Estimation of Finite Population Mean Using Auxiliary ...

Research Article Poonam Singh1, Nitesh Kumar Adichwal 2, Rajesh Singh3

Estimation of Finite Population Mean Using Auxiliary Attribute in Sample Surveys Abstract

1,2,3

Department of Statistics, Banaras Hindu University, Varanasi, India. Correspondance to: Mr. Rajesh Singh, Department of Statistics, Banaras Hindu University, Varanasi, India. E-mail Id: [email protected]

This article presents improved estimation of population mean using auxiliary attribute. The bias and mean square error of proposed estimators are obtained up to first order of approximation and it is shown that proposed class of estimator under optimum conditions is more efficient than the other estimators considered in this article. The theoretical results are supported by empirical study considering two data sets.

Keywords: Auxiliary information, Auxiliary attribute, Simple random sampling, Bias, Mean square error.

1. Introduction Several estimators of population mean are available in the literature. Cochran [3] introduced a ratio estimator of population mean Y . He showed the contribution of known auxiliary information in improving the efficiency of the estimator of population mean in survey sampling.

X yr  y    x

(1)

Bahl and Tuteja (1991) suggested following ratio type exponential estimator of Y.

Xx  y p  y exp Xx

How to cite this article: Singh P, Adichwal NK, Singh R. Estimation of Finite Population Mean Using Auxiliary Attribute in Sample Surveys. J Adv Res Appl Math Stat 2016; 1(2): 39-44. ISSN: 2455-7021

(2)

In some situations, in place of auxiliary variable we have information on qualitative variable. For example, the height of a person (y) may depend on gender (  ). In such situations by taking advantage of bi-serial correlation between study variable and auxiliary attribute we can construct efficient estimator. Shabbir and Gupta (2007), Singh et al. (2008), Singh et al. (2010) and Abd-Elfattah et al. (2010) have considered the problem of estimating population mean Y taking into consideration the point bi-serial correlation between auxiliary attribute and study variable. Let us consider a sample of size n is drawn by SRSWOR from a population of size N. Further let y i and  i denote the observations on variable y and  respectively for the ith unit (i = 1,2,3,…..,N). We note that  = 1, if ith unit of the population possesses attribute and  = 0, if ith unit of the population does not possess attribute. The usual unbiased estimator for population mean of the study variable Y is defined by

t0  y 

1 n  yi n i 1

(3)

Where y is the sample mean of the study variable Y. Using information on the population proportion P of the auxiliary attribute  P. Naik and Gupta (1996) and Singh et al. [11] respectively, proposed following estimators:

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Singh P et al.

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

t1  y P p 

(4)

MSE(t 2 )  Y 2

Pp  t 2  y exp Pp

(5)

To obtain the bias and MSE, we write

y  Y 1  e 0  and p  P 1  e 1 

(6)

Ee 0   Ee1   0

 

 

The variance of the usual unbiased estimator y under SRSWOR is given by,

Nn 2 Cy Nn

(7)

N

S

2

where C y 

2 y

Y2

2

 (y

, Sy 

i

 Y) 2 , C 2p 

i 1

N 1

S 2 P2

S 

 (

 P) 2

i

i 1

N 1

,

k

 py C y Cp

,

 py 

In this section, we have suggested some estimators of population mean based on usual unbiased estimator, usual ratio estimators and estimators due to Singh and Kumar (2011). The properties of the suggested estimators have been obtained up to the first degree of approximation.

2.1.

Estimators based on t 0 and t 1

Taking the arithmetic mean (AM), geometric mean (GM) and harmonic mean (HM) of the estimators t 0 and t 1 , we get the estimator of the population mean respectively as

,

N 2 

Motivated by Singh et al. (2011), we have proposed some estimators of population mean using auxiliary information in form of attribute based on arithmetic mean, geometric mean and harmonic mean of the estimators t 0 , t 1  , t 0 , t 2  , t 1 , t 2  and t 0 , t 1 , t 2  .

2. Suggested Estimator

Nn 2 Nn 2 2 E e 20   C y , E e 1   C p ,  Nn   Nn  Nn E e 0 e1     py C p C y  Nn 

MSE ( y)  Y 2

t (3AM ) 

1  y  P  (t 0  t 1 )   1   2  2  p 

S y S yS

1

t

( GM ) 3

1/ 2

 t 0 t 1 

N

 (y S y 

i

 Y )( i  P)

i 1

N 1

t (3HM ) 

.

To the first degree of approximation the biases and mean squared errors (MSE’s) of t 1 and t 1 are respectively given by

Nn 2 [C p 1  k ] Nn

(8)

Y Nn 2 [C p 3  4k ] 8 Nn

(9)

Bias ( t 1 )  Y Bias ( t 2 ) 

MSE ( t 1 )  Y 2

ISSN: 2455-7021

(11)

Properties of the suggested estimators are studied under large sample approximation. An empirical study is carried out in support of the present study.

such that

and

2  N  n  2 Cp C  1  4k   y Nn  4 

Nn 2 [C y  C 2p (1  2k )] Nn

(10)

P  y  p

2  1 1     t t  0 1 



(12)

2

(13)

2y p  1   P 

(14)

To the first order of approximation, the bias and the ( AM )

mean squared errors of t 3 respectively given by

( GM )

( HM )

, t3

and t 3

are

Bias(t (3AM) ) 

Y Nn 2 C p (1  k) 2 Nn

Bias ( t (3GM ) ) 

Y Nn 2 C p (3  4 k ) 8 Nn





(16)

Bias ( t (3HM ) ) 

Y Nn 2 C p (1  2k ) 4 Nn



(17)







(15)

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

Singh P et al.

MSE ( t 3( AM ) )  MSE ( t (3GM ) )  MSE ( t (3HM ) )  Y 2

2  N  n  2 Cp 1  4k  C y  Nn  4 

(18)

2.2. Estimator Based on t 0 and t 2 The estimator of Y based on AM, GM and HM of the estimators t 0 and t 2 are respectively defined as

t (4AM ) 

  1 t 0  t 2   y 1  exp P  p   2 2  P  p  1

t

( GM ) 4

 t 0 t 2 

12

t (4HM ) 

  P  p    y exp   P  p 

2 1 1     t0 t2 



(19)

  p  P    1  exp  P  p  



MSE t





Y Nn 2 C p (3  4k ) 16 Nn



(22)





Y Nn 2 C p (5  8k ) 32 Nn

(23)





Y Nn 2 C p (1  2k ) 8 Nn

Bias t (4AM ) 



2

2y

( AM ) 4

To the first degree of approximation, the biases and the MSEs of t (4AM ) , t (4GM ) and t (4HM ) are respectively given by

  MSEt

( GM ) 4

(20)

Bias t (4GM ) 

(21)

Bias t (4HM ) 

  MSEt

( HM ) 4











(24)

2  N  n  2 Cp Y 1  8k  C y  Nn  16  2

(25)

2.3. Estimator based on t 1 and t 2 The estimator of Y based on AM, GM and HM of the estimators t 1 and t 2 are respectively defined as

t (5AM ) 

  1 t 1  t 2   y  P  exp P  p   2 2p  P  p  1

t

( GM ) 5

12

 t 1 t 2 

t (5HM ) 

 P   P  p      y  exp  p   P  p  

2 1 1     t1 t 2 



p  p  P     exp  p  P  P



MSE t





Y Nn 2 C p 11  12 k  16 Nn

(29)





3Y N  n 2 C p 7  8k  32 Nn



(30)





Y Nn 2 C p 5  6k  8 Nn



(31)

Bias t (5AM ) 

2

2y

( AM ) 5

(26)

To the first degree of approximation, the biases and the MSEs of t (5AM ) , t 5( GM ) and t (5HM ) are respectively given by

  MSE t

( GM ) 5

(27)

Bias t (5GM ) 

(28)

Bias t (5HM ) 

  MSE t

2.4. Estimator Based on t 0 , t 1 and t 2 The estimator of Y based on AM, GM and HM of the estimators t 0 , t 1 and t 2 are respectively defined as

( HM ) 5



t (6AM) 







2  N  n  2 3C p Y 3  8k  C y  Nn  16  2

(32)

  1 t 0  t1  t 2   y 1  P   exp P  p  (33) 3 3 p  P  p 

t (6GM)  t 0 t 1t 2 

13

41



 P   P  p      y  exp  p   P  p  

1 3

(34)

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Singh P et al.

t (6HM) 

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

3 1 1 1      t 0 t1 t 2 



3y

MSEs of t (6AM ) , t (6GM ) and t (6HM ) are respectively given by

(35)

 p  p  P   1   exp  P  p   P

Y Nn 2 C p 11  12 k  24 Nn





y Nn 2 C p (3  4k ) 8 Nn





y Nn 2 C p (7  12 k ) 24 Nn

Bias t (6GM )  Bias t (6HM ) 



MSE t

  MSE t

( GM ) 6

  MSE t

( HM ) 6









2  N  n  2 Cp Y 1  4k  C y  Nn  4  2

(36)

(37)



(38)

(39)

2

We have considered a class of estimator t p such that

For

( t 0 , t1 , t 2 )   , where  is set of all possible

w

i

 1, wi  

(41)

i 0

where w 0 , w 1 and w 2 are the constants used for

estimators for estimating population mean Y . By definition set  is a linear variety if

t p  w 0 t 0  w 1 t1  w 2 t 2   ,





To the first degree of approximation, the biases and the

( AM ) 6





Bias t (6AM ) 

reducing the bias in the set of real numbers. and  denotes the set of real numbers.

(40) Expressing the estimator t p in terms of e’s, we have

 w  w   3     t p  Y1  e 0   w 1  2 e1   w 1  w 2 e12   w 1  2 e 0 e1  2  8 2       

(42)

Taking expectations of both sides of (42) and then subtracting Y from both sides, we get the biases of the estimators, up to the first order of approximation as

From (42), we have

1   B( t p ) = Ye0  we1  3w  w1 e12  we0e1  4   w where w  w 1  2  2w  2w 1  w 2 2

Squaring both sides of (45) and then taking expectations, we get the MSE of the estimator t, up to the first order of approximation as

MSE(t p ) = Y 2

(43)



Nn 2 C y  w 2 C 2p  2 w yp C p C y Nn

 

 py C y Cp

(45)

(44)



2 w 1  w 2  2T = T (say)

 

ISSN: 2455-7021

(49)

(47)

Putting the value of w (=T) in (46), we get the minimum MSE of the estimator t p , as Min. MSE t p = Y 2

(46)

From (44) and (47), we have

MSE t p will be minimum, when

w

t p  Y  Y e 0  we1 

Nn 1   2py C 2y Nn





(48)

From (41) and (49), we have only two equations in three unknowns. It is not possible to find unique values of wi 's (i  0,1,2). In order to get unique values for wi 's , we shall impose the linear restriction

w 0 B( t 0 )  w 1 B( t 1 )  w 2 B( t 2 )  0

(50)

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

Singh P et al.

Equations (41), (49) and (50) can be written in matrix form as

1 1  1 0 2 1   0 B( t 1 ) B( t 2 )

In this section, we compare the performance of the estimator t p with the other estimators considered in

 w0   1   w  = 2T   1    w2   0 

(51)

Using (51) we get the unique values of wi 's (i  0,1,2) as

w0 

4. Empirical Study

k0 k1 k2 , w1  , w2  kr kr kr

this article using two population data sets. The description of population data sets are as follows.

Population I [Source: Sukhatme & Sukhatme, p.256] y = Number of villages in the circle.

 = A circle consisting of more than five villages. N = 89, n = 23, Y = 3.36, P = 0.124,  pb  0.766 ,

where k 0  2(1  T) B( t 2 )  (1  2T) B( t 1 ),

C y  0.601, C p  2.678 .

k 1  2TB( t 2 ), k r  2B(t 2 )  B( t 1 ).

Population II [Source: Sukhatme & Sukhatme, p.256]

k 2  2TB( t 1 ),

y = Number of villages in the circle.

3. Efficiency Comparison

 = A circle consisting more than five villages.

From Eqs. (7), (10), (11), (18), (25), (32), (39) and (49), we have

N = 89, n = 23, Y = 1102, P = 0.124,  pb  0.624 ,

MSE ( t 0 )  MSE ( t1 ) if k>1/2

(52)

MSE ( t 0 )  MSE ( t 2 ) if k>1/4

(53)

C y  0.65 , C p  2.678 ( j)

MSE ( t 0 )  MSE ( t 3j ) if k>1/4; j= (AM, GM, HM) (54) j

MSE ( t 0 )  MSE ( t 4 ) if k>1/8; j= (AM, GM, HM) (55) MSE ( t 0 )  MSE ( t 5j ) if k>3/8; j= (AM, GM, HM) (56) j MSE ( t 0 )  MSE ( t 6 ) if k>1/4; j= (AM, GM, HM) (57)

MSE ( t 0 )  MSE ( t p ) if  yp  1

(58)

where t = t 1 , t 2 and t i , (I = 3, 4, 5, 6; j = AM, GM, HM). We have computed the percent relative efficiencies (PREs) of the various estimators of Y with respect to usual unbiased estimator y by using the formula:

PRE ( t , y) 

MSE( y)  100 MSE( t )

(59)

( j)

where t = t 1 , t 2 , t i (I = 3, 4, 5, 6; j = AM, GM, HM) and

t p and the findings are as follows:

w 's

Constant

w0

w1 w2

Table 1.Value of i (I = 0,1,2) Population I 0.258674

Population II 0.334475

-0.39751

-0.36261

1.138838

1.028138

The values of w0 , w1 and w2 are obtained by solving the system of equations (41), (49) and (50) which filters

43

the bias up to first order of approximation from the linear variety of estimators.

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Singh P et al.

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

Table 2.Percentage Relative Efficiency of Estimators with respect to y Estimators Population I Population II PRE PRE 100 100 t 0

t1 t2

7.13

7.79

39.20

37.41

j 3

39.20

37.41

t 4j

187.14

128.95

j 5

14.19

44.35

t 6j

39.20

37.41

tp

241.99

163.77

t

t

(j = AM, GM, HM)

Table 2 shows that estimator t p under optimum conditions has maximum PRE among all proposed estimators. So it is advisable to consider estimator t p over t1 , t 2 , t 3j , t 4j , t 5j and t 6j .

5. Conclusion From empirical study, we conclude that the proposed estimator t p under optimum conditions performs better than other estimators that we have considered in this article. The expressions of MSEs and biases are obtained up to the first order of approximation. Theoretical conditions have also been considered under which proposed class of estimators performs better than other estimators. Thus we can conclude that when condition (58) is fulfilled, estimator t p is more efficient than the usual estimator and it is recommended for use in practice.

References 1. Adichwal NK, Sharma P, Singh R. Generalised class of estimators for population variance using information on two auxiliary variables. Int J Appl Comput Math 2015: 1-10. 2. Adichwal NK, Sharma P, Singh R. Estimation of finite population variances using auxiliary attribute in sample surveys. Columbia Int Pub J of Advanced Computing 2015; 4(2): 88-100. 3. Cochran WG. Sampling Techniques. Wiley Eastern

ISSN: 2455-7021

Limited, 1977. 4. Isaki CT. Variance estimation using auxiliary information. Jour Amer Stat Assoc 1983; 78: 117-23. 5. Jhajji HS, Sharma MK, Grover LK. An efficient class of chain estimators of population variance under sub-sampling scheme. J Japan Statistics Soc 2005; 35: 273-86. 6. Malik S, Singh R. A family of estimators of population mean using information on point biserial and phi correlation coefficient. Int J of Statistics and Economics 2013; 10(1): 75-89. 7. Murthy MN. Sampling Theory and Methods. Calcutta, India: Statistical Pub. Society, 1967. 8. Sharma P, Singh R. Efficient estimators of population mean in stratified random sampling using auxiliary attribute. World Applied Sciences Journal 2013; 27(12): 1786-91. 9. Sharma P, Singh R. Improved estimators in simple random sampling when study variable is an attribute. J Stat Appl Pro Lett 2015; 2(1): 1-8. 10. Singh HP, Solanki RS. Improved estimation of population mean in simple random sampling using information on auxiliary attribute. Applied Mathematics and Computation 2012; 218: 7798812. 11. Singh R, Chauhan P, Sawan N et al. Auxiliary information and a priori values in construction of improved estimators. Renaissance High Press, 2007. 12. Singh R, Smarandache F. The efficient use of supplementary information in finite population sampling. Education Publishing, 2014: 42-51.

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