Mean Estimation Using Qualitative Prior

Research Article Viplav Kumar Singh1, Jong Min Kim2, Rajesh Singh3 1,3

Department of Statistics, Banaras Hindu University, Varanasi221005, India. 2

Statistics, Division of Science and Mathematics, University of Minnesota, Morris. Correspondance to: Mr. Viplav Kumar Singh, Banaras Hindu University, Varanasi. E-mail Id: Viplavkumarsingh0802@ gmail.com

Mean Estimation Using Qualitative Prior Information under Stratified Random Sampling Scheme Abstract Adapting [14] and [15], we proposed modified classes of estimators for the unknown mean of study variable when auxiliary information is available in the form of attribute. The bias and mean square error of the estimators belonging to the classes are obtained and the expressions for the optimum parameters minimizing the asymptotic mean square error (MSE) are given in closed form. An empirical study is carried to judge the merits of the suggested estimators over [2], regression estimator and other known estimators. Both theoretical and empirical study results present the soundness and usefulness of the suggested estimators in practice.

Keywords: Proposed class, Regression estimator, Stratified random sampling scheme, Efficiency comparisons.

1. Introduction In sample survey, prior information on the finite population is quite often available from previous experience, survey or administrative databases. Wide literature of basic statistics was given by [7]. The sampling literature describes a wide variety of techniques for using auxiliary information to improve the sampling design and/or obtain more efficient estimators. It is well known that when auxiliary information is to be used at the estimation stage, the ratio, product, regression methods are widely employed. Many authors such as [3] and [5] have suggested a difference class of estimator and a combined ratio-product type estimators using stratified random sampling, where the auxiliary information is in the form of auxiliary variates. In many practical situations instead of existence of auxiliary variables, there exist some auxiliary attributes  (say), which are highly correlated with the study variable Y such as (a) The height of a person (y) may depend on gender (ϕ) (b) Amount of milk produced (y) may depend on particular breed of the cow (ϕ) (c) The yield of wheat crop produced may depend on a particular variety of wheat, etc. Taking into consideration the point biserial correlation coefficient between auxiliary attribute  and the study variable y, several authors including [4, 6, 8–13] envisaged a How to cite this article: Singh VK, Kim JM, Singh R. Mean Estimation Using Qualitative Prior Information under Stratified Random Sampling Scheme. J Adv Res Appl Math Stat 2016; 1(2): 87-99. ISSN: 2455-7021

large number of improved estimators for the population mean Y of the study variable y. There are some situations when in place of one, we have information on two qualitative variables and in support of such type of study, authors including [14–16] have suggested different classes of estimators using two auxiliary attributes in stratified random sampling. Here, we assume that both auxiliary attributes have significant bi-serial correlation with the study variable y and there is significant phi-correlation between the two auxiliary attributes. In this article, we proposed different classes using single and multi-auxiliary attributes and compared them with some well-known estimators in literature.

© ADR Journals 2016. All Rights Reserved.

Singh VK et al.

J. Adv. Res. Appl. Math. Stat. 2016; 1(2)

Consider a finite population U of size N, stratified into L th strata with i stratum containing N i units, where I = 1,

E (d 0

 d )



L i 1

th

stratum such that

Wi2 g i S yp1i

1

2, 3…L. Let a simple random sample of size n i be drawn without replacement from the i

L i 1

E (d 0 d 2

 )

 r110

YP1 L i 1

Wi2 g i S yp2i YP2

 r101

ni  n

Let ( y ij ,  kij ) ( k  1, 2) denote the values of the study

and E (d 1 d 2

 )

variable (y) and the auxiliary attributes  k (k  1, 2) respectively in the i

th

stratum.

 kij  1 , if jth unit in the stratum I possesses the attribute  k

L

i 1

L

Wi p ki and Pk  i 1 Wi Pki be the

sample and population proportions of the units of the auxiliary attributes

2 yi

Ni

S   j1

 r011

( y ij  Yi ) 2 ( N i  1)

,S

2 p1i

Ni

  j1

( kij  Pki ) 2 ( N i  1)

And N

Yi   ji1

where

p ki 

P1 P2

be the population variance of the study variable and the th auxiliary variables respectively in the i stratum.

= 0, otherwise



Wi2 g i S p1p2i

Where

Suppose

Let p k 

L i 1

 y ki 

a ki A , Pki  ki and ni ni Ni

Ni

A ki   j1  kij , a ki   j1  kij Large sample properties of the classes are obtained according to stratified random sampling scheme. In so doing, we use the following notations in the rest of the article: L

y st  i1 Wi y i  Y (1  d 0 )

y ij Ni

, S 2yk i 



Ni

( y ij  Yi )( kij  Pki )

j1

( N i  1)

,

S yki S ki S yi

be the population bi-covariance and point bi-serial correlation between the study variable and the auxiliary th attributes, respectively in the i stratum. Also N

S 12i   ji1

 1 2i 

(1ij  P1i )( 2ij  P2i )

S 12i S 1iS  2i

( N i  1)

and

be the population bi-covariance and

L

phi-correlation coefficient between the auxiliary th attributes in the i stratum respectively.

L

2. Estimators in Literature

p1st  i1 Wi p1i  P1 (1  d1 ) p 2st  i1 Wi p 2i  P2 (1  d 2 ) Such that, E (d 0 )  E (d 1 )  E ( d 2 )  0



Also E ( d 2 )  0

L i 1

W i 2 g i S 2yi Y

E (d

2 1

 )

E (d

2 2

 )

L i 1

2 i

2

W g iS

2 p1i

P12 L

i 1

ISSN: 2455-7021

Wi2 g i S 2p 2i P22

 r 200

 r020  r002

Koyuncu [14] proposed the combined ratio estimator of population mean in stratified random sampling using auxiliary attribute, given by

ˆ   y st Y R   p 1st

  P1 

(1)

ˆ R , to the first order of The bias and MSE of Y approximation, are respectively given by

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ˆ )  Y (r  r ) B( Y R 020 110

(2)

ˆ )  Y 2 (r  r  2r ) MSE ( Y R 200 020 110

(3)

or



ˆ 2 MSE (Y g (st ) )  Y r200 1   g ( st )



(7)

2 r110 (say) r020 r200

Following [6], a general combined class in stratified random sampling is defined by Koyuncu [14] as

where  g ( st ) 

ˆ Y g ( st )  g ( y st , U st )

A combined regression estimator in stratified random sampling using auxiliary attribute, is given by

where U st 

(4)

p1st and g ( y st , U st ) is a function of y st P1

and U st .

ˆ Y yp1  y st  b yp1 ( P1  p1st )

(8)

ˆ

The MSE of Yyp1 , to the first order of approximation,

ˆ

The bias and MSE of Yg ( st ) , to the first order of approximation, are given by



ˆ )  g r  g Y r  g Y2 r B( Y g ( st ) 2 020 3 110 4 200



(5)

given by

  2yp1 r020 2 yp1 r110  ˆ 2 MSE( Yyp1 )  Y r200    (9) R 1  R 12  where  yp1 

for ( g 2 , g 3 , g 4 ) see [14].



ˆ 2 2 MSE (Y g (st ) )  Y r200  g 1 r020  2g 1 Yr110



Y r110 , the r020 ˆ minimum MSE of the estimators in the class Y is *

By using optimum value of g 1  



L

i 1

Wi2 g i S yp1i



L

i 1

Wi2 g i S 2p1i

A simple general class of biased regression estimators has been proposed by [2]. Thus following [2], we suggested the same in stratified random sampling using auxiliary attribute as

ˆ   y   (P  p ) Y RA 1st st 2 st 1 1st

(10)

g ( st )

found as

with 1st and  2st constants to be properly determined.

2 ˆ )  Y 2 r 1  r110  MSE (Y g ( st ) 200    r020 r200 

(6)

ˆ

The MSE of YRA , up to the first order of approximation, given by

2 ˆ )  Y 2 1   2 (1  r )   2st r  2  2  r110  MSE (Y RA  1st 200 020 1st 1st 2 st  R1  R 12 

ˆ

Minimization of MSE ( YRA ) is achieved for the optimum choice of the constants 1st and  2st

r020   2 r020 (1  r200 )  r110 * 1

 *2 

2 110

r

(11)

r110 R 1  r020 (1  r200 )

Singh [17] proposed a family of estimators for the population mean in stratified random sampling using information on auxiliary attribute, as

  ˆt SS  w 1 y st  w 2 (P1  p1st )exp  st (P1  p1st )    st (P1  p1st )  2 st 

(12)

where w 1 and w 2 are suitable constants.

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MSE of ˆt SS see [17], up to the first order of approximation, given as 3 2   r 2  3r r 2  r 2 r  4 2 r020  8r020 r110 MSE(ˆt SS )  Y 2 1  2 0202 2 020 1104 020 2200 2 2   r020  r020 r200  r110  2r020 r200  2r020 r110  r020 r200 r110 

(13)

or

MSE ( ˆt SS )  Y 2 1   SS 

 SS 

Where

(say) and  

2 r020

(14)

2 2 2 3 2 r020  3 r020 r110  r020 r200  4  2 r020  8  r020 r110 2 2 4 2 2 2  r020 r200  r110  2 r020 r200  2 r020 r110  r020 r200 r110

There are some situations when in place of one; we have information on two qualitative variables. Using multi-auxiliary information Koyuncu [14] proposed the following estimator

st P1 . For detail see [13]. 2( st P1   st )

 2st   P  1st  a st ( p1st  P1 )   P2   c st ( p 2st  P2 )  ˆ 1 YK  y st k 1st   exp    k 2st   exp     p 1st   P1  b st ( p1st  P1 )   p 2st   P2  d st ( p 2st  P2 ) 

(15)

Where k 1st , k 2 st ,  1st ,  2st are suitable constants. Also

Under the condition k 1st  k 2 st  1 , bias and MSE of

a st , b st , c st and d st are either real numbers or the function of the auxiliary attribute.

YK (see [14]), is given as

ˆ )  Y k 1  r  a st (a st  2b st )   a   1st (  1st  1)   (a   ) r B( Y   1st K 020  1st st st 1st 110 2! 2      (   1)   c ( c  2d st )  k 2st 1  r002  st st   2st c st  2st 2st   ( c st   2st ) r101  1 2! 2   

(16)

 B1D1 2  A 1E 12  2C1 E 1 D1  MSE ( YK )  Y 1   2 A1 B1  C1  

(17)

2

or

MSE ( YK )  Y 2 1   K  2

where  K 

(18)

2

B1 D1  A1 E1  2C1E1 D1 A 1B1  C1

2

(say)

and









A 1  1  2a st2  2  12st  2a st b st  4 1st a st   1st r020  r200  4a st   1st ) r110  B1  1  2c st2  2  22 st  2c st d st  4  2 st c st   2 st r002  r200  4c st   2 st ) r101 

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Singh VK et al.

  (  1)    a (a  2b st ) C1  1  1  r020  st st   1st a st  1st 1st   (a st   1st )r110   2(c   2st )r101 2! 2      (  1)   c (c  2d st )  (a st c st   1st c st   2st a st   1st  2st )r011   st st   2st c st  2st 2st  r002  r200 2! 2    (   1)   a (a  2b st ) D1  1  r020  st st   1st a st  1st 1st   (a st   1st )r110 2! 2    (   1)   c (c  2d st ) E1  1  r002  st st   2st c st  2st 2st   (c st   2st )r101 2! 2   3. Generalized Class of Estimators

 2( pk st )  i 1 Wi  2 (i ) ( k ) and correlation coefficient

3.1. Inspired by [2], [12] and [14], we define the following estimators of population mean Y as

 yp k ( st )  i1 Wi  ypk (i ) and  st being constant

L

L

which takes finite values for designing the different estimators. Some members of the proposed class of estimators [for different values of ( n 1st , m1st ,  )] have been summarized in Table 3 in appendix.

st   n 1st P1  m1st   ˆ 1 ˆ YP  1st Yyp1   2st y st    (19)   n 1st p1st  m1st  

Expressing Eq. (19) in terms of d i ' s , we have

where ( ( 1st ,  2st ) are suitably chosen scalars such that

ˆ1

MSE of YP is minimum, ( n 1st , m 1st ) are either

ˆ 1   Y(1 d )  byp1 d1    Y(1 d )1 T d st (20) Y  2 P 1st  0 0 h 1 R1  

constants or functions of some known population parameters such as coefficient of variation L

C p kst  i1 Wi C p ki (k=1), Coefficient of skewness

where R 1 

L

1( p kst )  i1 Wi1( i ) ( k ) , Coefficient of kurtosis



T1   2 p1 P1  2 p1 p1st   yp1





1

the following possible values



2



; T3  1p1 P1 1p1 p1st   yp1

2



; T6  P1 p 1st   2 p1

; T2   2 p1 P1  2 p1 p 1st   yp1

T4   22 p1 P1  22 p1 p1st   yp1



1

Y 1 and Th  n 1st P1 n 1st p1st  m1st  has P1



; T5  P1 p 1st   yp1



1

1





2



1



1

;

;



1

T7   yp1 P1  yp1 p1st  1 . We assume Th d 1  1 , so that the term 1  Th d 1 

  st

is

expandable. Thus by expanding the right hand side of

Eq. (20) and neglecting terms of d i ' s having power greater than two, we have

ˆ 1  Y  1  d  b yp1 d 1    1  d  T  d  T  d d   st ( st  1) T 2 d 2  Y  1st   P 0 2 0 h st 1 h st 0 1 h 1  R1  2     or

ˆ 1  Y  Y  1  d  b yp1 d 1    1  d  T  d  T  d d   st ( st  1) T 2 d 2   1 (21) Y  1st    P 0 2 0 h st 1 h st 0 1 h 1 R1  2     

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Taking expectations on both sides of Eq. (21), we get the

ˆ1

where AB( YP ) denotes the asymptotic mean square error and  ( ) denotes the terms of order greater than two.

ˆ1 bias of YP to the first degree of approximation as ˆ 1 )  AB( Y ˆ 1 )  (  ) B( Y P P

ˆ 1 )  Y     1  T  r   st ( st  1) T 2 r   1 AB ( Y P 2 h st 110 h 020   1st  2    

(22)

Squaring both sides of Eq. (21) and neglecting terms of d i ' s having power greater than two, we have 2 2 ˆ 1  Y ) 2  Y 2 1   2 1  d 2  b yp1 d 1  2d  2b yp1 d 1  2b yp1 d 0 d 1    2 1  d 2  T 2  2 d 2 (Y  P 1st  0 0 2st 0 h st 1 R1 R1 R 12   



b yp d 1    2Th  st d 1  4Th  st d 0 d 1   st ( st  1)Th2 d 12  21st 1  d 0  1   2 2st 1  d 0   st Th d 1 R1  



 Th  st d 0 d 1 

b yp d 0 d 1   st ( st  1) 2 2  Th d 1   21st  2st 1  2d 0  Th  st d 1  d 20  2Th  st d 0 d 1  1 2 R1   

b yp1  st Th d 12 R1

 st ( st  1) 2 2  Th d 1  2 



ˆ1

(23)

ˆ1

Analogously, using Eq. (23) the MSE of YP can be expressed as

where AMSE(YP ) denotes the asymptotic mean square error

ˆ 1 )  AMSE( Y ˆ 1 )   ( ) MSE (Y P P



ˆ 1 )  Y 2 1   2 A   2 B  2  C  2  2 D AMSE( Y P 1st 2 2 st 2 1 2 2 1 2 2



(24)

where

A 2  1  r200 

b 2yp1 r020 R

2 1



2b yp1 r110 R1

B 2  1  r200  Th2  st2 r020  4Th  st r110   st ( st  1)Th2 r020

C 2  1  r200  2Th  st r110 

b yp1 r110 R1



D 2  1  Th  st r110 

b yp1  st Th r020 R1



 st ( st  1) 2 Th r020 2

 st ( st  1) 2 Th r020 2

ˆ1

Minimization of AMSE(YP ) is achieved for the optimum choice of the constants 1 and  2

 B  C2D2  1*   2 2   A 2 B2  C 2 

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 A D  C2  *2   2 2 2   A 2 B2  C 2 

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1st ,  2st , y st , n 1st , m1st ,  st   1,0,,,

which leads to 2 ˆ 1 )  Y 2 1  (B2  2C 2 D 2  A 2 D 2 )  (25) AMSE( Y P   ( A 2 B2  C 22 )  

or



ˆ 1 )  Y 2 1  1 AMSE(Y P P



(26)

(B 2  2C 2 D 2  A 2 D 22 where   (say) (A 2 B 2  C 22 )

3.2. Further motivated by Koyuncu [14], we suggest another efficient estimator using two auxiliary attributes, given as 'st   ˆY2 ' Y ˆ expP1 p1 ' Y ˆ  n2stP2 m2st   (27)   2st yp2   P 1st yp1  P1 p1  n2stp2st m2st  

ˆ

1 P

where Yyp 2  y st  b yp2 (P2  p 2st ) is the well-known

We would like to mention here that the proposed class

ˆ 1 may reduce to some well-known class of population Y P mean Y by putting different values of 1st ,  2st , y st , n 1st , m1st ,  st  . i.e.,

ˆ1 Y ˆ ; for Y P R

1st ,  2st , y st , n 1st , m1st ,  st   0,1,1,0,1

regression estimator that depends on study variable y and auxiliary attribute p 2 . Also ( '1st , ' 2 st ) are

ˆ

2

suitably chosen constants such that MSE of YP is minimum, ( n 2 st , m 2 st ) are either constants or function of some known population parameters such as coefficient of variation, kurtosis, skewness and correlation coefficient as already defined in section 3.1 (except here we take value k=1,2). Some members of proposed class of estimators

ˆ 2 [ for differentvaluesof (n , m , ' , ' , ' )] Y P 2st 2st 1st 2st st

ˆ1 Y ˆ ; for Y P P

have been summarized in Table 4 in appendix.

1st ,  2st , y st , n 1st , m1st ,  st   0,1,1,0,1

Expressing Eq. (27) in terms of d i ' s , we have

ˆ1 Y ˆ ; for Y P reg   d1 d 12  ˆ 2 YP  '1st Y (1  d 0 )  b yp1 P1d1 exp    ' 2st Y (1  d 0 )  b yp2 P2 d 2 1  Th' d 2 4   2



We assume that

1  T d  ' h

  'st

2





Th' d 2  1 , so that the term





  'st

(28)

right-hand side of Eq. (28) and neglecting the terms of d i ' s having power greater than two, we have

is expandable. Thus by expanding the

2 2 ˆ 2  Y' 1  d  d 1  3d 1  d 0 d 1  b yp1  d  d 1   ' Y 1  d   ' T ' d Y  P 1st  0 1 2 st 0 st h 2 2 8 2 R 1  2  



  ' st Th' d 0 d 2 

  'st ( ' st 1) 2 2 b yp1 T' h d 2  d 2   'st Th' d 22  2 R2 





(29)

Or 2 2 ˆ 2  Y  Y' 1  d  d 1  3d 1  d 0 d 1  b yp1  d  d 1 Y P 1st  0 1 2 8 2 R 1  2 

  'st Th' d 0 d 2 

93

   ' 2st Y 1  d 0   ' st Th' d 2 



  ' st ( 'st 1) 2 2 b yp1 T'h d 2  d 2   ' st Th' d 22   Y 2 R2 





(30)

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where R 2 


Y and Th' has the following possible values P2



T1'  P2 p 2st   yp2





1

'



2

; T2  P2 p 2st   yp 2





1

'

2



1

 

2

; T3  1p 2 P2 1p 2 p 2st   yp2

2 T5'   22p 2 P2  22 p2 p 2st  1 ; T6'   22 p2 P2  22p 2 p 2st  12 p2

Taking expectation on both sides of Eq. (30) we get the

ˆ 2 to the first degree of approximation as bias of Y P



1



T4'   2p 2 P2 p 2st   yp2



1

;

1

ˆ

2

where AB( YP ) denotes the asymptotic bias and  ( ) denotes the terms of order greater than

ˆ 2 )  AB( Y ˆ 2 )   ( ) B( Y P P  byp r020   3r r  ' ( ' 1) 2  AB (YˆP2 )  Y  '1st 1  020  110  1    '2 st 1   'st Th'r202  st st T 'h r020 two. 8 2 2 R1  2   

b yp1 ' st Th' r020      1 R2  

(31)

Squaring both sides of Eq. (30) neglecting terms of d i ' s having power greater than two and then taking expectations, we have

ˆ 2 )  AMSE(Y ˆ 2 )   ( ) MSE (Y P P ˆ

where AMSE( YP2 ) denotes the asymptotic mean square error



ˆ 2 )  Y 2 1  ' 2 A  ' 2 B  2  C  2  D  2   E AMSE ( Y P 1st 3 2 st 3 1 3 2 3 1 2 3



(32)

where

  b 2yp1 r020 2b yp1 2b yp1 A 3  1  r200  r020  2r110   r  r 020 110  R1 R1 R 12    b 2yp 2 r002 4 b yp2 ' st Th' r002 2 2 ' 2 B 3  1  r200  'st T' h r002   4' st Th r101  ' st (' st 1)T ' h r002  R2 R 22  

2 b yp2 r101   3r020 r110 b yp1 r020     C 3  1   R2  8 2 2R 1  

  ' st ( ' st 1) 2 b yp1  'st Th' r002  ' D 3  1  T ' h r002   ' st Th r101   2 R2    b yp1 r020 b yp 2  ' st Th' r002 3r020 ' st ( ' st 1) 2 E 3  1  r200   T ' h r002    r110  2 ' st Th' r101 8 2 2 R R 1 2 

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

b yp1 r110 R1

ˆ



b yp2 r101 R2

Singh VK et al.



b yp1 ' st Th' r011 R1



b yp2 r011 2R 2



' st Th' r011 b yp1 b yp 2 r011    2 R 1R 2 

2

Minimization of AMSE( YP ) is achieved for the optimum choice of constants '1 and ' 2

 B C  D 3E 3  *  A 3D 3  C3E 3  '1*   3 3 ' 2   2  2   A 3 B3  E 3   A 3B3  E 3  which leads to 2 2 2 2 2 2 2 2 3 ˆ 2 )  Y 2 1  A 3 B 3 C 3  A 3 D 3 E 3  A 3 B 3 D 3  B 3 C 3 E 3  2C 3 D 3 E 3  2A 3 B 3 C 3 D 3 E 3  (33) AM min ( Y P   (A 3 B 3  E 23 ) 2  

or



ˆ 2 )  Y 2 1  2 AM min (Y P P



(34)

A 3 B 32 C 32  A 3 D 32 E 32  A 32 B 3 D 32  B 3 C 23 E 23  2C 3 D 3 E 33  2A 3 B 3 C 3 D 3 E 3 where   (say) (A 3 B 3  E 32 ) 2 2 P

ˆ

1     r 1     0 1 P

2

and AM min (YP ) shows the minimum asymptotic

ˆ mean square error of proposed estimator YP2 .

ˆ 1 )  MSE( Y ˆ ) , if AMSE(Y P yp1  b 2yp1 r020 2 b yp1 r110  1     r200   0 R 1  R 12 

ˆ 1 )  MSE(Y ˆ ) , if AMSE( Y P R 200



 r020  2r110 )  0

1 P



(35)

ˆ 1 )  MSE ( Y ˆ ) , if AMSE(Y P RA

ˆ 1 )  MSE( Y ˆ ) , if AMSE(Y P g ( st )

  22st r  2 1    1  1st (1  r200 )  2 r020  21st  21st  2st 110   0 R1  R1 



1 P



ˆ 1 )  MSE (ˆt ) , if AMSE( Y P SS

1     1     0

1     1     0

 2P   K

2 P

SS

or 1P   SS From Eqs. (18) and (34), we have

ˆ 2 )  MSE ( Y ˆ ) , if AMSE(Y P K 95

(37)

exists only R 1  0 .

1

for  P  0

1 P

(36)

1

From Eqs. (3), (7), (9), (11), (14) and (26), we have

1 P

g ( st )

or  P   g (st ) .

4. Efficiency Comparison

1     (r

200

(39)

(38)

K

(40)

Equations (35) to (40) are the conditions under which

ˆ

ˆ



proposed classes YP1 , YP2 have less mean square error, i.e., more efficient than simple mean estimator

ˆ , combined regression y st , usual ratio estimator Y R

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Singh VK et al.


ˆ

ˆ

ˆ

estimator Yyp1 , Rao class YRA , Singh et al. YSS and

ˆ

ˆ

Koyunsu generalized class of estimators ( Yg (st ) , YK ) .

The

MSE

values

for

existing

estimators

ˆ ,Y ˆ ,Y ˆ ,Y ˆ ,Y ˆ ) and adapted estimator (Y R g (st ) yp1 RA SS ˆ 1 ˆ 2 ( Y , Y ) are given in Table 1. When we examine Table P

P

5. Empirical Study

1, we observe that ( Y12P , Y32P , Y52P , Y62P ) have the

To evaluate the performance of proposed estimators

smallest MSE values among ( YP2 , YP1 ) and all existing

ˆ

Yˆ , Yˆ  1 P

2 P

in comparison to other estimators, we have considered the same data set as considered by Koyuncu [14]. Here the apple production amount in 1999 is specified as study variable and apple production amount less than 1000 tons in 1998 as auxiliary attributes in 854 villages of Turkey (Source: Institute of Statistics, republic of Turkey). The data set we have considered here is firstly stratified into six different regions of Turkey (as 1: Agean 2: Central Anatolia 3: Black Sea 4: East and southeast Anatolia 5: Mediterranean 6: Marmara) and from each stratum; we have randomly selected the samples whose sizes are computed by using Neyman allocation method. The summary of the data is given in Table 5 in appendix.

ˆ

ˆ

ˆ

ˆ

ˆ

ˆ

estimators defined in Sec. 2 including YK . Taking usual

ˆ

regression estimator Yyp1 as base, we have also given PRE values for some efficient estimators [among

ˆ 1, Y ˆ 2 ) ] and existing estimators in Table 1. For (Y P P computing the PRE value, we use the following expression

PRE () 

ˆ ) MSE ( Y yp1

* 100 MSE () ˆ1 ˆ2 ˆ ˆ ˆ ˆ ˆ where   YP , YP , YR , Yg (st ) , YRA , YSS and YK

Estimators

Table 1.MSE Values for Adapted and Existing Estimators MSE/ PRE Estimators

ˆ Y R ˆ Y

ˆ1 490041/79.14 Y4 P ˆ 367247/105.6 Y 1

ˆ Y yp1 ˆ Y

387835.5/100* Y61P 371116.6/104.50 Y1P

ˆt SS

ˆ2 346120.0/112.05 Y 2P

ˆ Y K ˆ Y1

248035.4/156.36** Y32P

ˆ1 Y 2P ˆ Y1

341079.6/113.48 Y52P

MSE/ PRE 339709.4/114.17 340622.8/113.86

5P

g ( st )

341079.6/113.70

ˆ

248309.8/156.19**

ˆ2

RA

305562.5/126.93

ˆ

ˆ

180535.1/214.83** 279827.3/138.59

2

309538.5/125.29* Y4 P

1P

223829.9/173.27**

ˆ ˆ

193190.9/200.75**

2

344776.8/112.48 Y6 P

3P

*corresponds to the MSE/PRE of the estimators using single auxiliary attribute; **for two auxiliary attributes

Table 2 establishes the conditions obtained in Sec. 4 empirically. It shows that all conditions are satisfied for the given data set. Table 2.Empirical Study of Theoretical Conditions Explained in Sec. 4

ˆ ) MSE ( Y R

ˆ 1)  AMSE( Y P ˆ MSE (Y )

ˆ 1)  AMSE( Y P ˆ ) MSE( Y

ˆ 1)  AMSE( Y P ˆ ) MSE( Y

ˆ 1)  AMSE( Y P MSE (ˆt SS )

ˆ 2)  AMSE( Y P ˆ ) MSE (Y

Condition (35) −0.02102 < 0

Condition (36) -0.006722 < 0

Condition (37) 0.00912 < 0

Condition (38) -0.00717 < 0

Condition (39) 0.963947 > 0.959686

Condition (40) 0.978962 > 0.959093

ˆ 1)  AMSE(Y P

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yp1

RA

K

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6. Conclusion Section 4 provides the theoretical conditions under

ˆ1 ˆ2 which proposed classes ( YP , YP ) are more efficient than other estimators. Table 2 establishes those conditions empirically and it shows that all theoretical conditions obtained in Sec. 4 are satisfied for the given data set (see Sec. 5). Table 1 exhibits mean square error and percent relative efficiency of proposed and existing classes with respect to usual regression estimator

ˆ . It can be observed that using single auxiliary Y yp1 ˆ 1 has highest percent relative efficiency and attribute Y

7. 8.

9.

10.

1P

ˆ 2 for two auxiliary attributes. Thus, it respectively Y 3P can be concluded that if conditions obtained in Sec. 4

ˆ 1, Y ˆ 2 ) are much are satisfied the proposed classes ( Y P P efficient for application point of view in comparison to other methods References 1. Udny YG. On the methods of measuring association between two attributes. J the Roy Stat and Stat Soc 1912; 75: 579-642. 2. Rao TJ. On certain methods of improving ratio and regression estimators. Comm Stat Theory and Methods 1991; 20: 3325-40. 3. Diana G. A class of estimators of the population mean in stratified random sampling. Statistica 1993; 53(1): 59-66. 4. Naik VD, Gupta PC. A note on estimation of mean with known population proportion of an auxiliary character. Jour of Ind Soc of Agr Stat 1996; 487(2): 151-58. 5. Singh HP, Vishwakarma GK. Combined ratio-product estimator of finite population mean in stratified sampling. Metodologia de Encuestas 2005; 8: 35-44. 6. Jhajj HS, Sharma MK, Grover LK. A family of estimators of population mean using information

11.

12.

13.

14.

15.

16.

17.

on auxiliary attribute. Pak Journal of Stat 2006; 22(1): 43-50. Gujarati DN, Sangeetha. Basic Stat. Theory and Meth. Econometrics 2007; 38(14): 2398-417. Shabbir J, Gupta S. A new estimator of population mean in stratified sampling. Commun Stat Theo Meth 2007; 35: 1201-209. Singh R, Chauhan P, Sawan N et al. Ratio estimators in simple random sampling using information on auxiliary attribute. Pak J Stat Oper Res 2008; 4(1): 47-53. Abd-Elfattah AM, El-Sherpieny EA, Mohamed SM et al. Improvement in estimating the population mean in simple random sampling using information on auxiliary attribute. Appl Math and Comp 2010. Shabbir J, Gupta S. Estimation of the finite population mean in two phase sampling when auxiliary variables are attributes. Hacet J Math and Stat 2010; 39(1): 121-29. Singh HP, Rathour A, Solanki RS. An improvement over regression method of estimation. Statistica 2012; anno LXXII: n. 4. Singh HP, Solanki RS. Improved estimation of population mean in simple random sampling using information on auxiliary attribute. Appl Math and Comp 2012; 218: 7798-812. Koyuncu N. Improved estimation of population mean in stratified random sampling using information on auxiliary attribute. Proceeding of 59th ISI World Statistics Congress, Hong Kong, China, Aug 2013: 25-30. Malik S, Singh R. An improved estimator using two auxiliary attributes. Appli Math Compt 2013; 219: 10983-86. Malik S, Singh R. A family of estimators of population mean using information on point biserial and phi correlation coefficient. IJSE 2013; 10(1): 75-89. Singh R, Sharma P. Efficient estimators of population mean in stratified random sampling using auxiliary attribute. World Applied Sciences Journal 2013; 27(12): 1786-91.

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Appendix

ˆ

1

ˆ

2



In the following Table 3, we have given some family of estimators of proposed class YP , YP as

ˆ1

Estimators

Table 3.Some Particular Members of YP Parameters

 st

n1st

m1st

ˆ1 Y 1P ˆ Y1

1

  2 p1

 yp1

1

1

  2yp1

ˆ1 Y 3P ˆ1 Y

1

 2 p1

  2yp1

1

 1p1

 2yp1

ˆ1 Y 5P ˆ1 Y

1

  22 p1

 yp1

1

1

 2yp1

2P

4P

6P

ˆ

Estimators

2

Table 4.Some Particular Members of YP Parameters

 'st

n 2 st

m 2 st

ˆ2 Y 1P ˆ Y2

1

1

  yp 2

1

1

  2yp2

ˆ2 Y 3P ˆ Y2

1

 12p 2

 yp 2

1

  2 p2

 yp 2

ˆ2 Y 5P ˆ Y2

1

  22 p2

1

1

  22 p2

12p 2

2P

4P

6P

Table 5.Parameters of the Population under Study

N1  106

N 2  106

N 3  94

N 4  171

N 5  204

N 6  173

n 1  13

n 2  24

n 3  55

n 4  95

n 5  10

n6  3

S y1  6425.0872

S y 2  11551.53

S y3  29907.48

S y 4  28643.42

S y 5  2389.77

S y 6  945 .7486

Y1  1536.773585

Y2  2212.594

Y3  9384.309

Y4  5588.012

Y5  966.9559

Y6  404.3988

S p1 (1)  0.432298

S p1 ( 2)  0.45705

S p1 (3)  0.501656

S p1 ( 4)  0.501254

S p1 (5)  0.481975

S p1 (6)  0.320681

P1 (1)  0.24528

P1 (1)  0.29245

P1 (1)  0.46808

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Singh VK et al.

P1 (1)  0.48538

P1 (5)  0.36274

P1 (6)  0.11561

S p 2 (1)  0.4205

S p 2 ( 2)  0.43777

S p 2 (3)  0.49338

S p 2 ( 4)  0.49085

S p 2 (5)  0.21569

S p 2 (1)  0.08671

P2 (1)  0.22641

P2 (2)  0.25472

P2 (3)  0.40426

P2 (1)  0.39766

P2 (5)  0.21569

P2 (6)  0.08671

 yp1 (1)  0.3588

 yp1 ( 2)  0.26587

 yp1 (3)  0.32486

 yp1 ( 4)  0.19113

 yp1 (5)  0.38986

 yp1 (6)  0.67316

 yp2 (1)  0.38094

 yp2 ( 2)  0.29294

 yp 2 (3)  0.36653

 yp 2 ( 4)  0.22694

 yp 2 (5)  0.51919

 yp 2 (6)  0.71532

 p1p 2 (1)  0.8442

 p1p 2 ( 2)  0.86172

 p1p 2 (3)  0.83468

 p1p 2 ( 4)  0.7649

 p1p2 (5)  0.62069

 p1p2 (6)  0.59525

 2 ( p1(1) )  0.56846

 2 (p 1( 2 ) )  1.16549

 2 ( p1( 3) )  2.02669

 2 ( p 1( 4 ) )  2.02028

 2 (p 1( 5 ) )  1.68578

 2 ( p 1( 6 ) )  3.927807

1 ( p 1(1) )  1.20109

1 ( p 1( 2 ) )  0.92567

1 ( p 1( 3) )  0.1300

1 ( p 1( 4) )  0.05902

1 ( p 1( 5) )  0.57519

1 ( p 1( 6) )  2.42539

 2 ( p 2 (1) )  0.24603

 2 ( p 2 ( 2 ) )  0.70924

 2 ( p 2 ( 3) )  1.88328

 2 ( p 2 ( 4 ) )  1.84369

 2 ( p 2 (5 ) )  0.06085

 2 ( p 2 (6 ) )  6.8594

1 ( p 2(1) )  1.32626

1 ( p 2( 2) )  1.14215

1 ( p 2(3) )  0.39656

1 ( p 2 ( 4 ) )  0.42192

1 ( p 2 ( 5 ) )  1.39278

1 ( p 2 ( 6 ) )  2.96314

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