An improved estimator using two auxiliary attributes

Applied Mathematics and Computation 219 (2013) 10983–10986

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An improved estimator using two auxiliary attributes Sachin Malik, Rajesh Singh ⇑ Department of Statistics, Banaras Hindu University, Varanasi 221005, India

a r t i c l e

i n f o

a b s t r a c t In the present study, we propose two new estimators for population mean using Kadilar and Cingi (2005) [2] and Lu et al. (2010) [9] estimators in the case when the information is available in form of attributes. Expressions for the MSE’s of the proposed estimator are derived up to the first degree of approximation. The theoretical conditions have also been verified by a numerical example. It has been shown that the proposed estimators are more efficient than usual regression estimators. Ó 2013 Elsevier Inc. All rights reserved.

Keywords: Simple random sampling Auxiliary attribute Point bi-serial correlation Phi correlation Efficiency

1. Introduction The use of auxiliary information can increase the precision of an estimator when study variable y is highly correlated with auxiliary variable x. There exist situations when information is available in the form of attribute, which is highly correlated with y. Naik and Gupta [10] introduced a ratio estimator when study variable and the auxiliary attribute are positively correlated. Shabbir and Gupta [6], Singh et al. [8], Abd-Elfattah et al. [1] and Singh et al. [7] have considered the problem of estimating population mean Y taking into consideration the point bi-serial correlation between auxiliary attribute and study variable. There are some situations when in place of one auxiliary attribute, we have information on two qualitative variables. For illustration, to estimate the hourly wages we can use the information on marital status and region of residence Gujarati [3]. Here we assume that both auxiliary attributes have significant point bi-serial correlation with the study variable and there is significant phi-correlation Yule [4] between the two auxiliary attributes. Suppose that an auxiliary attribute p, correlated with study variable y, is obtained for each unit in the sample and that the population proportion P of the p is known. The regression estimator for estimating the unknown population mean of y, is

t1 ¼ y þ bðP  pÞ

ð1:1Þ

where b is an estimate of the change in y when p is increased by unity. The MSE expression of regression estimator is:

  MSEðt 1 Þ ¼ f1 Y 2 C 2y 1  q2

ð1:2Þ

Nn , Nn

where f1 ¼ n is the sample size, N is the population size, (Cy, Cp) are the coefficients of variation of the variates (y, p), respectively. When there are two auxiliary attributes P1 andP2, the regression estimator of Y is:

t2 ¼ y þ b1 ðP1  p1 Þ þ b2 ðP2  p2 Þ where b1 ¼

syp1 s2p

and b2 ¼

1

syp2 . s2p 2

ð1:3Þ

Here s2p1 and s2p2 are the sample variances of p1 and p2, respectively, syp1 and syp2 are the sample

covariance’s between y and p1 and between p2, respectively. The MSE expression of this estimator is: ⇑ Corresponding author. E-mail address: [email protected] (R. Singh). 0096-3003/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.05.014

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  MSEðt 2 Þ ¼ f1 Y 2 C 2y 1  q2yp1  q2yp2 þ 2qyp1 qyp2 q/

ð1:4Þ

2. The proposed estimator Adapting Lu et al. [5] estimator, we propose a multivariate ratio estimator using information on two auxiliary attributes as:

 a m1 P 1 þ m2 P 2 t3 ¼ y m1 p1 þ m2 p2

ð2:1Þ

where m1 and m2 are weights that satisfy the condition

m1 þ m2 ¼ 1: To obtain the MSE of t3 to the first degree of approximation, we define

e0 ¼

yY ; Y

e1 ¼

p1  P1 ; P1

e2 ¼

p2  P2 P2

Such that, Eðei Þ ¼ 0 ; i ¼ 0; 1; 2:

Eðe20 Þ ¼ f1 C 2y ;

Eðe21 Þ ¼ f1 C 2p1 ;

Eðe0 e1 Þ ¼ f1 K yp1 C 2p1 ; K yp1 ¼ qyp1

Cy ; C p1

Eðe22 Þ ¼ f1 C 2p2 ;

Eðe0 e2 Þ ¼ f1 K yp2 C 2p2 ;

K yp2 ¼ qyp2

Cy ; C p2

Eðe1 e2 Þ ¼ f1 K / C 2p2 ;

K / ¼ q/

C p1 : C p2

The MSE expression of the estimator t3 is given by:

  MSEðt 3 Þ ¼ f1 C 2y þ m21 a2 h2 S2p1 þ m22 a2 h2 S2p2  2m1 haSyp1  2m2 ahSyp2 þ 2m1 m2 a2 h2 Sp1 p2

ð2:2Þ

Y where h ¼ m1 P1 þm . 2 P2 Minimising expression (2.2) with respect to m1 and m2 we get the optimum values as

m1 ¼

ahSp1p2  ahS2p2  Syp1 þ Syp2 

ah S2p1 þ S2p2  2Sp1p2



and m2 ¼ 1  m1

The minimum MSE of t3 is:

  2 2 2 2 2 2 2     2 2 MSEmin ðt 3 Þ ¼ f1 C 2y þ m2 1 a h Sp1 þ m2 a h Sp2  2m1 haSyp1  2m2 ahSyp2 þ 2m1 m2 a h Sp1p2

ð2:3Þ

Following Singh et al. [8] we propose another estimator as:

t4 ¼ y exp

 b  b P1  p1 1 P2  p2 2 exp P1 þ p1 P2 þ p2

ð2:4Þ

where b1 and b2 are real constant. Following Kadilar and Cingi [2], we suggest using the ratio estimator given in (2.4) instead of sample mean of study variable in (1.3). By this way we obtain the following estimator as

tp ¼ y exp

 b  b P1  p1 1 P2  p2 2 exp þ b1 ðP1  p1 Þ þ b2 ðP2  p2 Þ P1 þ p1 P2 þ p2

ð2:5Þ

Expressing equation (2.5) in terms of e’s, we have

tp ¼ Y ð1 þ e0 Þ exp " ¼ Y 1 þ e0 



e1 2 þ e1

b1

exp



e2 2 þ e2

b2 !

 b1 e1 P1  b2 e2 P2

# b1 e1 b21 e21 b2 e2 b1 b2 e1 e2 b22 e22 b2 e0 e2 b1 e0 e1  b1 e1 P1  b2 e2 P2 þ   þ   2 4 2 4 4 2 2

ð2:6Þ

Retaining the term’s up to single power of e’s in (2.6) we have

tp  Y ¼

   b e1 b e2  b1 e1 P1  b2 e2 P2 Y e0  1  2 2 2

ð2:7Þ

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S. Malik, R. Singh / Applied Mathematics and Computation 219 (2013) 10983–10986 Table 1 MSE values of estimators. Estimators

MSE values

y t1 t2 t3 tp

655.28 402.58 592.71 358.67 356.87

Squaring both sides of (2.7) and then taking expectations, we get the MSE of the estimator tp up to the first order of approximation, as

( MSEðt p Þ ¼ f1 Y

" 2

C 2y

þ

b21 C 2p1 4

þ

b22 C 2p2 4 "

þ

b1 b2 k/ C 2p2 2

# 

b1 kyp1 C 2p1



b2 kyp2 C 2p2

þ B21 P21 C 2p1 þ B22 P22 C 2p2

þ 2B1 B2 P 1 P2 k/ C 2p2  2Y B1 P 1 kyp1 :C 2p1 þ B2 P2 kyp2 C 2p2  where B1 ¼

Syp1 S2p

b1 B1 P1 C 2p1 2

and B2 ¼

1

Syp2 S2p



b2 B2 P2 C 2p2 2



b1 B2 P2 k/ C 2p2 2



b2 B1 P1 k/ C 2p2

#) ð2:8Þ

2

.

2

Minimising equation (2.8) with respect to b1 and b2 we get the optimum values as 2

A1  YC p1 b1 A1  A2 k/ i and b2 ¼ b1 ¼ h 2 2 2 Yk/ C 2p2 Y C p1  k/ C p2 where A1 ¼ 2Ykyp1 C 2p1  2P1 B1 C 2p1  2P 2 B2 k/ C 2p2 and

A2 ¼ 2Ykyp2 C 2p2  2P2 B2 C 2p2  2P1 B1 k/ C 2p2 : 3. Numerical illustration 3.1. Data: (Source: Government of Pakistan (2004)) The population consists rice cultivation areas in 73 districts of Pakistan. The variables are defined as: Y = rice production (in 000’ tonnes, with one tonne = 0.984 tonne) during 2003, P1 = production of farms where rice production is more than 20 tonnes during the year 2002, and P2 = proportion of farms with rice cultivation area more than 20 hectares during the year 2003. For this data, we have

N ¼ 73;

Y ¼ 61:3;

¼ 0:621;

P1 ¼ 0:4247;

P2 ¼ 0:3425;

S2y ¼ 12371:4;

S2/1 ¼ 0:225490;

S2/2 ¼ 0:228311;

qyp1

qyp2 ¼ 0:673; q/ ¼ 0:889:

4. Conclusion In this paper, we have proposed some new estimators for estimating unknown population mean of study variable using information on two auxiliary attributes. From Table 1 we observe that the proposed estimator tp is best followed by the Lu et al. [5] type estimator t3. References [1] A.M. Abd-Elfattah, E.A. El-Sherpieny, S.M. Mohamed, O.F. Abdou, Improvement in estimating the population mean in simple random sampling using information on auxiliary attribute, Applied Mathematics and Computation (2010), http://dx.doi.org/10.1016/j.amc.2010.12.041. [2] C. Kadilar, H. Cingi, A new estimator using two auxiliary variables, Applied Mathematics and Computation 162 (2005) 901–908. [3] D.N. Gujarati, Sangeetha, Basic Econometrics, Tata McGraw – Hill, 2007. [4] G.U. Yule, On the methods of measuring association between two attributes, Journal of the Royal Society 75 (1912) 579–642.

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[5] J. Lu, Z. Yan, C. Ding, Z. Hong, The chain ratio estimator using two auxiliary information, International Conference on Computer and Communication Technologies in Agriculture Engineering (2010) 586–589. [6] J. Shabbir, S. Gupta, On estimating the finite population mean with known population proportion of an auxiliary variable, Pakistan Journal of Statistics 23 (1) (2007) 1–9. [7] R. Singh, M. Kumar, F. Smarandache, Ratio estimators in simple random sampling when study variable is an attribute, World Applied Science Journal 11 (5) (2010). [8] R. Singh, P. Chauhan, N. Sawan, On linear combination of Ratio-product type exponential estimator for estimating finite population mean, Statistics in Transition 9 (1) (2008) 105–115. [9] S. Bahl, R.K. Tuteja, Ratio and product type exponential estimator, Journal of Information and Optimization Sciences XII I (1991) 159–163. [10] V.D. Naik, P.C. Gupta, A note on estimation of mean with known population proportion of an auxiliary character, Journal of the Indian Society of Agricultural Statistics 48 (2) (1996) 151–158.