & Shukla, D. (1994). A class of chain ratio type estimators with two auxiliary variables under double sampling scheme. Sankhya, 56(2B), 209-221.

Sitter, R. R. (1997). Variance estimation for the regression estimator in two-phase sampling. Journal of the American Statistical

Association, 37, 95-102.

Tabasum, R., & Khan, I. A. (2004). Double sampling for ratio estimation with non- response. Journal of the Indian Society of

Agricultural Statistics, 58, 300-306.

Tabasum, R., & Khan, I. A. (2006). Double sampling ratio estimator for the population mean in presence of non-response.