Population Mean Estimation with Sub Sampling the Non-Respondents Using Two Phase Sampling

Volume 13 | Issue 1 Article 12

5-1-2014

Population Mean Estimation with Sub Sampling

the Non-Respondents Using Two Phase Sampling

Sunil Kumar

Alliance University, Karnataka, India, [email protected]

M Viswanathaiah

Shantha Group of Institutions, Karnataka, India, [email protected]

Follow this and additional works at:http://digitalcommons.wayne.edu/jmasm

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Recommended Citation

Kumar, Sunil and Viswanathaiah, M (2014) "Population Mean Estimation with Sub Sampling the Non-Respondents Using Two Phase Sampling," Journal of Modern Applied Statistical Methods: Vol. 13 : Iss. 1 , Article 12.

DOI: 10.22237/jmasm/1398917460

Sunil Kumar is an Associate Professor. Email at [email protected]. M.

Viswanathaiah is a Professor. Email at [email protected].

Population Mean Estimation with Sub

Sampling the Non-Respondents Using Two

Phase Sampling

Sunil Kumar

Alliance University Karnataka, India

M. Viswanathaiah

Shantha Group of Institutions Karnataka, India

The problem of non-response in double (or two phase) sampling is dealt with combined ratio, product and regression estimators. Expressions of bias and MSE for these estimators are obtained. Comparisons of a proposed strategy with a usual unbiased estimator and other estimators are carried out and results obtained are illustrated numerically using an empirical sample.

Keywords: Study variable, auxiliary variable, bias, mean squared error,

non-response

Introduction

In surveys regarding human populations, it is common for some information to be missing, even after some callbacks. Hansen and Hurwitz (1946) considered the problem of non-response while estimating a population mean by taking a sub sample from the non-respondent group and proposed an estimator by considering the information available from response and non-response groups. In estimating population parameters such as the mean, total or ratio, product and regression, sample survey experts sometimes use auxiliary information to improve the precision of estimates. Using Hansen and Hurwitz’s (1946) technique, several authors including Cochran (1977), Rao (1986, 1987), Khare and Srivastava (1993, 1995, 1997), Okafor and Lee (2000), Lundström and Särndal (2001), Särndal and Lundström (2005), Tabasum and Khan (2004, 2006), Singh and Kumar (2008, 2009a, b, 2010), and Singh, et al. (2010) have suggested improvements to the population mean estimation procedure in the presence of non-response using an auxiliary variable.

Following Singh and Ruiz Espejo (2007), a class of ratio-product estimators in two phase sampling in the presence of non-response is suggested in this article, and its properties studied. An estimator was studied by using one auxiliary variable for two phase sampling, which is the combined regression with Okafor and Lee’s (2000) estimator and Singh and Ruiz Espejo’s (2007) estimator for no information case. The conditions for attaining minimum mean squared error of the proposed classes of estimators were obtained. A comparison of the proposed estimator with other estimators was conducted and a numerical illustration is provided to support the proposed estimator.

Double Sampling Ratio, Product and Regression Estimator

Let y and x be the study and auxiliary variables with population means Y and X respectively. The population is divided into N (responding) and ₁ N (non-₂ responding) units such that N N1 2 N. When the population mean X of the

auxiliary variable x is unknown, it is suggested that a first phase sample of size

n _{be selected from the population of size N using the simple random sampling}

without replacement (SRSWOR) method, and observing the information on variable x. From these selected n units, a second phase sample size n n



 



is selected for the study variable _{y , and it is observed that} n units respond and 1 n 2

units do not respond in the sample of size n. Further, from n non-responding 2

units, select a sub sample of size _r n2 _{;k 1}

k

_  _

 

  using SRSWOR. Hence, there are n r1 responding units on y . Consequently, to estimate Y using the sub

sampling scheme suggested by Hansen and Hurwitz (1946),

*

1 1 2 2r

y w y w y ,

where w1



n n w1



, 2 



n n2



;y and 1 y denotes the sample means of the 2r y

variable based on n and 1 r units, respectively.

Similarly, to estimate the population mean X of the auxiliary variable x, the estimator _{x ,} *

*

1 1 2 2r

with variance

 

* 2 2





2 (2) 1 1 1 x x W k Var x S S n N n    _  _    where 2 x S and 2 (2) x

S are the population mean square of the auxiliary variable y for the entire population and for the non-responding portion of the population.

Khare and Srivastava (1993) proposed ratio and product methods for estimators respectively as:

* 1R x* t y x     _{ }   and * * 1P x . t y x    _{ }   

The MSE’s of the estimators t and 1R t to the first degree of approximation, are 1P

 













_

_

_

_

2 2 2 1 2 2 2 (2) (2) (2) 1 1 ₂ 1 1 1 2 R y yx x y y yx x MSE t S R R S S n n n N W k S R R S n       _  _   _  _             _  (2)

 













_

_

_

_

2 2 2 1 2 2 2 (2) (2) (2) 1 1 ₂ 1 1 1 2 P y yx x y y yx x MSE t S R R S S n n n N W k S R R S n       _  _   _  _             _  (3) where R Y X





, Syx yx y xS S , Syx(2) yx(2)S Sy(2) x(2), yx 



S Syx x2



,



2



(2) (2) (2) yx yx x

K  S S , and yx and yx(2) respectively denote the correlation

coefficients between x and y for the whole population and for the non-response group of the population.

Okafor and Lee (2000) proposed a double (two phase) sampling regression estimator in the presence of non-response on study as well as auxiliary variables, as





* * 3Re t  y b x x  (4) where



_{* *}2



_{* * *}2 _{2 2 *} 1 1 1 1 1 1 ˆ _{, ,} 1 1 n r n r xy x xy i i i i x i i i i i i b s s s x y k x y nxy s x k x nxx n   n         _   _  _   _  _





_  _





_.

The MSE of the estimator t is 3Re

 



2



2 2 2 



2





2



3Re (2) (2) (2) 1 1 1 ₁ 1 1 ₂ yx y y y yx yx yx x W k MSE t S S S S n n  n N n          _  _  _  _    _         (5)

Singh and Ruiz Espejo (2007) defined an estimator in presence of non-response as





* * * * 1 SR x x t y x x       _   _    (6)

where  is any suitably chosen constant. For  0,1, the class of estimators *

SR

t reduces to the Khare and Srivastava (1993, 1995) and Tabasum and Khan (2004) product and ratio type estimators, that is, t and _1P t . _1R

The MSE of the estimator *

SR

t to the first degree of approximation is

 





 













_

_

_{ }

_

_

_

_

* 2 2 2 2 2 2 (2) (2) (2) 1 1 1 1 _{2 1 2 1 2} 1 2 1 2 1 2 SR y y yx x y yx x MSE t S S R R S n N n n W k _{S R R S} n           _{ }  _ _  _                  _  (7) which is the minimum, when

* 1 1 2 D RD   _  _  ,

where









2 2 2 2 * 2 2 (2) (2) (2) 1 1 1 1 _, 1 1 x x yx x yx x W k W k D S S D K S K S n n n n n n   _{ }  _{ }  _  _  _ _  _  _          .

Thus, the minimum MSE of *

SR t is given by









* * * 2 2 2 ( ) * * 2 2 2 (2) (2) (2) 1 1 1 1 ₂ 1 2 . SR opt y y yx x y yx x D D MSE t S S S n N n n D D W k _S D D _S n D D          _  _ _  _  _  _{ }                        _  _  _{ }       (8)

The Proposed Estimator

An estimator was developed using one auxiliary variable for two phase sampling for estimating the population mean Y of a study variable y in the presence of non-response. Okafor and Lee’s (2000) estimator is combined with the estimator

*

SR

t . Thus, the proposed estimator is:

















* * * * * * 3Re * 1 1 CR x x t y b x x x x x x t x x            _   _        _   _    (9)

where  is any suitably chosen constant and t is defined at (3Re 4).

To obtain the bias and mean squared error of *

CR t ,

















2





* * * * 2 0 1 2 3 4 1 , 1 , 1 , _{xy xy} 1 , _{x x} 1 , y Y  x  X  x X  s S  s S  such that

 

i 0 0 4 E    i to ;

 

2 2 2





2 0 (2) 1 1 1 y y W k E S S n N n  _  _     ;

 





2 2 2 2 1 (2) 1 1 1 x x W k E S S n N n  _  _     ;

 

2 2 2 1 1 x E S n N  _  _    ;









2 0 1 (2) 1 1 1 yx yx W k E S s n N n   _  _     ;



0 2



1 1 yx E S n N   _  _    ;





2 1 2 1 1 x E S n N   _  _    ;





_



_



_

21 2





21(2) 1 3 1 1 2 xy xy N N n W k E N N nXS n XS           ;





_



_



_

21 2 3 _{1 2} xy N N n E N N n XS          ;





_



_



_

30 2





30(2) 1 4 2 2 1 1 2 x x N N n W k E N N nXS n XS           ;





_



_



_

30 2 4 _{1 2} 2 x N N n E N N n XS          ; where



 



1 1 N _{r s} rs i i i x X y Y N   



  _{; (2)} 1 2



₂

 

₂



1 2 1 N N N r s rs i i i x X y Y N     



  ; 1 2 2 1 2 1 N N N i i X x N    



; 2 1 2 1 2 1 N N N i i Y y N    



;

 

r s , being non negative integers.

Expanding * CR t in terms of 's results in    



1





   1   1



* 0 0 2 1 3 4 2 1 1 2 1 1 1 1 1 1 1 1 CR t Y  A                          (10)

where A0 



 R R



; 



Y X



.

Assume that 4 1, 1 1 and 2 1 so that





1 4 1    ,



1 1



1   and





1 2 1  

 are expandable in terms of 's. Expanding the right hand side of (10) in terms of 's _{and neglecting terms of} 's with power greater than two results in:











 











* 2 2 2 0 2 1 2 1 2 0 2 0 1 2 1 1 2 0 2 0 1 2 2 2 2 0 2 1 1 2 0 2 1 2 1 1 2 2 4 1 4 2 3 1 3 2 2 2 2 2 . CR t Y Y A A                                                                  (11) Taking expectations of both sides of (11) results in the bias of *

CR t to the first degree of approximation, as

 



















_{ }

_

_

_

_

_











2 2 * 2 (2) (2) 2 21(2) 30(2) 2 30 21 2 2 1 1 1 _{1 2 2} 1 1 2 2 . 1 1 1 1 2 yx yx x CR yx yx x yx xy x xy x S R R S X n n W k B t S R S nX W k N K N N n n S S n S S            _{ }   _  _            _            _{    }      _{       }       _{    }_ _ _ _     (12)

Squaring both sides of (11) and neglecting terms of 's with power greater than two results in





















































2 2 * 2 0 2 1 0 2 1 2 2 2 2 2 2 2 2 0 2 1 1 2 0 2 1 1 2 2 2 0 2 0 1 0 2 1 1 2 0 0 2 0 1 2 1 2 1 2 2 2 2 1 2 2 1 2 2 . CR t Y Y A Y A A A                                                       (13)

Taking expectations of both sides of (13) results in the MSE of *

CR

t to the first degree of approximation as:

 



   



 

_

_{   }

_

2 * 2 2 2 2 0 0 2 2 2 2 2 (2) 0 (2) 0 (2) 1 1 1 1 _{2 1 2 2 1} 1 2 1 2 2 1 CR y y x yx y x yx MSE t S S A R S A RS n N n n W k S A R S A RS n         _{ }  _ _  _                      _ (14)

which is the minimum * 0 1 1 2 D A RD   _   _  .

The resulting minimum mean squared error of *

CR

t is therefore given by:













* * * 2 2 2 ( ) * * 2 2 2 (2) (2) (2) * ( ) 1 1 1 1 ₂ 1 2 . CR opt y y yx x y yx x SR opt D D MSE t S S S n N n n D D W k _S D D _S n D D MSE t          _  _ _  _  _  _{ }                        _  _  _{ }        (15) From (1), (2), (3), (5), (7) and (14),

 

_*

 

_*







2 ₂





_*



0 0 2 1 2 2 1 CR Var y MSE t   A R D R  A D (16)

 

 

_*



_{2 *}





2 ₂





_*



1R CR 2 2 1 0 2 2 1 0 MSE t MSE t  R D RD   A R D  A RD (17)

 

 

_*



_{2 *}





2 ₂





_*



1P CR 2 2 1 0 2 2 1 0 MSE t MSE t  R D RD   A R D  A RD (18)

 

 

*



2 2



*



2



3Re CR 4 4 yx MSE t MSE t    R D R RD D D R D (19)

 

*

 

*



2



* 0 0 0 0 2 4 2 SR CR MSE t MSE t RD A A  A  A D (20) The differences given by (16), (17), (18), (19) and (20) are positive, respectively, if









* 0 * 0 1 0 1 2 1 1 0 2 D either A RD D or A RD        _     _{  } (21)

* * 2 1 2 2 2 1 2 2 yx yx yx yx D D either RD R D D or R RD             _{ }        _      _    (22) * * 1 2 2 1 2 2 yx yx yx yx D either R RD R D or RD R R            _  _        _       _    (23) * * 1 1 2 2 1 1 2 2 D either RD D or RD        _  _       _       _    (24) * 0 * 0 1 0 1 2 2 2 1 1 0 2 2 2 A D either RD A D or RD        _  _ _      _ _{   }   _    (25)

The proposed estimator *

CR

t is more robust than estimators _{y ,} * 1R

t , t and _1P *

SR

t respectively, if (21) – (25) hold true.

Empirical Study

To examine the robustness of the proposed estimators, consider the following data sets (Khare & Sinha, 2004, p. 53) from a survey on physical growth of an upper socioeconomic group of 95 Varanasi school children under an Indian Council of Medical Research (ICMR) study, Department of Pediatrics, Banaras Hindu University (BHU) during 1983-1984. The first 25% (24 children) were considered non-response units.

The parameter values related to the study variable _{y (weight in kg) and the} auxiliary variable x (chest circumference in cm) were:

19.4968 Y  , X 55.8611w , S_y 3.0435 , S_x 3.2735 , Sy(2) 2.3552 , 5137 . 2 ) 2 (  x S , _yx 0.8460 , yx(2) 0.7290 , R0.3490 , yx 0.7865 , 6829 . 0 ) 2 (  yx  , W2 0.25, N2 24, N1 71, N 95, n35, n70.

The percent relative efficiencies (PREs) of different suggested estimators were computed with respect to a usual unbiased estimator _{y for different values} *

of k.

Table 1: Percent relative efficiency of different Y estimators with respect to y*.

Estimators _{(1/5) (1/4)} (1/k) _{(1/3) (1/2)} * y 100.00 100.00 100.00 100.00 1R t 165.65 165.35 164.95 164.41 3Re t 218.74 220.25 222.29 225.16 * * ( ) = ( ) SR opt CR opt t t 220.00 221.16 222.81 225.18

Table 1 shows that

(i) the PRE’s of the estimators t and 3Re tCR opt* ( ) increase as the value of

k increases, while the PRE’s of the estimator t decrease as the 1R

value of k increases.

(ii) the performance of the proposed estimator * ( )

CR opt

t is the best among all other estimators _{y ,} *

1R

t and t because it has the largest gain in 3Re

efficiency.

Based on these study results, the proposed estimator *

CR

t is recommended for use in practice.

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