Viplav Kumar Singh, Rajesh Singh Department of Statistics, Banaras Hindu University Varanasi-221005, India
Florentin Smarandache University of New Mexico, Gallup, USA
Some Improved Estimators for Population Variance Using Two Auxiliary Variables in Double Sampling
Published in: Rajesh Singh, Florentin Smarandache (Editors) ON IMPROVEMENT IN ESTIMATING POPULATION PARAMETER(S) USING AUXILIARY INFORMATION Educational Publishing (Columbus) & Journal of Matter Regularity (Beijing), USA - China, 2013 ISBN: 978-1-59973-230-5 pp. 54 - 64
Abstract In this article we have proposed an efficient generalised class of estimator using two auxiliary variables for estimating unknown population variance S 2y of study variable y .We have also extended our problem to the case of two phase sampling. In support of theoretical results we have included an empirical study.
1. Introduction Use of auxiliary information improves the precision of the estimate of parameter .Out of many ratio and product methods of estimation are good example in this context. We can use ratio method of estimation when correlation coefficient between auxiliary and study variate is positive (high), on the other hand we use product method of estimation when correlation coefficient between auxiliary and study variate is highly negative. Variations are present everywhere in our day-to-day life. An agriculturist needs an adequate understanding of the variations in climatic factors especially from place to place (or time to time) to be able to plan on when, how and where to plant his crop. The problem of estimation of finite population variance S y2 , of the study variable y was discussed by Isaki (1983), Singh and Singh (2001, 2002, 2003), Singh et al. (2008), Grover (2010), and Singh et al. (2011). 54
Let x and z are auxiliary variates having values ( x i , z i ) and y is the study variate having values ( y i ) respectively. Let Vi (i = 1,2,.......N) is the population having N units such that y is positively correlated x and negatively correlated with z. To estimate S 2y , we assume that S 2x and S 2z are known, where
S 2y =
1 N 1 N 1 N ( y i − Y ) 2 , S 2x = ( x i − X ) 2 and S 2z = (z i − Z) 2 . N i =1 N i =1 N i =1
Assume that N is large so that the finite population correction terms are ignored. A sample of size n is drawn from the population V using simple random sample without replacement.
Usual unbiased estimator of population variance S 2y is s 2y , where, s 2y =
n 1 ( y i − y) 2 . (n − 1) i =1
Up to the first order of approximation, variance of s 2y is given by 2 y
var(s ) =
S 4y n
∂ *400
where, ∂ *400 = ∂ 400 − 1 ,
μ pqr =
(1.1)
∂ pqr =
μ pqr μ
p/2 200
r/2 μ q020/ 2 μ 002
, and
1 N ( y i − Y ) p ( x i − X ) q (z i − Z) r ; p, q, r being the non-negative integers. N i =1
2. Existing Estimators Let s 2y = S 2y (1 + e 0 ) , s 2x = S 2x (1 + e1 ) and s 2z = S 2z (1 + e 2 ) where, s 2x =
and x =
n n 1 1 2 2 ( x − x ) , s = i (z i − z ) 2 z (n − 1) i =1 (n − 1) i =1
1 n 1 n x , z = i zi . n i =1 n i =1
Also, let E (e1 ) = E (e 2 ) = 0
55
E (e 02 ) =
∂ *400 ∂* ∂* , E (e12 ) = 040 and E (e 22 ) = 004 n n n
E (e 0 e 1 ) =
1 * 1 1 ∂ 220 , E (e1e 2 ) = ∂ *022 , E (e 0 e 2 ) = ∂ *202 n n n
Isaki (1983) suggested ratio estimator t 1 for estimating S 2y ast 1 = s 2y
S 2y s 2x
; where s 2x is unbiased estimator of S 2x
(1.2)
Up to the first order of approximation, mean square error of t 1 is given by, MSE ( t 1 ) =
S 4y
[∂ n
* 400
+ ∂ *040 − 2∂ *220
]
(1.3)
Singh et al. (2007) proposed the exponential ratio-type estimator t 2 as S 2 − s 2x t 2 = s 2y exp 2x 2 Sx + s x
(1.4)
And exponential product type estimator t 3 as s 2 − S 2x t 3 = s 2y 2x 2 s x + Sx
(1.5)
Following Kadilar and Cingi (2006), Singh et al. (2011) proposed an improved estimator for estimating population variance S 2y , as S 2 − s 2x s 2x − S 2x + − t 4 = s 2y k 4 exp 2x ( 1 k ) 4 2 2 2 Sx + s x s x + S x
(1.6)
where k 4 is a constant. Up to the first order of approximation mean square errors of t 2 , t 3 and t 4 are respectively given by
MSE ( t 2 ) =
S 4y * ∂ *040 ∂ + − ∂ *220 400 n 4
(1.7)
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S 4y * ∂ *004 ∂ + − ∂ *202 MSE ( t 3 ) = 400 n 4
(1.8)
* * S 4y * k (1 − k 4 ) * 2 ∂ 040 2 ∂ 004 ∂ + + − − k 4 ∂ *220 + (1 − k 4 )∂ *202 − 4 ∂ 022 (1.9) k ( 1 k ) MSE ( t 4 ) = 400 4 4 2 n 4 4
where k 4 =
(∂ *004 / 2) + ∂ *220 + ∂ *022 . 2(∂ *040 + ∂ *004 + ∂ *022
3. Improved Estimator Using Singh and Solanki (2011), we propose some improved estimators for estimating population variance S 2y as cS 2 − Ds 2x t5 = s x 2 (c − d )S x
p
2 y
(a + b)S 2z t6 = s 2 2 aS x + bs x
(1.10)
q
2 y
(1.11)
q cS 2 − Ds 2 p (a + b)S 2z x x t 7 = s k 7 + (1 − k 7 ) 2 2 2 (c − d )S x aS x + bs x 2 y
(1.12)
where a, b, c, d are suitably chosen constants and k 7 is a real constant to be determined so as to minimize MSE’s. Expressing t 5 , t 6 and t 7 in terms of e i ' s , we have
[
t 5 = S 2y 1 − x 1 pe1 + e 0 − x 1 pe 0 e1 where, x 1 =
]
(1.13)
d . (c − d )
t 6 = S 2y [1 − qx 2 e 2 + e 0 − qx 2 e 0 e 2 ] where, x 2 =
[
(1.14)
b . (a + b )
t 7 = S 2y 1 + e 0 + px 1 k 7 (e1' − e1 ) + (k 7 − 1)qx 2 e '2
]
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(1.15)
The mean squared error of estimators are obtained by subtracting S 2y from each estimator and squaring both sides and than taking expectations-
MSE ( t 5 ) =
S 4y
[∂ n
* 400
+ x 12 p 2 ∂ *040 − 2 x 1 p∂ *220
]
(1.16)
Differentiating (1.16) with respect to x 1 , we get the optimum value of x 1 asx 1 ( opt ) =
∂ *220 . p∂ *040
MSE ( t 6 ) =
S 4y n
[∂
* 400
+ x 22 q 2 ∂ *004 − 2 x 1∂ *202
]
(1.17)
Differentiating (1.17) with respect to x 2 , we get the optimum value of x 2 as – x 2 ( opt ) =
∂ *202 . q∂ *004
MSE ( t 7 ) =
S 4y n
[A + k B + (1 − k 2 7
7
)C − 2k 7 (1 − k 7 )E − 2(1 − k 7 )F
]
(1.18)
Differentiating (1.18) with respect to k 7 , we get the optimum value k 7 of as – k 7 ( opt ) =
C+D−F−E . B + C − 2E
where A = ∂ *400 ,
B = x 12 p 2 ∂ *040 ,
C = x 22 q 2 ∂ *004 ,
D = x 1 p∂ *220 ,
E = x 1 x 2 pq∂ *022 ,
F = x 2 q∂ *202 .
2. Estimators In Two Phase Sampling In certain practical situations when S 2x is not known a priori, the technique of
two phase
sampling or double sampling is used. Allowing SRSWOR design in each phase, the two – phase sampling scheme is as follows: The first phase sample s 'n (s 'n ⊂ V ) of a fixed size n’ is drawn to measure only x and z
in order to formulate the a good estimate of S 2x and S 2z , respectively. 58
Given s 'n ,the second phase sample s n (s n ⊂ s 'n ) of a fixed size n is drawn to measure y
only.
Existing Estimators Singh et al. (2007) proposed some estimators to estimate S 2y in two phase sampling, as: s ' 2 −s 2 t '2 = s 2y exp 2x x2 s' x +s x
(2.1)
s' 2 −s 2 t 3' = s 2y exp 2z z2 s ' z +s z
(2.2)
s' 2 −s 2 s' 2 −s 2 t '4 = s 2y k '4 exp 2x x2 + (1 − k '4 ) exp 2z z2 s ' x +s x s' z +s z
(2.3)
MSE of the estimator t '2 , t 3' and t '4 are respectively, given by ∂ *400 1 1 1 * 1 1 MSE ( t ) = S + − ∂ 040 + − ∂ *220 4 n n' n' n n
(2.4)
∂ *400 1 1 1 * 1 1 MSE ( t ) = S + − ∂ 004 − − ∂ *202 4 n n' n' n n
(2.5)
' 2
' 3
4 y
4 y
[
]
MSE ( t '4 ) = S 4y A'+ k ' 24 B'+(1 − k '4 ) 2 C'+ k '4 D'+ (1 − k '4 )E'
And k '4 ( opt ) =
(2.6)
2C'+ E '−D' 2(B'+C' )
Where, ∂ *400 1 * 11 1 ∂ 004 , B' = − ∂ *040 C' = n 4 n n' 4n ' 1 1 1 D' = − ∂ *202 , E ' = ∂ *202 n' n' n
A' =
Proposed estimators in two phase sampling The estimator proposed in section 3 will take the following form in two phase sampling; cs' 2 −ds 2x t 5' = s 2y x 2 (c − d )s' x
(2.7)
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(a + b)S 2z t =s 2 2 aS z + bs' z
(2.8)
2 ' cs' 2x −ds 2x ' (a + b)S z + − t = s k 7 ( 1 k ) 7 2 2 2 (c − d )s' x aS z + bs' z
(2.9)
' 6
2 y
' 7
2 y
Let, s 2y = S 2y (1 + e 0 ) , s 2x = S 2x (1 + e1 ), s' 2x = S 2x (1 + e1' )
s 2z = S 2z (1 + e 2 ) , s' 2z = S 2z (1 + e '2 ) Where, 1 n' 1 n' 2 2 − = ( x x ' ) , s ' i (z i − z ' ) 2 z (n '−1) i =1 (n '−1) i =1
s' 2x =
and x ' =
1 n' 1 n' x , z ' = zi i n' n ' i =1 i =1
Also, E (e1' ) = E (e '2 ) = 0 ∂ *040 ∂* 2 , E (e '2 ) = 004 n' n' 1 1 1 E (e 0 e1' ) = ∂ *220 , E (e 0 e '2 ) = ∂ *202 , E (e1e1' ) = ∂ *040 n' n' n' 1 1 1 E (e 2 e '2 ) = ∂ *004 , E(e1e '2 ) = ∂ *022 , E (e1 ' e '2 ) = ∂ *022 n' n' n' 2
E (e1' ) =
Writing estimators t 5' , t '6 and t '7 in terms of e i ' s we have ,respectively
[
t 5' = S 2y 1 + e 0 + px 1 (e1' − e1 )
[
t '6 = S 2y 1 + e 0 − qx 2 e '2
[
]
(2.10)
]
(2.11)
t '7 = S 2y 1 + px 1 k '7 ((e1' − e1 ) + e 0 + (k '7 − 1)qx 2 e '2
]
(2.12)
Solving (1.10),(1.11) and (1.12),we get the MSE’S of the estimators t 5' , t '6 and t '7 , respectively as ∂* 1 1 1 1 MSE ( t 5' ) = S 4y 400 + p 2 x 12 − ∂ *040 + 2px 1 − ∂ *220 n n' n' n n
Differentiate (2.13) w.r.t. x 1 , we get the optimum value of x 1 as-
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(2.13)
x 1 ( opt ) =
∂ *220 p∂ *040
∂* 1 1 MSE ( t '6 ) = S 4y 400 + q 2 x 22 ∂ *004 + 2qx 2 ∂ *202 n' n' n
(2.14)
Differentiate (2.14) with respect to x 2 , we get the optimum value of x 2 as – x 2 ( opt ) =
∂ *202 . q∂ *004
[
2
MSE ( t '7 ) = S 4y A 1 + B1 k '7 + (k '7 − 1) 2 C1 + 2k '7 D1 + 2(k '7 − 1)E 1
]
Differentiate (2.15) with respect to k '7 , we get the optimum value of k '7 as – k '7 ( opt ) =
C1 − D1 − E 1 B1 + C1
Where, A1 =
∂ *400 q 2 x 22 * 1 1 , B1 = p 2 x 12 − ∂ *040 , C1 = px 1 ∂ 004 n n' n n'
∂* 1 1 D1 = px 1 − ∂ *220 , E 1 = qx 2 202 n' n' n
5. Empirical Study In support of theoretical result an empirical study is carried out. The data is taken from Murthy(1967): ∂ 400 = 3.726, ∂ 040 = 2.912, ∂ 044 = 2.808 ∂ 022 = 2.73, ∂ 202 = 2.979, ∂ 220 = 3.105 c x = 0.5938, c y = 0.7531, c z = 0.7205 ρ yz = 0.904, ρ xy = 0.98, n = 7, n ' = 15 X = 747.5882, Y = 199.4412, Z = 208.8824
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(2.15)
ď ś In Table 5.1 percent relative efficiency of various estimators of S 2y is written with respect to s 2y
ďƒ˜ Table 5.1: PRE of the estimator with respect to s 2y
Estimators
PRE
s 2y
100
t1
636.9158
t2
248.0436
t3
52.86019
t4
699.2526
t5
667.2895
t6
486.9362
t7
699.5512
62
ď ś In Table 5.2 percent relative efficiency of various estimators of S 2y is written with respect to s 2y in two phase sampling:
ďƒ˜
Table 5.2: PRE of the estimators in two phase sampling with respect to
Estimators
s 2y
PRE
s 2y
100
t2'
142.60
t3 '
66.42
t4'
460.75
t5 '
182.95
t6 '
158.93
t7 '
568.75
6. Conclusion In Table 5.1 and 5.2 percent relative efficiencies of various estimators are written with respect to s 2y . From Table 5.1 and 5.2 we observe that the proposed estimator
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under optimum condition performs better than usual estimator, Isaki (1983) estimator and Singh et al. (2007 ) estimator.
7. References Bahl, S. and Tuteja, R.K. (1991): Ratio and Product type exponential estimator. Information and Optimization sciences, Vol. XII, I, 159-163. Grover, L. K . (2010): A correction note on improvement in variance estimation using auxiliary information. Commun. Stat. Theo. Meth. 39:753–764. Isaki, C. T. (1983): Variance estimation using auxiliary information. Journal of American Statistical Association. Kadilar,C. and Cingi,H. (2006) : Improvement in estimating the population mean in simple random sampling. Applied Mathematics Letters 19 (2006), 75–79. Murthy, M. N. (1967).: Sampling Theory and Methods, Statistical Publishing Soc., Calcutta, India. Singh, H. P. and Singh, R. (2001): Improved Ratio-type Estimator for Variance Using Auxiliary Information. J.I.S.A.S.,54(3),276-287. Singh, H. P. and Singh, R. (2002): A class of chain ratio-type estimators for the coefficient of variation of finite population in two phase sampling. Aligarh Journal of Statistics, 22,1-9. Singh, H. P. and Singh, R. (2003): Estimation of variance through regression approach in two phase sampling. Aligarh Journal of Statistics, 23, 13-30. Singh, H. P, and Solanki, R. S. (2012): A new procedure for variance estimation in simple random sampling using auxiliary information. Statistical papers, DOI 10.1007/s00362-012-0445-2. Singh, R., Chauhan, P., Sawan, N. and Smarandache, F. (2011): Improved exponential estimator for population variance using two auxiliary variables. Italian Jour. Of Pure and Applied. Math., 28, 103110. Singh, R., Kumar, M., Singh, A.K. and Smarandache, F. (2011): population
A family of estimators of
variance using information on auxiliary attribute. Studies in sampling techniques and
time series analysis. Zip publishing, USA. Singh, R., Chauhan, P., Sawan, N. and Smarandache, F. (2008): Almost unbiased ratio and product type estimator of finite population variance using the knowledge of kurtosis of an auxiliary variable in sample surveys. Octogon Mathematical Journal, Vol. 16, No. 1, 123-130.
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