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Incorporating the auxiliary information available in variance estimation
Applied Mathematics and Computation 160 (2005) 387–399 www.elsevier.com/locate/amc
Incorporating the auxiliary information available in variance estimation A. Arcos a,*, M. Rueda a, M.D. Martınez a, S. Gonz alez a,b, Y. Rom an a a
Department of Statistics and Operational Research, University of Granada, Estadistica, 18071 Granada, Spain b University of Jaen, Spain
Abstract A difference estimator using an auxiliary variable x is defined to estimate the finite population variance Sy2 of the study character y. Classical difference type estimators use auxiliary information based on a single auxiliary parameter, specifically the parameter of interest, associated with the auxiliary variable. In practice, however, several parameters for auxiliary variables are available. This paper discusses how such estimators can be modified to improve the usual methods if information related to other parameters associated with an auxiliary variable is known. A simulation study is carried out to demonstrate the superiority of the suggested estimator over the others. 2003 Elsevier Inc. All rights reserved. Keywords: Auxiliary information; Finite population variance
1. Introduction The use of supplementary information for improving estimators in sample surveys has been dealt with at great length. It is common practice to use auxiliary information on a character x in the estimation of the finite population parameter Sy2 of a character y under study.
*
Corresponding author. E-mail address: [email protected] (A. Arcos).
0096-3003/$ - see front matter 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2003.11.010
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When auxiliary information is used in the estimation stage, it is very common for several parameters associated with the auxiliary variable to be available. In other situations, the population data associated with the auxiliary variables are obtained from the census, administrative files, etc., and these sources often provide different parameters of these auxiliary variables. The purpose of this paper is to apply a general difference method to incorporate the available information associated with auxiliary variables in the estimation stage when the population variance is estimated.
2. Estimation of finite population variance with auxiliary information Assume a sample s with size n from a population U with size N , selected by a specific sampling design. Let y be the variable which is the object of study and x, the available auxiliary variable. We consider that s is drawn by simple random sampling without replacement. To estimate the variance of a variable of interest y Sy2 ¼
N 1 X 2 ðyj Y Þ ; N 1 j¼1
if the variance of an auxiliary variable x is known, Sx2 , the simple estimator s2y ¼
n 1 X ðyj y Þ2 n 1 j¼1
is modified in order to construct ratio and difference estimators, Isaki [7] 2
S 2 b ¼ s2y 2x ; S IR sx
2 b S ID ¼ s2y þ d Sx2 s2x :
Nevertheless, in practice the population variance of the auxiliary variable, Sx2 , is not often known if its population mean, X is not also known. Indeed, the following expression: Sx2 ¼
N X 1 2 ðxj xt Þ ; N ðN 1Þ j6¼t
which does not require the calculation of X , is rarely used. It is then useful to provide estimation methods which exploit the knowledge of the population mean, X , to estimate the variance Sy2 with a known variance of x, Sx2 . Das and Tripathi [5] have considered sampling strategies for estimating the population variance of study variable y by using information on auxiliary variable x, studying their properties in the situation where the population
A. Arcos et al. / Appl. Math. Comput. 160 (2005) 387–399
389
parameter X or Sx2 is known. These strategies, based on simple random sampling with replacement, are a 2 a X 2 2 Sx b b S 12 ¼ s2y ; S 2 ¼ sy 2 ; x sx b S 32 ¼
s2y X ; X þ a x X
b S 42 ¼
s2y Sx2 ; Sx2 þ a s2x Sx2
where a is a suitable constant. Following the idea of Isaki, Singh et al. [12] propose two difference type estimators b S 52 ¼ W1 s2y W2 ðx X Þ;
b S 62 ¼ W1 s2y W2 ðs2x r2x Þ;
where W1 , W2 , W1 and W2 are constants to be chosen to minimise the MSE of the estimators. Other important works related to variance estimation include Das and Tripathi [4], Srivastava and Jhajj [14], Singh and Kataria [13], Gupta et al. [6] and Agrawal and Sthapit [2].
3. The proposed method Assume a sample s with size n from a population U of size N , selected by a specific sampling design. Let y be the variable which is the object of study and x1 ; . . . ; xk , the available auxiliary variables with (for the sake of simplicity) two known population parameters Hi ¼ ðSxi2 ; hi2 Þ, i ¼ 1; . . . ; k. The difference method can be applied to estimate the population parameter Sy2 by using observations of the variables y, x1 ; . . . ; xk in the sample s, and the known population values H1 ; . . . ; Hk associated with the auxiliary variables. Suppose that unbiased estimators of the parameters Sxi2 and hi2 , b S xi2 and b h i2 2 2 b are provided and let S y be an unbiased estimator of Sy . A difference estimator b S D2 associated with the parameter Sy2 is given by b S D2 ¼ b S y2 þ
k k X X ci Sxi2 b di hi2 b S xi2 þ h i2 : i¼1
ð1Þ
i¼1
0 0 2 2 2 ; h12 ; . . . ; Sxk ; hk2 Þ and p ¼ ð b S x1 ; By denoting B ¼ ðc1 ; d1 ; . . . ; ck ; dk Þ , P ¼ ðSx1 0 2 b b b h 12 ; . . . ; S xk ; h k2 Þ estimator (1) can be written as 0 b S D2 ¼ b S y2 þ ðP pÞ B:
By calculations similar to those used by Rao [10], it is possible to obtain the B-value that provides the minimum variance of the estimator (1). It can be seen
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that Bopt ¼ minB V ð b S D2 Þ ¼ R1 r and Vmin ð b S D2 Þ ¼ V ð b S y2 Þ r0 R1 r, where Rð2k2kÞ ¼ ðaij Þ with aii ¼ V ð b S 2 Þ if i is odd, aii ¼ V ðb h i2 Þ if i is even, aij ¼ covð^h i 2 ; ^h j Þ ð2k2kÞ
xi
2
22
if i and j are even, aij ¼ covð^ h2i 2 ; b S xj2 Þ if i is even and j odd, aij ¼ covð b S xi2 ; b h j2 Þ if 2 2 b2 b i is odd and j is even and aij ¼ covð S xi ; S xj Þ if i and j are odd. Similarly r ¼ ðcovð b S 2; b S 2 Þ; covð b S 2; b h 12 Þ . . . ; covð b S 2; b S 2 Þ; covð b S 2; b h k2 ÞÞ0 . y
x1
y
y
xk
y
Consequently, this unbiased estimator is always more accurate than the simple estimator b S y2 . Note that no model has been used to arrive at this estimator. The optimal B-value depends on unknown population characteristics, so the optimal estimator cannot be used. In the absence of good a priori knowledge of these characteristics, we replace the optimal B-value by sample-based estib and r ^, we obtain mates. After replacing R and r by their unbiased estimators R the estimator b2 ¼ b b 1 r ^: S S 2 þ ðP pÞ R ð2Þ Dopt
y
b ¼ ð^ ^ii ¼ Vb ð b S xi2 Þ if i is odd, ^aii ¼ Vb ð^hi2 Þ if i is even, ^aij ¼ where R aij Þ with a cd ovð^ h2i 2 ; ^ hj 2 Þ if i and j are even, ^ aij ¼ cd ovð^ h2i 2 ; b S xj2 Þ if i is even and j odd, ^aij ¼ 2 b2 ; ^ cd ovð S h j Þ if i is odd and j is even, ^ aij ¼ cd ovð b S2 ; b S 2 Þ if i and j are odd, and xi
22
xi
xj
0 2 2 ^ ¼ ðd S x1 Þ; cd ovð b S y2 ; ^ h12 Þ . . . ; cd ovð b S y2 ; b S xk Þ; cd ovð b S y2 ; ^hk2 ÞÞ . r covð b S y2 ; b
4. Asymptotic theory The estimator (1) is asymptotically unbiased and normal. The asymptotic unbiasedness of the proposed estimator is easily derived by its linear expression and the fact that the estimators b S y2 , b S xi2 and ^hi2 are unbiased of their respective 2 2 parameters. Similarly, if b Sy , b S xi and ^ hi2 are asymptotically normal, the estimator (1) is also asymptotically normal. Next, we consider the estimator b S D2 opt , which is obtained by replacing the unknown parameters with their sampling estimators. Randles [9] derived the limit distribution for such statistics. Following his notation, we denote the b 1 r ^. estimator b S D2 opt as Tn ð^ kÞ with ^ k¼R ^ We replace k in Tn ðÞ with a variable 1. Now we calculate the limit of the expectation of the statistic Tn ðkÞ when the current value of the parameter k is lðkÞ ¼ lim Ek ðTn ð1ÞÞ ¼ Sy2 ^ and Tn ðkÞ have and then its derivative on 1 ¼ k is zero. It now follows that Tn ðkÞ the same limit distribution, i.e., b S D2 opt and b S D2 ¼ b S y2 þ ðP pÞ0 R1 r have the same limit distribution. Thus b S D2 opt is asymptotically unbiased for Sy2 and has an asymptotically minimum variance. If b S y2 and p are asymptotically normal, then the estimator
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b S y2 Þ S D2 opt is asymptotically normal with expectation Sy2 and variance V ð b r0 R1 r.
5. Estimating the population variance Sy2 using Sx2 and X under
SRSWOR
Estimator (1) in the finite population variance estimation case, Sy2 , when hi2 ¼ X i is b S D2 ¼ s2y þ c Sx2 s2x þ dðX xÞ: ð3Þ The classical difference estimator is 2 b S ID ¼ s2y þ d Sx2 s2x ; which only uses one known population parameter for variable x, Sx2 and the unbiased estimators s2y and s2x of Sy2 and Sx2 , respectively. Observe that estimator (3) uses auxiliary information which is often not fully exploited. Thus 0 1 V s2x cov s2x ; x B C R¼@ A and 2 cov sx ; x V ðxÞ 0 r ¼ cov s2y ; s2x ; cov s2y ; x ; detðRÞ ¼ V ðxÞV ðs2x Þð1 q2 ðx; s2x ÞÞ and the optimum estimator can be evaluated except for q2 ðx; s2x Þ ¼ 1. The determination of the optimum estimator and its minimum variance by Vmin ð b S D2 Þ ¼ V ðs2y Þ r0 R1 r then only requires us to compute the variances and covariances for the unbiased estimator of the parameters. The problem is then reduced to determining V ðxÞ, covðx; s2x Þ, V ðs2x Þ, covðs2y ; xÞ and covðs2y ; s2x Þ, i.e., the variances and covariances of the unbiased estimators of the parameters. To obtain the minimum variance by Vmin ð b S D2 Þ ¼ V ðs2y Þ r0 R1 r the same conditions must be applied. In this section we develop estimator (3) under simple random sampling without replacement (S R S W O R ), because with this sampling design it is possible to evaluate the variances and covariances of the unbiased estimators for the parameters depending on certain population values. Indeed, given a sample with size n obtained by this design from a population of size N , by denoting lrs ¼
N 1 X r s ðyi Y Þ ðxi X Þ ; N i¼1
^rs ¼ l
n 1X r s ðyi y Þ ðxi xÞ n i¼1
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and writing Sx2 ¼
N l ¼ K02 ; N 1 02
s2x ¼
n b 02 ; ^ ¼K l n 1 02
Sy2 ¼
N l ¼ K20 N 1 20
and s2y ¼
n b 20 ; ^ ¼K l n 1 20
we obtain, Kendall and Stuart [8] that 1 1 1 1 b b K 02 ; V ðxÞ ¼ K02 ; V ðxÞ ¼ n N n N 2 N n l04 2 K 2 M ; V sx ¼ K02 ðN 2Þn l02 N n l40 2 2 K 2 M ; V sy ¼ K20 ðN 2Þn l20 1 1 1 1 2 2 cov x; sx ¼ K03 ; cov y ; sx ¼ K12 ; n N n N N n l22 l211 2 2 K K2 cov sy ; sx ¼ K20 K02 þ K1 ðN 2Þn l20 l02 l20 l02 with K¼
ðN 1ÞðNn N n 1Þ ; ðn 1ÞN ðN 3Þ
M¼
N 2 n 3N 2 þ 6N 3n 3 ; ðn 1ÞN ðN 3Þ
K03 ¼
N2 l ; ðN 1ÞðN 2Þ 03
b 03 ¼ K
n2 ^ ; l ðn 1Þðn 2Þ 03
K12 ¼
N2 l ; ðN 1ÞðN 2Þ 12
b 12 ¼ K
n2 ^ ; l ðn 1Þðn 2Þ 12
K1 ¼
2ðN 1ÞðN n 1Þ ; ðn 1ÞN ðN 3Þ
K2 ¼
N 2 n 2Nn N 2 þ 2N n 1 : ðn 1ÞN ðN 3Þ
From the above expressions, we can evaluate the theoretical asymptotic variance for the estimator (3) and also provide an estimator of this value.
6. A more general class of estimators Motivated by Srivastava and Jhajj [14] and Allen et al. [1], we suggest a class of estimators of Sy2 :
A. Arcos et al. / Appl. Math. Comput. 160 (2005) 387–399
o G¼ b S g2 ; b S g2 ¼ Gð b S y2 ; u1 ; . . . uk ; v1 . . . ; vk Þ ; n
393
ð4Þ
bS ^ where GðÞ is a function of b S y2 , ui ¼ S 2xi , vi ¼ hhi2i2 continuous in a closed convex 2
xi
sub-space, P , containing the point 1 ¼ (1, . . . ,1), such that: • GðSy2 ; 1Þ ¼ Sy2 ; • G1 ðSy2 ; 1Þ ¼ 1 where G1 ðSy2 ; 1Þ denoting the first partial derivative of GðÞ with respect to b S y2 , and • the first- and second-order partial derivatives of G exist and are also continuous in P . Any parametric function G satisfying these conditions can generate an asymptotically acceptable estimator. Note that the usual ratio and difference 2 2 estimators, b S IR and b S ID , the usual estimator b S y2 ¼ s2y and b S i2 , i ¼ 1; . . . ; 6 are included in (4). Moreover, if the information on hi2 , i ¼ 1; . . . ; k is not used, then the class of estimators (4) reduces to the class of estimators of Sy2 as n o S h2 ¼ G b H¼ b S h2 ; b S y2 ; u1 ; . . . ; uk ; i.e., a classical, well-known class of estimators that uses the same parameter but is associated with an auxiliary variable as supplementary information. Parti2 S R2 , b S ID and b S 62 . cular cases of estimators in this class are b S 22 , b 2 By expanding G about the point ðSy ; 1Þ in a second order Taylor series, it is found that: X oG 2 2 2 b b S g ¼ S y Sy þ 2 ou i ðSy ;1Þ i¼1;...;k X oG þ ovi i¼1;...;k
ðSy2 ;1Þ
^ hi2 1 hi2
b S xi2 1 Sxi2
!
! þ Oðn1 Þ
and so the bias of b S g2 is of the order n1 . By squaring both sides in the last expression and taking expectations, we obtain the variance of b S g2 to the first degree of approximation. Following Allen et al. [1], to terms of the order n1 , we obtain that for all estimators in class (4) V b S g2 P V b S y2 r0 R1 r:
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It is interesting to note that the lower bound of the asymptotic variance of b S g2 is 2 b the variance of the difference estimator S Dopt with the value estimates. Thus, asymptotically, b S D2 opt is an optimal estimator in this class, in the sense that it has a lower asymptotic variance and, moreover, is better than any estimator of the class b S h2 . In conclusion, the proposed estimator is asymptotically better than the other estimators analyzed in this paper.
7. Numerical comparison The following examples reflect the potential gains that would accrue from the use of the proposed estimator, b S D2 opt , instead of the customary simple 2 S ID . estimator, s2y , and the classical difference estimator, b The first population (S U G A R C A N E ), originally used by Chambers and Dunstan [3], consists of 338 sugar cane farms surveyed in 1982 in Queensland, Australia. The principal variable is the income y, and the planted area x is the auxiliary variable. The M U 2 8 4 population consists of 284 municipalities in Sweden. The variable of interest is RMT85, the revenues from 1985 municipal taxation (in millions of kronor) and the auxiliary variable is the number of Social-Democratic seats in the municipal councils, SS82. The M U 2 8 1 population consists of all but the three largest municipalities, with a value of RMT85. These data are taken from S€ arndal et al. [11]. The B E E F population consists of 430 farms. This population was also originally used by Chambers and Dunstan [3]. In this case, x is the number of beef cattle in each farm and y is the income received. A description of the C A N C E R , H O S P I T A L , C O 6 0 and C O 7 0 populations can be seen in Valliant et al. [15] and the populations are available at the John Wiley worldwide web site. For all populations and for n ¼ 10; 20; 30; 40 and 50 as sample sizes, the theoretical asymptotic variances of the compared estimators in estimating the population variance are computed. Root of ratio, R, of the variance V ð^hÞ to the variance of the simple estimator, V ð b S y2 Þ ¼ V ðs2y Þ, is computed as a measure of efficiency of any of the estimators: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ RðhÞ ¼ V ð^ hÞ=V ð b S y2 Þ: Table 1 shows R for these sample sizes and the populations considered, and reveals that the proposed difference estimator, b S D2 , improves on the behaviour 2 of the simple estimator and the classical difference estimator, b S ID , for all of the sample sizes. The error reduction, in comparison with the classical difference estimator, is not so marked as with the simple estimator. The error reduction with respect to
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395
Table 1 Theoretical asymptotic efficiency of the compared estimators of a population variance Estimator
SUGAR
CANCER
MU284
CO60
MU281
CO70
BEEF
HOSPITAL
n ¼ 10 s2y b S2
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
0.3789
0.3294
0.9767
0.0604
0.8639
0.3440
0.8969
0.7528
b S D2
0.3788
0.3271
0.9688
0.0599
0.8248
0.3391
0.8492
0.7523
n ¼ 20 s2y b S2
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
0.3752
0.3288
0.9758
0.0600
0.8615
0.3443
0.8968
0.7597
0.3751
0.3262
0.9684
0.0594
0.8230
0.3390
0.8476
0.7589
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
0.3740
0.3287
0.9755
0.0598
0.8606
0.3444
0.8968
0.7620
0.3739
0.3259
0.9683
0.0592
0.8224
0.3389
0.8471
0.7610
n ¼ 40 s2y b S2
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
0.3735
0.3286
0.9754
0.0598
0.8602
0.3444
0.8968
0.7631
b S D2
0.3734
0.3257
0.9682
0.0591
0.8222
0.3389
0.8469
0.7620
n ¼ 50 s2y b S2
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
0.3731
0.3285
0.9753
0.0597
0.8600
0.3444
0.8968
0.7638
b S D2
0.3730
0.3256
0.9682
0.0591
0.8220
0.3389
0.8467
0.7626
ID
ID b2 S D
n ¼ 30 s2y b S2 ID b S D2
ID
ID
the difference estimator is greatest (about 4% for all sample sizes) for the M U 2 8 1 and B E E F populations.
8. Simulation study In order to test the real behaviour of the proposed estimator, we carried out a simulation study with the M U 2 1 8 and B E E F populations. For these two populations and for all sample sizes, 5000 samples ðsi ; i ¼ 1; . . . ; 1000Þ were chosen in accordance with S R S W O R . As a measure of real efficiency, we computed 2 P5000 ^ hðsi Þ Sy2 i¼1 b ^ Rð hÞ ¼ P 2 5000 b 2 2 ðs Þ S S i i¼1 y y
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2 for the simple estimator b S y2 ¼ s2y , the classical difference estimator, b S ID and the 2 b proposed difference estimator, S Dopt .
Table 2 Real efficiency ð b r Þ of the compared estimators of a population variance Est.
n ¼ 10
n ¼ 20
n ¼ 30
n ¼ 40
n ¼ 50
1.0000 0.8362 0.7903
1.0000 0.8636 0.8167
1.0000 0.8617 0.8191
1.0000 0.8629 0.8162
1.0000 0.8620 0.8095
1.0000 0.8275 0.7730
1.0000 0.8460 0.7979
1.0000 0.8543 0.8129
1.0000 0.8698 0.8125
1.0000 0.8728 0.8147
MU281
s2y 2 b S ID b S D2 BEEF
s2y 2 b S ID b S D2
n = 10
n = 20
Simple Difference Proposed
Simple Difference Proposed
n = 30
Simple Difference Proposed
n = 40
Simple Difference Proposed n = 50
Simple Difference Proposed Fig. 1. Box-plots for standard, difference and proposed estimators for the
BEEF
population.
A. Arcos et al. / Appl. Math. Comput. 160 (2005) 387–399
397
n = 10
n = 20
Simple Difference Proposed
Simple Difference Proposed
n = 30
Simple Difference Proposed
n = 40
Simple Difference Proposed n = 50
Simple Difference Proposed Fig. 2. Box-plots for standard, difference and proposed estimators for the
MU281
population.
b for the populations considered and for sample sizes n ¼ 10, Table 2 shows R 20, 30, 40 and 50 in the B E E F and M U 2 8 1 populations. 2 Note it is not possible to ensure b S D2 opt > 0 for all sample s (analogy b S ID > 0). 2 b However, our simulation study reveals that for sample size n P 20, S Dopt > 0 2 and b S ID > 0, whereas b S D2 opt < 0 only for 1% of samples of size n ¼ 10. Table 2 clearly shows the superiority of the proposed estimator b S D2 opt over 2 the sample variance s2y and the classical difference estimator SbID , with error reductions of 4.1–5.8%. Figs. 1 and 2 show the box and whisker plots for the three estimators in the M U 2 8 1 and B E E F populations, respectively. These diagrams enable us to visualize the behaviour of the estimators more clearly. In surveying, the b S D2 opt estimator should be considered as a serious alterna2 tive to b S ID when selecting among variance estimators.
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9. Conclusions The difference method provides alternative estimators to the classical difference estimator (which only uses one auxiliary parameter, the parameter of interest, but associated with the auxiliary variable). The proposed estimators are valid, i.e., they always improve on the usual estimators, reducing the errors obtained. Their use is appropriate if information is available about other parameters associated with the auxiliary variable or variables because these estimators are optimal, with minimal variance in the class G which contains all known estimators b S i2 , i ¼ 1; . . . ; 6. The proposed method is simple to implement and the estimated variance is computed from sample data and serves as an indicator of the quality of a survey estimate. The proposed estimator has been developed for use when both the population variance of an auxiliary variable and the population mean are known (as is usual in practice). However, the method can be used with other parameters such as the coefficient of variation or the kurtosis coefficient, and its extension to more than two parameters is very easy.
References [1] J. Allen, H.P. Singh, S. Singh, F. Smarandache, A general class of estimators of population median using two auxiliary variables in double sampling, Interstat. [2] M.C. Agrawal, A.B. Sthapit, Unbiased ratio-type variance estimation, Statistics and Probability Letters 25 (1995) 361–364. [3] R.L. Chambers, R. Dunstan, Estimating distribution functions from survey data, Biometrika 73 (1986) 597–604. [4] A.K. Das, T.P. Tripathi, Admissible estimators for quadratic forms in finite populations, Bulletin Institute International Statistics 47 (1977) 132–135. [5] A.K. Das, T.P. Tripathi, Use of auxiliary information in estimating the finite population variance, Sankhya 40 (1978) 139–148. [6] R.K. Gupta, S. Singh, N.S. Mangat, Some chain ratio type estimators for estimating finite population variance, Aligarh Journal of Statistics 13 (1993) 65–69. [7] C.T. Isaki, Variance estimation using auxiliary information, Journal of the American Statistical Association 78 (1983) 117–123. [8] M. Kendall, A. Stuart, The Advanced Theory of Statistics, Griffin, London, 1977. [9] R.H. Randles, On the asymptotic normality of statistics with estimated parameters, The Annals of Statistics 10 (1982) 462–474. [10] J.N.K. Rao, Estimating totals and distribution functions using auxiliary information at the estimation stage, Journal of Official Statistics 10 (1994) 153–165. [11] C.E. S€ arndal, B. Swensson, J. Wretman, Model Assisted Survey Sampling, Springer-Verlag, New York, 1992. [12] H.P. Singh, L.N. Upadhyaya, U.D. Namjoshi, Estimation of finite population variance, Current Science 57 (1988) 1331–1334.
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[13] S. Singh, P. Kataria, An estimator of finite population variance, Journal of Indian Society of Agricultural Statistics 42 (1990) 186–188. [14] S.K. Srivastava, H.S. Jhajj, A class of estimators using auxiliary information for estimating finite population variance, Sankhya 42 (1980) 87–96. [15] R. Valliant, A.H. Dorfman, R.M. Royall, Finite Population Sampling and Inference, John Wiley and Sons, New York, 2000.
Incorporating the auxiliary information available in variance estimation A. Arcos a,*, M. Rueda a, M.D. Martınez a, S. Gonz alez a,b, Y. Rom an a a
Department of Statistics and Operational Research, University of Granada, Estadistica, 18071 Granada, Spain b University of Jaen, Spain
Abstract A difference estimator using an auxiliary variable x is defined to estimate the finite population variance Sy2 of the study character y. Classical difference type estimators use auxiliary information based on a single auxiliary parameter, specifically the parameter of interest, associated with the auxiliary variable. In practice, however, several parameters for auxiliary variables are available. This paper discusses how such estimators can be modified to improve the usual methods if information related to other parameters associated with an auxiliary variable is known. A simulation study is carried out to demonstrate the superiority of the suggested estimator over the others. 2003 Elsevier Inc. All rights reserved. Keywords: Auxiliary information; Finite population variance
1. Introduction The use of supplementary information for improving estimators in sample surveys has been dealt with at great length. It is common practice to use auxiliary information on a character x in the estimation of the finite population parameter Sy2 of a character y under study.
*
Corresponding author. E-mail address: [email protected] (A. Arcos).
0096-3003/$ - see front matter 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2003.11.010
388
A. Arcos et al. / Appl. Math. Comput. 160 (2005) 387–399
When auxiliary information is used in the estimation stage, it is very common for several parameters associated with the auxiliary variable to be available. In other situations, the population data associated with the auxiliary variables are obtained from the census, administrative files, etc., and these sources often provide different parameters of these auxiliary variables. The purpose of this paper is to apply a general difference method to incorporate the available information associated with auxiliary variables in the estimation stage when the population variance is estimated.
2. Estimation of finite population variance with auxiliary information Assume a sample s with size n from a population U with size N , selected by a specific sampling design. Let y be the variable which is the object of study and x, the available auxiliary variable. We consider that s is drawn by simple random sampling without replacement. To estimate the variance of a variable of interest y Sy2 ¼
N 1 X 2 ðyj Y Þ ; N 1 j¼1
if the variance of an auxiliary variable x is known, Sx2 , the simple estimator s2y ¼
n 1 X ðyj y Þ2 n 1 j¼1
is modified in order to construct ratio and difference estimators, Isaki [7] 2
S 2 b ¼ s2y 2x ; S IR sx
2 b S ID ¼ s2y þ d Sx2 s2x :
Nevertheless, in practice the population variance of the auxiliary variable, Sx2 , is not often known if its population mean, X is not also known. Indeed, the following expression: Sx2 ¼
N X 1 2 ðxj xt Þ ; N ðN 1Þ j6¼t
which does not require the calculation of X , is rarely used. It is then useful to provide estimation methods which exploit the knowledge of the population mean, X , to estimate the variance Sy2 with a known variance of x, Sx2 . Das and Tripathi [5] have considered sampling strategies for estimating the population variance of study variable y by using information on auxiliary variable x, studying their properties in the situation where the population
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389
parameter X or Sx2 is known. These strategies, based on simple random sampling with replacement, are a 2 a X 2 2 Sx b b S 12 ¼ s2y ; S 2 ¼ sy 2 ; x sx b S 32 ¼
s2y X ; X þ a x X
b S 42 ¼
s2y Sx2 ; Sx2 þ a s2x Sx2
where a is a suitable constant. Following the idea of Isaki, Singh et al. [12] propose two difference type estimators b S 52 ¼ W1 s2y W2 ðx X Þ;
b S 62 ¼ W1 s2y W2 ðs2x r2x Þ;
where W1 , W2 , W1 and W2 are constants to be chosen to minimise the MSE of the estimators. Other important works related to variance estimation include Das and Tripathi [4], Srivastava and Jhajj [14], Singh and Kataria [13], Gupta et al. [6] and Agrawal and Sthapit [2].
3. The proposed method Assume a sample s with size n from a population U of size N , selected by a specific sampling design. Let y be the variable which is the object of study and x1 ; . . . ; xk , the available auxiliary variables with (for the sake of simplicity) two known population parameters Hi ¼ ðSxi2 ; hi2 Þ, i ¼ 1; . . . ; k. The difference method can be applied to estimate the population parameter Sy2 by using observations of the variables y, x1 ; . . . ; xk in the sample s, and the known population values H1 ; . . . ; Hk associated with the auxiliary variables. Suppose that unbiased estimators of the parameters Sxi2 and hi2 , b S xi2 and b h i2 2 2 b are provided and let S y be an unbiased estimator of Sy . A difference estimator b S D2 associated with the parameter Sy2 is given by b S D2 ¼ b S y2 þ
k k X X ci Sxi2 b di hi2 b S xi2 þ h i2 : i¼1
ð1Þ
i¼1
0 0 2 2 2 ; h12 ; . . . ; Sxk ; hk2 Þ and p ¼ ð b S x1 ; By denoting B ¼ ðc1 ; d1 ; . . . ; ck ; dk Þ , P ¼ ðSx1 0 2 b b b h 12 ; . . . ; S xk ; h k2 Þ estimator (1) can be written as 0 b S D2 ¼ b S y2 þ ðP pÞ B:
By calculations similar to those used by Rao [10], it is possible to obtain the B-value that provides the minimum variance of the estimator (1). It can be seen
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that Bopt ¼ minB V ð b S D2 Þ ¼ R1 r and Vmin ð b S D2 Þ ¼ V ð b S y2 Þ r0 R1 r, where Rð2k2kÞ ¼ ðaij Þ with aii ¼ V ð b S 2 Þ if i is odd, aii ¼ V ðb h i2 Þ if i is even, aij ¼ covð^h i 2 ; ^h j Þ ð2k2kÞ
xi
2
22
if i and j are even, aij ¼ covð^ h2i 2 ; b S xj2 Þ if i is even and j odd, aij ¼ covð b S xi2 ; b h j2 Þ if 2 2 b2 b i is odd and j is even and aij ¼ covð S xi ; S xj Þ if i and j are odd. Similarly r ¼ ðcovð b S 2; b S 2 Þ; covð b S 2; b h 12 Þ . . . ; covð b S 2; b S 2 Þ; covð b S 2; b h k2 ÞÞ0 . y
x1
y
y
xk
y
Consequently, this unbiased estimator is always more accurate than the simple estimator b S y2 . Note that no model has been used to arrive at this estimator. The optimal B-value depends on unknown population characteristics, so the optimal estimator cannot be used. In the absence of good a priori knowledge of these characteristics, we replace the optimal B-value by sample-based estib and r ^, we obtain mates. After replacing R and r by their unbiased estimators R the estimator b2 ¼ b b 1 r ^: S S 2 þ ðP pÞ R ð2Þ Dopt
y
b ¼ ð^ ^ii ¼ Vb ð b S xi2 Þ if i is odd, ^aii ¼ Vb ð^hi2 Þ if i is even, ^aij ¼ where R aij Þ with a cd ovð^ h2i 2 ; ^ hj 2 Þ if i and j are even, ^ aij ¼ cd ovð^ h2i 2 ; b S xj2 Þ if i is even and j odd, ^aij ¼ 2 b2 ; ^ cd ovð S h j Þ if i is odd and j is even, ^ aij ¼ cd ovð b S2 ; b S 2 Þ if i and j are odd, and xi
22
xi
xj
0 2 2 ^ ¼ ðd S x1 Þ; cd ovð b S y2 ; ^ h12 Þ . . . ; cd ovð b S y2 ; b S xk Þ; cd ovð b S y2 ; ^hk2 ÞÞ . r covð b S y2 ; b
4. Asymptotic theory The estimator (1) is asymptotically unbiased and normal. The asymptotic unbiasedness of the proposed estimator is easily derived by its linear expression and the fact that the estimators b S y2 , b S xi2 and ^hi2 are unbiased of their respective 2 2 parameters. Similarly, if b Sy , b S xi and ^ hi2 are asymptotically normal, the estimator (1) is also asymptotically normal. Next, we consider the estimator b S D2 opt , which is obtained by replacing the unknown parameters with their sampling estimators. Randles [9] derived the limit distribution for such statistics. Following his notation, we denote the b 1 r ^. estimator b S D2 opt as Tn ð^ kÞ with ^ k¼R ^ We replace k in Tn ðÞ with a variable 1. Now we calculate the limit of the expectation of the statistic Tn ðkÞ when the current value of the parameter k is lðkÞ ¼ lim Ek ðTn ð1ÞÞ ¼ Sy2 ^ and Tn ðkÞ have and then its derivative on 1 ¼ k is zero. It now follows that Tn ðkÞ the same limit distribution, i.e., b S D2 opt and b S D2 ¼ b S y2 þ ðP pÞ0 R1 r have the same limit distribution. Thus b S D2 opt is asymptotically unbiased for Sy2 and has an asymptotically minimum variance. If b S y2 and p are asymptotically normal, then the estimator
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b S y2 Þ S D2 opt is asymptotically normal with expectation Sy2 and variance V ð b r0 R1 r.
5. Estimating the population variance Sy2 using Sx2 and X under
SRSWOR
Estimator (1) in the finite population variance estimation case, Sy2 , when hi2 ¼ X i is b S D2 ¼ s2y þ c Sx2 s2x þ dðX xÞ: ð3Þ The classical difference estimator is 2 b S ID ¼ s2y þ d Sx2 s2x ; which only uses one known population parameter for variable x, Sx2 and the unbiased estimators s2y and s2x of Sy2 and Sx2 , respectively. Observe that estimator (3) uses auxiliary information which is often not fully exploited. Thus 0 1 V s2x cov s2x ; x B C R¼@ A and 2 cov sx ; x V ðxÞ 0 r ¼ cov s2y ; s2x ; cov s2y ; x ; detðRÞ ¼ V ðxÞV ðs2x Þð1 q2 ðx; s2x ÞÞ and the optimum estimator can be evaluated except for q2 ðx; s2x Þ ¼ 1. The determination of the optimum estimator and its minimum variance by Vmin ð b S D2 Þ ¼ V ðs2y Þ r0 R1 r then only requires us to compute the variances and covariances for the unbiased estimator of the parameters. The problem is then reduced to determining V ðxÞ, covðx; s2x Þ, V ðs2x Þ, covðs2y ; xÞ and covðs2y ; s2x Þ, i.e., the variances and covariances of the unbiased estimators of the parameters. To obtain the minimum variance by Vmin ð b S D2 Þ ¼ V ðs2y Þ r0 R1 r the same conditions must be applied. In this section we develop estimator (3) under simple random sampling without replacement (S R S W O R ), because with this sampling design it is possible to evaluate the variances and covariances of the unbiased estimators for the parameters depending on certain population values. Indeed, given a sample with size n obtained by this design from a population of size N , by denoting lrs ¼
N 1 X r s ðyi Y Þ ðxi X Þ ; N i¼1
^rs ¼ l
n 1X r s ðyi y Þ ðxi xÞ n i¼1
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and writing Sx2 ¼
N l ¼ K02 ; N 1 02
s2x ¼
n b 02 ; ^ ¼K l n 1 02
Sy2 ¼
N l ¼ K20 N 1 20
and s2y ¼
n b 20 ; ^ ¼K l n 1 20
we obtain, Kendall and Stuart [8] that 1 1 1 1 b b K 02 ; V ðxÞ ¼ K02 ; V ðxÞ ¼ n N n N 2 N n l04 2 K 2 M ; V sx ¼ K02 ðN 2Þn l02 N n l40 2 2 K 2 M ; V sy ¼ K20 ðN 2Þn l20 1 1 1 1 2 2 cov x; sx ¼ K03 ; cov y ; sx ¼ K12 ; n N n N N n l22 l211 2 2 K K2 cov sy ; sx ¼ K20 K02 þ K1 ðN 2Þn l20 l02 l20 l02 with K¼
ðN 1ÞðNn N n 1Þ ; ðn 1ÞN ðN 3Þ
M¼
N 2 n 3N 2 þ 6N 3n 3 ; ðn 1ÞN ðN 3Þ
K03 ¼
N2 l ; ðN 1ÞðN 2Þ 03
b 03 ¼ K
n2 ^ ; l ðn 1Þðn 2Þ 03
K12 ¼
N2 l ; ðN 1ÞðN 2Þ 12
b 12 ¼ K
n2 ^ ; l ðn 1Þðn 2Þ 12
K1 ¼
2ðN 1ÞðN n 1Þ ; ðn 1ÞN ðN 3Þ
K2 ¼
N 2 n 2Nn N 2 þ 2N n 1 : ðn 1ÞN ðN 3Þ
From the above expressions, we can evaluate the theoretical asymptotic variance for the estimator (3) and also provide an estimator of this value.
6. A more general class of estimators Motivated by Srivastava and Jhajj [14] and Allen et al. [1], we suggest a class of estimators of Sy2 :
A. Arcos et al. / Appl. Math. Comput. 160 (2005) 387–399
o G¼ b S g2 ; b S g2 ¼ Gð b S y2 ; u1 ; . . . uk ; v1 . . . ; vk Þ ; n
393
ð4Þ
bS ^ where GðÞ is a function of b S y2 , ui ¼ S 2xi , vi ¼ hhi2i2 continuous in a closed convex 2
xi
sub-space, P , containing the point 1 ¼ (1, . . . ,1), such that: • GðSy2 ; 1Þ ¼ Sy2 ; • G1 ðSy2 ; 1Þ ¼ 1 where G1 ðSy2 ; 1Þ denoting the first partial derivative of GðÞ with respect to b S y2 , and • the first- and second-order partial derivatives of G exist and are also continuous in P . Any parametric function G satisfying these conditions can generate an asymptotically acceptable estimator. Note that the usual ratio and difference 2 2 estimators, b S IR and b S ID , the usual estimator b S y2 ¼ s2y and b S i2 , i ¼ 1; . . . ; 6 are included in (4). Moreover, if the information on hi2 , i ¼ 1; . . . ; k is not used, then the class of estimators (4) reduces to the class of estimators of Sy2 as n o S h2 ¼ G b H¼ b S h2 ; b S y2 ; u1 ; . . . ; uk ; i.e., a classical, well-known class of estimators that uses the same parameter but is associated with an auxiliary variable as supplementary information. Parti2 S R2 , b S ID and b S 62 . cular cases of estimators in this class are b S 22 , b 2 By expanding G about the point ðSy ; 1Þ in a second order Taylor series, it is found that: X oG 2 2 2 b b S g ¼ S y Sy þ 2 ou i ðSy ;1Þ i¼1;...;k X oG þ ovi i¼1;...;k
ðSy2 ;1Þ
^ hi2 1 hi2
b S xi2 1 Sxi2
!
! þ Oðn1 Þ
and so the bias of b S g2 is of the order n1 . By squaring both sides in the last expression and taking expectations, we obtain the variance of b S g2 to the first degree of approximation. Following Allen et al. [1], to terms of the order n1 , we obtain that for all estimators in class (4) V b S g2 P V b S y2 r0 R1 r:
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It is interesting to note that the lower bound of the asymptotic variance of b S g2 is 2 b the variance of the difference estimator S Dopt with the value estimates. Thus, asymptotically, b S D2 opt is an optimal estimator in this class, in the sense that it has a lower asymptotic variance and, moreover, is better than any estimator of the class b S h2 . In conclusion, the proposed estimator is asymptotically better than the other estimators analyzed in this paper.
7. Numerical comparison The following examples reflect the potential gains that would accrue from the use of the proposed estimator, b S D2 opt , instead of the customary simple 2 S ID . estimator, s2y , and the classical difference estimator, b The first population (S U G A R C A N E ), originally used by Chambers and Dunstan [3], consists of 338 sugar cane farms surveyed in 1982 in Queensland, Australia. The principal variable is the income y, and the planted area x is the auxiliary variable. The M U 2 8 4 population consists of 284 municipalities in Sweden. The variable of interest is RMT85, the revenues from 1985 municipal taxation (in millions of kronor) and the auxiliary variable is the number of Social-Democratic seats in the municipal councils, SS82. The M U 2 8 1 population consists of all but the three largest municipalities, with a value of RMT85. These data are taken from S€ arndal et al. [11]. The B E E F population consists of 430 farms. This population was also originally used by Chambers and Dunstan [3]. In this case, x is the number of beef cattle in each farm and y is the income received. A description of the C A N C E R , H O S P I T A L , C O 6 0 and C O 7 0 populations can be seen in Valliant et al. [15] and the populations are available at the John Wiley worldwide web site. For all populations and for n ¼ 10; 20; 30; 40 and 50 as sample sizes, the theoretical asymptotic variances of the compared estimators in estimating the population variance are computed. Root of ratio, R, of the variance V ð^hÞ to the variance of the simple estimator, V ð b S y2 Þ ¼ V ðs2y Þ, is computed as a measure of efficiency of any of the estimators: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ RðhÞ ¼ V ð^ hÞ=V ð b S y2 Þ: Table 1 shows R for these sample sizes and the populations considered, and reveals that the proposed difference estimator, b S D2 , improves on the behaviour 2 of the simple estimator and the classical difference estimator, b S ID , for all of the sample sizes. The error reduction, in comparison with the classical difference estimator, is not so marked as with the simple estimator. The error reduction with respect to
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395
Table 1 Theoretical asymptotic efficiency of the compared estimators of a population variance Estimator
SUGAR
CANCER
MU284
CO60
MU281
CO70
BEEF
HOSPITAL
n ¼ 10 s2y b S2
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
0.3789
0.3294
0.9767
0.0604
0.8639
0.3440
0.8969
0.7528
b S D2
0.3788
0.3271
0.9688
0.0599
0.8248
0.3391
0.8492
0.7523
n ¼ 20 s2y b S2
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
0.3752
0.3288
0.9758
0.0600
0.8615
0.3443
0.8968
0.7597
0.3751
0.3262
0.9684
0.0594
0.8230
0.3390
0.8476
0.7589
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
0.3740
0.3287
0.9755
0.0598
0.8606
0.3444
0.8968
0.7620
0.3739
0.3259
0.9683
0.0592
0.8224
0.3389
0.8471
0.7610
n ¼ 40 s2y b S2
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
0.3735
0.3286
0.9754
0.0598
0.8602
0.3444
0.8968
0.7631
b S D2
0.3734
0.3257
0.9682
0.0591
0.8222
0.3389
0.8469
0.7620
n ¼ 50 s2y b S2
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
1.0000
0.3731
0.3285
0.9753
0.0597
0.8600
0.3444
0.8968
0.7638
b S D2
0.3730
0.3256
0.9682
0.0591
0.8220
0.3389
0.8467
0.7626
ID
ID b2 S D
n ¼ 30 s2y b S2 ID b S D2
ID
ID
the difference estimator is greatest (about 4% for all sample sizes) for the M U 2 8 1 and B E E F populations.
8. Simulation study In order to test the real behaviour of the proposed estimator, we carried out a simulation study with the M U 2 1 8 and B E E F populations. For these two populations and for all sample sizes, 5000 samples ðsi ; i ¼ 1; . . . ; 1000Þ were chosen in accordance with S R S W O R . As a measure of real efficiency, we computed 2 P5000 ^ hðsi Þ Sy2 i¼1 b ^ Rð hÞ ¼ P 2 5000 b 2 2 ðs Þ S S i i¼1 y y
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2 for the simple estimator b S y2 ¼ s2y , the classical difference estimator, b S ID and the 2 b proposed difference estimator, S Dopt .
Table 2 Real efficiency ð b r Þ of the compared estimators of a population variance Est.
n ¼ 10
n ¼ 20
n ¼ 30
n ¼ 40
n ¼ 50
1.0000 0.8362 0.7903
1.0000 0.8636 0.8167
1.0000 0.8617 0.8191
1.0000 0.8629 0.8162
1.0000 0.8620 0.8095
1.0000 0.8275 0.7730
1.0000 0.8460 0.7979
1.0000 0.8543 0.8129
1.0000 0.8698 0.8125
1.0000 0.8728 0.8147
MU281
s2y 2 b S ID b S D2 BEEF
s2y 2 b S ID b S D2
n = 10
n = 20
Simple Difference Proposed
Simple Difference Proposed
n = 30
Simple Difference Proposed
n = 40
Simple Difference Proposed n = 50
Simple Difference Proposed Fig. 1. Box-plots for standard, difference and proposed estimators for the
BEEF
population.
A. Arcos et al. / Appl. Math. Comput. 160 (2005) 387–399
397
n = 10
n = 20
Simple Difference Proposed
Simple Difference Proposed
n = 30
Simple Difference Proposed
n = 40
Simple Difference Proposed n = 50
Simple Difference Proposed Fig. 2. Box-plots for standard, difference and proposed estimators for the
MU281
population.
b for the populations considered and for sample sizes n ¼ 10, Table 2 shows R 20, 30, 40 and 50 in the B E E F and M U 2 8 1 populations. 2 Note it is not possible to ensure b S D2 opt > 0 for all sample s (analogy b S ID > 0). 2 b However, our simulation study reveals that for sample size n P 20, S Dopt > 0 2 and b S ID > 0, whereas b S D2 opt < 0 only for 1% of samples of size n ¼ 10. Table 2 clearly shows the superiority of the proposed estimator b S D2 opt over 2 the sample variance s2y and the classical difference estimator SbID , with error reductions of 4.1–5.8%. Figs. 1 and 2 show the box and whisker plots for the three estimators in the M U 2 8 1 and B E E F populations, respectively. These diagrams enable us to visualize the behaviour of the estimators more clearly. In surveying, the b S D2 opt estimator should be considered as a serious alterna2 tive to b S ID when selecting among variance estimators.
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9. Conclusions The difference method provides alternative estimators to the classical difference estimator (which only uses one auxiliary parameter, the parameter of interest, but associated with the auxiliary variable). The proposed estimators are valid, i.e., they always improve on the usual estimators, reducing the errors obtained. Their use is appropriate if information is available about other parameters associated with the auxiliary variable or variables because these estimators are optimal, with minimal variance in the class G which contains all known estimators b S i2 , i ¼ 1; . . . ; 6. The proposed method is simple to implement and the estimated variance is computed from sample data and serves as an indicator of the quality of a survey estimate. The proposed estimator has been developed for use when both the population variance of an auxiliary variable and the population mean are known (as is usual in practice). However, the method can be used with other parameters such as the coefficient of variation or the kurtosis coefficient, and its extension to more than two parameters is very easy.
References [1] J. Allen, H.P. Singh, S. Singh, F. Smarandache, A general class of estimators of population median using two auxiliary variables in double sampling, Interstat. [2] M.C. Agrawal, A.B. Sthapit, Unbiased ratio-type variance estimation, Statistics and Probability Letters 25 (1995) 361–364. [3] R.L. Chambers, R. Dunstan, Estimating distribution functions from survey data, Biometrika 73 (1986) 597–604. [4] A.K. Das, T.P. Tripathi, Admissible estimators for quadratic forms in finite populations, Bulletin Institute International Statistics 47 (1977) 132–135. [5] A.K. Das, T.P. Tripathi, Use of auxiliary information in estimating the finite population variance, Sankhya 40 (1978) 139–148. [6] R.K. Gupta, S. Singh, N.S. Mangat, Some chain ratio type estimators for estimating finite population variance, Aligarh Journal of Statistics 13 (1993) 65–69. [7] C.T. Isaki, Variance estimation using auxiliary information, Journal of the American Statistical Association 78 (1983) 117–123. [8] M. Kendall, A. Stuart, The Advanced Theory of Statistics, Griffin, London, 1977. [9] R.H. Randles, On the asymptotic normality of statistics with estimated parameters, The Annals of Statistics 10 (1982) 462–474. [10] J.N.K. Rao, Estimating totals and distribution functions using auxiliary information at the estimation stage, Journal of Official Statistics 10 (1994) 153–165. [11] C.E. S€ arndal, B. Swensson, J. Wretman, Model Assisted Survey Sampling, Springer-Verlag, New York, 1992. [12] H.P. Singh, L.N. Upadhyaya, U.D. Namjoshi, Estimation of finite population variance, Current Science 57 (1988) 1331–1334.
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[13] S. Singh, P. Kataria, An estimator of finite population variance, Journal of Indian Society of Agricultural Statistics 42 (1990) 186–188. [14] S.K. Srivastava, H.S. Jhajj, A class of estimators using auxiliary information for estimating finite population variance, Sankhya 42 (1980) 87–96. [15] R. Valliant, A.H. Dorfman, R.M. Royall, Finite Population Sampling and Inference, John Wiley and Sons, New York, 2000.