Estimation of Finite Population Variances using Auxiliary Attribute in ...

Columbia International Publishing Journal of Advanced Computing (2015) Vol. 4 No. 2 pp. 88-100 doi:10.7726/jac.2015.1006

Research Article

Estimation of Finite Population Variances using Auxiliary Attribute in Sample Surveys Nitesh K. Adichwal1, Prayas Sharma1*, and Rajesh Singh1 Received: 9 June 2015; Published online 27 June 2015 © The author(s) 2015. Published with open access at www.uscip.us

Abstract Singh and Kumar (2011) suggested some estimators for estimating the population variances using an auxiliary attribute. This paper suggest some improved class of estimators of population variance using auxiliary information in form of attribute of the study variable Y based on arithmetic mean, geometric mean and harmonic mean of the usual unbiased estimator, usual ratio estimators and estimators due to Singh and Kumar (2011) in case of simple random sampling. The expressions of bias and MSEs have been derived up to the first order of approximation. It has been shown that the performance of the proposed estimators is better than the usual unbiased estimator, usual ratio estimators and estimators due to Singh and Kumar (2011). In addition, an empirical study is carried out in the support of theoretical results. Keywords: Auxiliary information; Auxiliary Attribute; Simple random sampling; Bias; Mean Square Error; Arithmetic mean; Geometric mean; Harmonic mean

1. Introduction Over the last several decades statisticians are using supplementary information in sampling theory in order to get the better estimates of population parameters. It has been recognized that the use of auxiliary information in sample survey design results in efficient estimators of population parameters of interest. The problem of estimating the finite population variance using information on auxiliary attributes has been discussed in sampling literature. Several authors including Naik and Gupta (1996), Jhajj et al. (2006), Shabbir and Gupta (2007), Singh et al. (2007, 2008), AbdElfattah et al. (2010), Malik and Singh (2013), Singh and Malik (2013), Sharma and Singh (2013,2015) and Sharma et al. (2013 a, b) have paid their attention towards the estimation of population parameters in the presence of auxiliary attributes. ______________________________________________________________________________________________________________________________ *Corresponding e-mail: [email protected] 1 Department of Statistics, Faculty of Science, Banaras Hindu University, Varanasi (U.P.), India.

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Nitesh K. Adichwal, Prayas Sharma, and Rajesh Singh/ Journal of Advanced Computing (2015) Vol. 4 No. 2 pp. 88-100

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). It is assumed that attribute  takes only the two values 0 and 1 according as  =1, if ith unit of the population possesses attribute  =0, if otherwise. The usual unbiased estimator for population variance of the study variable Y is defined by 2

1 n t0  s   y i  y  n  1 i 1 2 y

where y 

(1.1)

n

y i 1

i

is the sample mean of the study variable Y.

Using information on the population variance S 2 of the auxiliary attribute  , Singh and Kumar (2011) proposed the following estimators



t 1  s 2y S2 s 2



(1.2)

 S2  s 2  t 2  s exp  2 2  S s     2 y

(1.3)

To obtain the bias and MSE, we write-

s 2y  S2y 1  e 0  and s 2  S2 1  e1  Such that Ee 0   Ee1   0

   n 1 , Ee    n 1 , Ee e    n 1

and E e 02 

2 1

40

04

22

0 1

We know that the mean squared error (MSE) of the usual unbiased estimator s 2y under SRSWOR is approximately, for large n,

MSE(s )  2 y

S 4y n

 40  1

(1.4) N

where  rs =

 rq  r20/ 2  q02/ 2

,

 rq 

 (y i 1

i

 Y ) r ( i  P ) q N 1

p, q and r being non negative integers.

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Nitesh K. Adichwal, Prayas Sharma, and Rajesh Singh/ Journal of Advanced Computing (2015) Vol. 4 No. 2 pp. 88-100

To the first degree of approximation the biases and mean squared errors (MSE’s) of t 0 and t 1 are respectively given by

Bias ( t 1 ) 

S 2y

Bias ( t 2 ) 

MSE( t1 )  MSE( t 2 ) 

n S 2y 8n

S4y n

 04  11  k 

(1.5)

 04  13  4k 

(1.6)

 40  1   04  11  2k 

(1.7)

40  1  1 404  11  4k 

(1.8)

S4y n

where k   22  1  04  1 Motivated by Singh et al. (2011), we proposed some estimators of population variance of study variable Y using auxiliary 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  . Properties of the suggested estimators are studied under large sample approximation. An empirical study is carried out in support of the present study.

2. Suggested Estimator In this section we have suggested some estimators of population variance 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 The 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 variance respectively as

t

( AM ) 3

 s y2  S2  1  ( t 0  t1 )   1  2   2  s  2    

t 3(GM )  t 0 t1 

1/ 2

 s 2y S s 

(2.1) (2.2)

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Nitesh K. Adichwal, Prayas Sharma, and Rajesh Singh/ Journal of Advanced Computing (2015) Vol. 4 No. 2 pp. 88-100

t

( HM ) 3



2 1 1     t 0 t1 



2s 2y

(2.3)

 s 2  1  2   S    

( AM ) ( GM ) ( HM ) To the first order of approximation, the bias and the mean squared errors of t 3 , t 3 and t 3

are respectively given by





 )  S

 4n 

Bias (t 3( AM) )  S 2y 2n  04  11  k 

(2.4)

Bias (t 3(GM ) )  S 2y 8n  04  13  4k 

(2.5)

 11  2k 

(2.6)

Bias (t 3( HM )

MSE( t

( AM ) 3

2 y

)  MSE( t

04

( GM ) 3

)  MSE( t

( HM ) 3

S4y    1 1  4k  )   40  1  04  n  4 

(2.7)

2.2 The estimator based on t 0 and t 2 The estimator of S 2y based on AM, GM and HM of the estimators t 0 and t 2 are respectively defined as

t

( AM ) 4

 s 2y 1  t 0  t 2     2 2 

t (4GM )  t 0 t 2 

12

t

( HM ) 4



2   2 1  exp  S x  s x  S2  s 2  x  x   1  S 2x  s 2x   2   s y exp   2  2 S  s 2  x    x

2 1 1     t0 t2 



2s 2y   s 2x  S 2x   2 1  exp   S  s2  x  x 

    

   

(2.8)

(2.9)

(2.10)

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

 Bias t

    S

 32n 

Bias t (4AM)  S 2y 16n  04  13  4k 

(2.11)

 15  8k 

(2.12)

( GM ) 4

2 y

04

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Nitesh K. Adichwal, Prayas Sharma, and Rajesh Singh/ Journal of Advanced Computing (2015) Vol. 4 No. 2 pp. 88-100



 







Bias t (4HM )  S 2y 8n  04  11  2k 







(2.13)



MSE t (4AM )  MSE t (4GM )  MSE t (4HM ) 

S 4y   04  1 1  8k   40  1  n  16 

(2.14)

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

t

( AM ) 5

 s 2y 1  t 1  t 2     2 2 

t

( GM ) 5

 t 1 t 2 

12

t 5( HM ) 

 S s  s   2 y

2 1 1     t1 t 2 



 S 2  S 2x  s 2x      exp  2 2   s 2 S  s x   x  

(2.15)

2    2   exp  1  S x  s x     2  S2  s 2   x     x

(2.16)

2s 2y 2   S 2x  s 2x  s  2   exp  2 2     S x  s x  S 

(2.17)

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

 Bias t Bias t

    3S   S



Bias t 5( AM)  S 2y 16n  04  111  12k 



( GM ) 5 ( HM ) 5

MSE t

( AM ) 5

2 y

2 y

(2.18)



32n  04  17  8k 

(2.19)



8n  04  15  6k 

  MSEt

( GM ) 5

  MSEt

( HM ) 5

(2.20)



S 4y  3. 04  1 3  8k    40  1  n  16 

(2.21)

2.4 The estimator based on t 0 , t 1 and t 2 The estimator of S 2y based on AM, GM and HM of the estimators t 0 , t 1 and t 2 are respectively defined as 92

Nitesh K. Adichwal, Prayas Sharma, and Rajesh Singh/ Journal of Advanced Computing (2015) Vol. 4 No. 2 pp. 88-100

t

( AM ) 6

 s 2y 1  t 0  t 1  t 2     3 3 

t

( GM ) 6

 t 0 t 1 t 2 

13

t (6HM ) 

 S s  s  

   

2 y

3 1 1 1      t 0 t1 t 2 



 S 2  S 2x  s 2x 1   2  exp 2  S  s2  s  x  x 

23

   

 1  S 2x  s 2x    exp   2  2 S  s2  x    x

2s 2y 2   S 2x  s 2x   s  2  1   exp  2 2     S x  s x   S 

(2.22)

(2.23)

(2.24)

To the first degree of approximation, the biases and the MSEs of t (6AM ) , t (6GM ) and t (6HM ) are respectively given by

 Bias t Bias t

    3S   S

 8n  24n 

Bias t (6AM)  S 2y 24n  04  111  12k  ( GM ) 6 ( HM ) 6



2 y

2 y





(2.25)

04

 13  4k 

(2.26)

04

 17  12k 

(2.27)







MSE t (6AM )  MSE t (6GM )  MSE t (6HM ) 

S 4y   04  1 1  4k   40  1  n  4 

(2.28)

3. Efficiency Comparison From (1.4), (1.7), (1.8), (2.7), (2.14), (2.21) and (2.28), we have

  MSE(t )  MSE(s )  S 4n   11  4k  MSE(t )  MSE(s )  S 4n   11  4k  MSE(t )  MSE(s )  S 16n   11  8k  MSE(t )  MSE(s )  3S 16n   13  8k  MSE(t )  MSE(s )  S 4n   11  4k  MSE(t1 )  MSE(s2y )  S4y n 04  11  2k  2

2 y

4 y

04

( j) 3

2 y

4 y

( j) 4

2 y

4 y

( j) 5

2 y

( j) 6

2 y

04

04

4 y

4 y

04

04

(3.1) (3.2) (3.3) (3.4) (3.5) (3.6)

where (j=AM, GM, HM) 93

Nitesh K. Adichwal, Prayas Sharma, and Rajesh Singh/ Journal of Advanced Computing (2015) Vol. 4 No. 2 pp. 88-100 ( j)

From (3.1)-(3.6) we get that the estimators t1 , t 2 and t i , (i=3, 4, 5, 6; j=AM, GM, HM) are better 2 than usual unbiased estimator s y respectively if

k  1 2 ,

(3.7)

k  1 4 ,

(3.8)

k  1 4 ,

(3.9)

k  1 8 ,

(3.10)

k  3 8 ,

(3.11)

k  1 4 .

(3.12)

( j)

From (3.7)-(3.12) it follows that the sufficient condition for the estimators t1 , t 2 and t i , (i=3, 4, 5, 2 6; j=AM, GM, HM) to be better than the usual unbiased estimator s y is k  1 2 .

From (1.7), (1.8), (2.7), (2.14), (2.21) and (2.28), we have





3  MSE(t 2 )  MSE(t1 )  S4y n 04  1 k   , 4 

(3.13)





(3.14)





(3.15)

3  MSE(t 3( j) )  MSE( t1 )  S4y n 04  1 k   4  5  MSE( t (4j) )  MSE( t1 )  S4y n 04  1 k   8 





7  MSE(t 5( j) )  MSE(t1 )  S4y 2n 04  1 k   8 

(3.16)



(3.17)



3  MSE(t (6j) )  MSE( t1 )  S4y n 04  1 k   4  where (j=AM, GM, HM)

( j)

From (3.13)-(3.17), it is observed that the estimators t 2 and t i , (i=3, 4, 5, 6; j=AM, GM, HM) are better than usual unbiased estimator t1 respectively if

k  3 4 ,

(3.18) 94

Nitesh K. Adichwal, Prayas Sharma, and Rajesh Singh/ Journal of Advanced Computing (2015) Vol. 4 No. 2 pp. 88-100

k  3 4 ,

(3.19)

k  5 8 ,

(3.20)

k  7 8 ,

(3.21)

k  3 4 ,

(3.22)

( j)

Thus it follows from (3.18)-(3.22) that the sufficient condition for the estimators t 2 and t i , (i=3, 4, 5, 6; j=AM, GM, HM) to be better than the usual unbiased estimator t1 if k  5 8 . From (1.8), (2.7), (2.14), (2.21) and (2.28), we have

MSE(t 3( j) )  MSE(t 2 )  0 ,

(3.23)

 )  MSE(t )  S

 16n 

MSE(t (4j) )  MSE(t 2 )  S4y 16n 04  18k  3 ,

(3.24)

 15  8k  ,

(3.25)

MSE(t 5( j)

4 y

2

04

MSE(t (6j) )  MSE(t 2 )  0

(3.26)

where (j=AM, GM, HM).

( j) ( j) From (3.23) and (3.26) it is observed that the estimators t 2 , t 3 and t 6 , (j=AM, GM, HM) are

( j)

( j) equally efficient. Further it is also observed from (3.24) and (3.25) that the estimators t 4 and t 5 ,

(j=AM, GM, HM) are better than the estimators t 2 respectively if

k  3 8 ,

(3.27)

k  5 8 .

(3.28)

From (2.7), (2.14) and (2.21), we have

 )  S

 16n 

MSE(t (4j) )  MSE(t 3( j) )  S4y 16n 04  18k  3 ,

(3.29)

 15  8k  ,

(3.30)

MSE(t 5( j) )  MSE(t 3( j)

4 y

04

where (j=AM, GM, HM).

95

Nitesh K. Adichwal, Prayas Sharma, and Rajesh Singh/ Journal of Advanced Computing (2015) Vol. 4 No. 2 pp. 88-100 ( j)

( j) From (3.29) and (3.30) it is observed that the estimators t 4 and t 5 are better than the estimators

t 3( j) , (j=AM, GM, HM) respectively if k  3 8 ,

(3.31)

k  5 8 .

(3.32)

From (2.14), (2.21) and (2.28), we have

 )  S

 16n 

MSE(t 5( j) )  MSE(t (4j) )  S4y 2n 04  11  2k  , MSE(t (6j) )  MSE(t (4j)

4 y

04

(3.33)

 13  8k  ,

(3.34)

where (j=AM, GM, HM). ( j) ( j) From (3.33) and (3.34) it is observed that the estimators t 5 and t 6 are better than the estimators

t (4j) , (j=AM, GM, HM) respectively if k  1 2 ,

(3.35)

k  3 8 .

(3.36)

( j) ( j) Thus it follows from (3.35)-(3.36) that the sufficient condition for the estimators t 5 and t 6 ,

(j=AM, GM, HM) to be better than the usual unbiased estimator t 4 if k  1 2 . ( j)

Further from (2.21) and (2.28), we have





MSE(t 5( j) )  MSE(t (6j) )  S4y 16n 04  15  8k 

(3.37)

which is positive if

k  5 8 .

(3.38)

( j) ( j) Thus the estimator t 6 are better than t 5 , ( j=AM, GM, HM ) if k  5 8 .

4. Empirical Study In this section we compare the performance of different estimators considered in this paper using two population data sets. The description of population data sets are as follows. Population I Source : Sukhatme and Sukhatme (1970), p. 256 . 96

Nitesh K. Adichwal, Prayas Sharma, and Rajesh Singh/ Journal of Advanced Computing (2015) Vol. 4 No. 2 pp. 88-100

y = Number of villages in the circle.

 = A circle consisting more than five villages. 2 2 N=89, n=23, Sy  4.074 , S   0.11 ,  22  3.996 ,  40  3.811 ,  04  6.162 .

Population II Source : Sukhatme and Sukhatme (1970), p. 256 . y = Number of villages in the circle.

 = A circle consisting more than five villages. 2 2 N=89, n=23, S y  513592 , S  0.11 ,  22  3.946 ,  40  4.741 ,  04  6.162 . 2 We have computed the percent absolute relative biases (PARBs) of different estimators of S y by

using the formula:

PARB( t ) 

Bias t   100 , S2y n



(4.1)



( j)

where t= t1 , t 2 and t i , (i=2, 3, 4, 5, 6; j=AM, GM, HM). 2 Further we have computed the percent relative efficiencies (PREs) of the various estimators of S y 2 with respect to usual unbiased estimator s y by using the formula:

PRE( t, s )  2 y

MSE(s 2y ) MSE( t )

 100 ,

(4.2)

( j)

where t= t1 , t 2 and t i , (i= 3, 4, 5, 6; j=AM, GM, HM) and the findings are as follows

Table 4.1 The PARB of different estimators Population

Population

I

II

PARB

PARB

t1

216.60

221.00

t2

43.78

46.28

Estimators

97

Nitesh K. Adichwal, Prayas Sharma, and Rajesh Singh/ Journal of Advanced Computing (2015) Vol. 4 No. 2 pp. 88-100

t 3( AM )

108.30

t 3( GM )

43.78

t 3( HM )

20.75

t (4AM )

21.89

t (4GM )

5.76

t (4HM )

10.38

t 5( AM )

130.29

t 5( GM )

114.06

t 5( HM )

97.93

t (6AM )

86.79

t (6GM )

131.33

t (6HM )

0.76

110.80 46.28 18.25 23.14 7.01 9.13 133.94 117.81 101.68 89.29 138.83 3.26

Table: 4.2 The PREs of different estimators Population Estimators

I

II

PRE

PRE

s 2y

100.00

100.00

t1

141.90

124.24

t2

254.27

179.38

t 3( j)

254.27

179.38

t (4j)

171.86

144.41

t 5( j)

230.29

168.09

t (6j)

254.27

179.38 98

Nitesh K. Adichwal, Prayas Sharma, and Rajesh Singh/ Journal of Advanced Computing (2015) Vol. 4 No. 2 pp. 88-100

( j=AM, GM, HM) Table 4.1 reveals that the estimator t (6HM ) has the least PARB as compared to all other estimators discussed above for both the population. It follows that the proposed estimator t (6HM ) is less biased among all the estimators. From table 4.2 we can observe that the proposed estimators t 2 , t 3( j) and

t (6j) have largest PRE. However the PARB of the estimator t (6HM ) is less than t1 , t 2 , t i( j) , (i=2, 3, 4, 5, ; j=AM, GM, HM), t (6AM ) and t (6GM ) in both the population set I and II. So the estimator t (6HM ) is to be ( j)

preferred over t1 , t 2 , t i , (i=2, 3, 4, 5, ; j=AM, GM, HM), t (6AM ) and t (6GM ) .

5. Conclusion In this article we have suggested some improved estimators of population variance using auxiliary information in form of attribute of the study variable Y based on arithmetic mean, geometric mean and harmonic mean of the usual unbiased estimator, usual ratio estimators and estimators due to Singh and Kumar (2011). The expressions of MSEs and biases are obtained to the first order of approximation. The theoretical conditions under which the proposed estimators are more efficient than the usual unbiased estimator, usual ratio estimators and estimators due to Singh and Kumar (2011) have been obtained. In theoretical and empirical efficiency comparisons, it has been shown that the performance of the proposed estimators are better than other existing estimators i.e. proposed estimators have greater PREs and least PARBs in comparisons to other estimators considered in this paper . Thus it can be concluded that when conditions in sec. 3 are satisfied proposed estimator t (6HM ) has higher efficiency in comparison to other estimators considered here and it is recommended for use in practice.

References Abd-Elfattah, A.M., Sherpeny E.A., Mohamed S.M., & Abdou O.F. (2010):Improvement estimating the population mean in simple random sampling using information on auxiliary attribute. Applied mathematics and computation. http://dx.doi.org/10.1016/j.amc.2009.12.041 Jhajj, H. S., Sharma, M. K. & Grover, L. K. (2006): A family of estimators of population mean using information on auxiliary attribute. Pak. J. Statist., 22 (1), 43-50. Malik, S. & Singh, R. (2013): An improved estimator using two auxiliary attributes. Appli. Math. Compt., 219, 10983-10986. http://dx.doi.org/10.1016/j.amc.2013.05.014 Naik,V.D. & Gupta, P.C., (1996): A note on estimation of mean with known population proportion of an auxiliary character. Jour. Ind. Soc. Agr. Stat., 48(2),151-158. Shabbir, J., & Gupta, S. (2007): On estimating the finite population mean with known population proportion of an auxiliary variable. Pak. Jour. of Statist. 23 (1) 1–9. Sharma, P., Verma, H., Sanaullah, A. & Singh, R. (2013 a): A study of some exponential ratio-product type estimators using information on auxiliary attributes under second order approximation. IJSE, 12(3), 5866. 99

Nitesh K. Adichwal, Prayas Sharma, and Rajesh Singh/ Journal of Advanced Computing (2015) Vol. 4 No. 2 pp. 88-100 Sharma, P., Singh, R. & Kim, J. M. (2013 b): Study of some improved ratio type estimators using information on auxiliary attributes under second order approximation. Jour. Sci. Res., 57, 138-146. Sharma, P. & Singh, R. (2013): Efficient estimators of Population Mean in stratified random sampling using auxiliary attribute. World Applied Sciences Journal, 27(12), 1786-1791. Sharma, P. & Singh, R. (2015): Improved estimators in simple random sampling when Study variable is an attribute, J. Stat. Appl. Pro. Lett. 2(1), 1-8. Singh R. & Kumar M. (2011): A family of estimators of population variance using information on auxiliary attribute. Studies in sampling techniques and time series analysis. Zip Publishing, 63-70. Singh, R. Cauhan, P., Sawan, N., & Smarandache F. (2007): Auxiliary information and a priori values in construction of improved estimators, Renaissance High Press. Singh, R. Cauhan, P., Sawan, N., & Smarandache F. (2008): Ratio estimators in simple random sampling using information on auxiliary attribute. Pak. J. Stat. Oper. Res. 4(1),47-53. http://dx.doi.org/10.18187/pjsor.v4i1.60 Singh, R., & Malik, S. (2013) : A family of estimators of population mean using information on two auxiliary attribute. Worl. App. Sci. Jour., 23(7), 950-955. Sukhatme, P.V. & Sukhatme, B.V. (1970): Sampling Theory of Surveys with Applications, Iowa state university Press, Ames, U.S.A.

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