Sachin Malik, Rajesh Singh 1Department of Statistics, Banaras Hindu University Varanasi-221005, India
Florentin Smarandache University of New Mexico, Gallup, USA
A General Procedure of Estimating Population Mean Using Information on Auxiliary Attribute
Published in: Rajesh Singh, F. Smarandache (Editors) SAMPLING STRATEGIES FOR FINITE POPULATION USING AUXILIARY INFORMATION The Educational Publisher, Columbus, USA, 2015 ISBN 978-1-59973-348-7 pp. 21 - 30
Sampling Strategies for Finite Population Using Auxiliary Information
Abstract This paper deals with the problem of estimating the finite population mean when some information on auxiliary attribute is available. It is shown that the proposed estimator is more efficient than the usual mean estimator and other existing estimators. The results have been illustrated numerically by taking empirical population considered in the literature.
Keywords
Simple random sampling, auxiliary attribute, point bi-serial correlation, ratio estimator, 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 are many situations when auxiliary information is available in the form of attributes, e.g. sex and height of the persons, amount of milk produced and a particular breed of cow, amount of yield of wheat crop and a particular variety of wheat (see Jhajj et. al. (2006)). Consider a sample of size n drawn by simple random sampling without replacement (SRSWOR) from a population of size N. Let y i and i denote the observations on variable y and respectively for i th unit ( i =1, 2,......, N). Let i =1; if the i th unit of the population possesses attribute = 0; otherwise. N
n
i 1
i 1
Let A= i and a= i , denote the total number of units in the population and sample respectively possessing attribute . Let P=A/N and p=a/n denote the proportion of units in the population and sample respectively possessing attribute . Naik and Gupta (1996) introduced a ratio estimator t NG when the study variable and the auxiliary attribute are positively correlated. The estimator t NG is given by
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Rajesh Singh ■ Florentin Smarandache (editors)
t NG y
P p
(1.1)
with MSE
MSE(t NG ) f1 S2y R 2S2 2RS y
where
(1.2)
f1
Nn , Nn
S2
1 N 1 N S , P i P y i Y . i y N 1 i 1 N 1 i 1
R
2 1 N Y , S2y yi Y , P N n i 1 2
(for details see Singh et al. (2008)) Jhajj et. al. (2006) suggested a family of estimators for the population mean in single and two phase sampling when the study variable and auxiliary attribute are positively correlated. Shabbir and Gupta (2007), Singh et. al. (2008) and Abd-Elfattah et. al. (2010) have considered the problem of estimating population mean Y taking into consideration the point biserial correlation coefficient between auxiliary attribute and study variable. The objective of this article is to suggest a generalised class of estimators for population mean Y and analyse its properties. A numerical illustration is given in support of the present study.
2. Proposed Estimator Let *i i mA , m being a suitably chosen scalar, that takes values 0 and 1. Then
q p mA p NmP , and
Q ( Nm 1)P, where q
N n b B , Q , B i and b i . n N i 1 i 1
Motivated by Bedi (1996), we define a family of estimators for population mean Y as
q t w1 y w 2 bP p Q
(2.1)
where w1 , w2 and are suitably chosen scalars. To obtain the Bias and MSE of the estimator t, we write
22
Sampling Strategies for Finite Population Using Auxiliary Information
y Y1 e 0 , p P1 e1 , s 2 S2 1 e 2 , s y S y 1 e 3 , b 1 e 3 1 e 2 1
such that
E(e i ) 0 , i=0,1,2,3 and
1 1 E(e 02 ) C 2y , n N
1 1 E(e12 ) C 2p , n N
1 1 E(e 0 e1 ) pb C y C p , n N
1 1 E(e1e 2 ) C p 03 , n N
1 1 E(e1e 3 ) C p n N
12 , pb
Expressing (2.1) in terms of e’s , we have e1 t Y w 1 1 e 0 w 2 e1 1 e 3 1 e 2 1 1 R Nm 1
(2.2)
e1 e1 We assume that e2 1 and 1 , so that ( 1 e2 ) 1 and 1 are expandable. Nm 1 Na 1 Expanding the right hand side of (2.2) and retaining terms up to second powers of e’s ,we have
e 0 e1 e1 e12 1 t Y Y[ w1 1 e 0 Nm 1 2 Nm 12 Nm 1 e12 w 2 e1 e1e 3 e1e 2 1] R Nm 1
(2.3)
Taking expectation of both sides of (2.3) , we get the bias of t to the first degree of approximation as :
1 B( t ) Y w1 1 w1 f1C py f1C 2p Nm 1 2Nm 12 w2
f1 C p R
12 C p 03 C 2p pb Nm 1
(2.4)
Squaring both sides of (2.3) and neglecting terms of e’s having power greater than two, we have 23
Rajesh Singh ■ Florentin Smarandache (editors)
t Y
2
2 1e12 4e 0 e1 2 e 2 2 1 Y w 1 1 2e 0 e0 2 Nm 1 Nm 1 Nm 1 2
w 22 R
2
e12
2e12 1 2w1w 2 e1 e1e 3 e 0 e1 e1e 2 R Nm 1
2 e 0 e1 1e1 e1 2w1 1 e 0 Nm 1 Nm 1 2Nm 12 e12 2w 2 e1 e1e 3 e1e 2 R Nm 1 (2.5) Taking expectation of both sides of (2.5), we get the MSE of t to the first degree of approximation as:
m m m 2 MSE ( t ) Y 1 w12 A1m w 2 A 2 2w1w 2 A 3 2w1A 4 2w 2 A 5 2
(2.6)
2
C p 2 1 2 A1m 1 f1 C y Nm 1 Nm 1 4k
where,
2
A 2 f1C 2p R
A 3m
2 2 Cp f1 C p k 12 C p 03 R Nm 1 pb
1 A 4m 1 f1 k Nm 1 2Nm 1 f C p C p C A 5m 12 p 03 R 1 Nm 1 pb 2
where , k pb
Cy Cp
.
The MSE(t) is minimised for w1
w2
A A A A A A A w A A A A w A A A m 4
2
m 1
m 3
m 1
m 3
m 5
2
m 3
m 4
m 1
2
m 3
(2.7)
10
2
m 5
2
24
20
(2.8)
Sampling Strategies for Finite Population Using Auxiliary Information
3. Members of the family of estimator of t and their Biases and MSE Table 3.1:
Different members of the family of estimators of t
Choice of scalars w2 w1
Estimator m
t1 y
1
0
0
0
w1
0
0
0
w1
0
m
w1
0
0
1
0
-1
0
1
1
-1
t 2 w1 y Searls (1964) type estimator q t 3 w 1 y Q p t 4 w y P P t 5 y p ,
0
Naik and Gupta (1996) estimator P t 6 y bP p p
Singh et. al. (2008) estimator
w1
w2
0
0
t 7 w 1 y w 2 bP p
w1
1
0
0
t 8 w 1 y bP p
w
w
0
0
t 9 w y bP p
1
1
0
0
t10 y bP p Regression estimator
The estimator t1 y is an unbiased estimator of the population mean Y and has the variance Var t1 f1S2y
(3.1)
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Rajesh Singh ■ Florentin Smarandache (editors)
To, the first degree of approximation the biases and MSE’s of t i ' s , i=1,2,.......,10 are respectively given by Bt 2 Yw1 1
(3.2)
1 2 Bt 3 Y w1 1 f1w1 pb C y C p C p 2Nm 1 Nm 1
(3.3)
1 2 Bt 4 Y w1 1 w1f1 pb C y C p C p 2
(3.4)
Bt 5 Yf1 C 2p pb C y C p
(3.5)
Bt 6 Yf1 (C 2p pb C y C p ) C p R Bt 7 Y w1 1 12 f1 C p R
Bt 8 Y w1 1 f1 C p R
12 C p 03 pb
12 C p 03 pb
12 C p 03 pb
Bt 9 Y w 1 wf 1 C p 12 C p 03 R pb Bt10 Y
f1 C p 12 C p 03 R pb
(3.6)
(3.7)
(3.8)
(3.9)
(3.10)
The corresponding MSE’s will be
1 w A 2w A
MSE t 2 Y 1 w 12 A100 2w 1A 400 2
MSE t 3 Y
2
2 m 1 1
m 4
1
MSE t 4 Y 1 w 12 A10 2w 1A 40 2
MSE t 5 Y 1 A101 2A 401 2
(3.11)
(3.12)
(3.13)
(3.14) 26
Sampling Strategies for Finite Population Using Auxiliary Information
MSE t 6 Y 1 A101 A 2 2A 30 1 2A 401 2A 50 1 2
MSE t 7 Y 1 w12 A100 w 22 A 2 2w1w 2 A 300 2w1A 400 2w 2 A 500 2
MSE t 8 Y 1 w12 A100 A 2 2w1 A 300 A 400 2A 500 2
MSE t 9 Y 1 w 2 A100 A 2 2A 300 2w A 400 A 500 2
MSE t 10 Y 1 A100 A 2 2 A 300 A 400 A 500 2
(3.15)
(3.16)
(3.17)
(3.18) (3.19)
The MSE’s of the estimaors of ti, i=2,3,4,7,8,9 will be minimised respectively, for
w1
A 400
A100
(3.20)
A 4m
w1
A1m
(3.21)
w1
w1
w2
w1
A 40
A10
(3.22)
A A A A A A A A A A A A A A 0 40
2
0 30
0 2 10
( 0) 3(0)
( 0) 4( 0)
0 50 0 2 30 ( 0) 1(0)
( 0) 2 1(0)
( 0) 5(0) ( 0) 2 3(0)
(3.23)
A A
w
0 30
A 100
0 40
(3.24)
A 40(0 ) A 500
A100 A 2 2A 300
(3.25)
Thus the resulting minimum MSE of ti , i= 2,3,4,7,8,9 are, respectively given by
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Rajesh Singh ■ Florentin Smarandache (editors)
0 2 A 2 40 min . MSE t 2 Y 1 A100
(3.26)
m 2 A 2 4 min . MSE t 3 Y 1 A1m
(3.27)
A 1
2
min . MSE t 4 Y
0 2 4 A10
(3.28)
0 2 0 0 0 0 A 2 A 40 2A 30 A 50 A10 A 50 2 min . MSE t 7 Y 1 2 A 2 A100 A 300
A 300 A 400 2 0 min . MSE t 8 Y 1 A 2 2A 50 A100
2
(3.29)
2
(3.30)
0 A 0 2 A 2 40 50 min . MSE t 9 Y 1 0 0 A10 A 2 2A 30
(3.31)
4. Empirical study The data for the empirical study is taken from natural population data set considered by Sukhatme and Sukhatme (1970): y = Number of villages in the circles and = A circle consisting more than five villages
N 89, Y 3.36, P 0.1236, pb 0.766, C y 0.6040, C p 2.190 04 6.1619, 40 3.810, 12 146.475, 03 2.2744
In the Table 4.1 percent relative efficiencies (PRE’s) of various estimators are computed with respect to y . 28
Sampling Strategies for Finite Population Using Auxiliary Information
Table 4.1: PRE of different estimators of Y with respect to y . Estimator t1 y
PRE’s 100.00
t2 t3
101.41
t4 t5
6.92 11.64
t6
7.38
t7
100.44
t8
243.39
t9
243.42
t10
241.98
90.35
Conclusion The MSE values of the members of the family of the estimator t have been obtained using (2.6). These values are given in Table 4.1. When we examine Table 4.1, we observe the superiority of the proposed estimators t2, t7, t8, t9 and t10 over usual unbiased estimator t1, t3, t4, Naik and Gupta (1996) estimator t5 and Singh et. al. (2008) estimator t6. From this result we can infer that the proposed estimators t8 and t9 are more efficient than the rest of the estimators considered in this paper for this data set. We would also like to remark that the value of the min. MSE(t10), which is equal to the value of the MSE of the regression estimator is 241.98. From Table 4.1 we notice that the value of MSE of the estimators t8 and t9 are less than this value, as shown in Table 4.1. Finally, we can say that the proposed estimators t8 and t9 are more efficient than the regression estimator for this data set.
References 1. Abd-Elfattah, A.M. El-Sherpieny, E.A. Mohamed, S.M. Abdou, O. F. (2010): Improvement in estimating the population mean in simple random sampling using information on auxiliary attribute. Appl. Mathe. and Compt. doi:10.1016/j.amc.2009.12.041 2. Bedi, P. K. (1996). Efficient utilization of auxiliary information at estimation stage. Biom. Jour, 38:973–976. 3. Jhajj, H.S., Sharma, M.K. and grover, L.K. (2006) : A family of estimators of population mean using information on auxiliary attribute. Pak. Journ. of Stat., 22(1), 43-50. 4. Naik,V.D. and Gupta,P.C.(1996): A note on estimation of mean with known population proportion of an auxiliary character. Journ. of the Ind. Soc. of Agr. Stat., 48(2), 151-158. 5. Searls, D.T. (1964): The utilization of known coefficient of variation in the estimation procedure. Journ. of the Amer. Stat. Assoc., 59, 1125-1126. 29
Rajesh Singh ■Florentin Smarandache (editors) 6. Singh, R. Chauhan, 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. 7. Shabbir, J. and Gupta, S.(2007): On estimating the finite population mean with known population proportion of an auxiliary variable. Pak. Journ. of Stat., 23 (1), 1-9. 8. Sukhatme, P.V. and Sukhatme, B.V. (1970): Sampling theory of surveys with applications. Iowa State University Press, Ames, U.S.A.
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