Rajesh Singh Department of Statistics,Banaras Hindu University(U.P.), India
Pankaj Chauhan, Nirmala Sawan School of Statistics, DAVV, Indore (M.P.), India
Florentin Smarandache Department of Mathematics, University of New Mexico, Gallup, USA
Improved Exponential Estimator for Population Variance Using Two Auxiliary Variables
Published in: Rajesh Singh, Jayant Singh, F. Smarandache (Editors) STUDIES IN STATISTICAL INFERENCE, SAMPLING TECHNIQUES AND DEMOGRAPHY InfoLearnQuest, Ann Arbor, USA, 2009 (ISBN-10): 1-59973-087-1 (ISBN-13): 978-1-59973-087-5 pp. 36 - 44
Abstract
In this paper exponential ratio and exponential product type estimators using two auxiliary variables are proposed for estimating unknown population variance S 2y . Problem is extended to the case of two-phase sampling. Theoretical results are supported by an empirical study. Key words: Auxiliary information, exponential estimator, mean squared error.
1.
Introduction
It is common practice to use the auxiliary variable for improving the precision of the estimate of a parameter. Out of many ratio and product methods of estimation are good examples in this context. When the correlation between the study variate and the auxiliary variate is positive (high) ratio method of estimation is quite effective. On the other hand, when this correlation is negative (high) product method of estimation can be employed effectively. Let y and (x,z) denotes the study variate and auxiliary variates taking the values yi and (xi,zi) respectively, on the unit Ui (i=1,2,‌‌,N), where x is positively correlated with
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y and z is negatively correlated with y. To estimate S 2y that S 2x
N 1 (x i X ) 2 ÂŚ ( N 1) i 1
and S 2z
N 1 ( y i y) 2 , it is assumed ÂŚ ( N 1) i 1
N 1 ( z i Z) 2 are known. Assume that ÂŚ ( N 1) i 1
population size N is large so that the finite population correction terms are ignored. Assume that a simple random sample of size n is drawn without replacement (SRSWOR) from U. The usual unbiased estimator of S 2y is
s 2y
where y
n 1 ( y i y) 2 ÂŚ ( n 1) i 1
(1.1)
1 n ÂŚ y i is the sample mean of y. ni1
When the population variance S 2x
N 1 ( x i X ) 2 is known, Isaki (1983) proposed a ÂŚ ( N 1) i 1
ratio estimator for S 2y as
tk
where s 2x
s 2y
S 2x s 2x
(1.2)
n 1 2 ( x i X ) 2 is an unbiased estimator of S x . ÂŚ ( n 1) i 1
Up to the first order of approximation, the variance of S 2y and MSE of tk (ignoring the finite population correction (fpc) term) are respectively given by
var s 2y
§ S 4y ¨ ¨ n Š
¡ ¸>w 400 1@ ¸ š
§ S 4y MSE t k ¨ ¨ n Š
¡ ¸>w 400 w 040 2w 220 @ ¸ š
(1.3)
(1.4)
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where G pqr
P pqr
P
P pqr p/2 200
q/2 r/2 P 020 P 002
,
1 N p q r ÂŚ y i Y x i X z i Z ; p, q, r being the non-negative integers. Ni1
Following Bahl and Tuteja (1991), we propose exponential ratio type and exponential product type estimators for estimating population variance S 2y as –
t1
ÂŞ S 2 s 2x Âş s 2y expÂŤ 2x 2 Âť ÂŹSx s x Âź
(1.5)
t2
ÂŞ s 2 S 2z Âş s 2y expÂŤ 2z 2 Âť ÂŹ s z Sz Âź
(1.6)
2. Bias and MSE of proposed estimators To obtain the bias and MSE of t1, we write s 2y
S 2y 1 e 0 , s 2x
S2x 1 e1
Such that E(e0) = E(e1) = 0 and E(e 02 )
1 w 400 1 , E(e12 ) n
1 w 040 1 , E(e 0 e1 ) n
1 w 220 1 . n
After simplification we get the bias and MSE of t1 as
B( t 1 ) #
S 2y ÂŞ w 040 w 220 3 Âş Âť n ÂŤÂŹ 8 2 8Âź
MSE( t 1 ) #
S 2y ÂŞ w 1Âş w 400 040 w 220 Âť ÂŤ n ÂŹ 4 4Âź
(2.1)
(2.2)
To obtain the bias and MSE of t2, we write s 2y
S 2y 1 e 0 , s 2z
S2z 1 e 2
Such that E(e0) = E(e2) = 0
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E(e 22 )
1 w 004 1 , E(e 0 e 2 ) n
1 w 202 1
n
After simplification we get the bias and MSE of t2 as
B( t 2 ) #
S 2y ª w 004 w 202 5 º » n «¬ 8 2 8¼
MSE( t 2 ) #
3.
(2.3)
S 2y ª w 9º w 400 004 w 202 » n «¬ 4 4¼
(2.4)
Improved Estimator
Following Kadilar and Cingi (2006) and Singh et. al. (2007), we propose an improved estimator for estimating population variance S 2y as-
t
ª  S 2 s 2x ½  s 2z S 2z ½º s 2y «D exp ® 2x ( 1 ) exp D ¾ ® 2 2 ¾» 2 ¯S x s x ¿ ¯ s x S z ¿¼» ¬«
(3.1)
where D is a real constant to be determined such that the MSE of t is minimum. Expressing t in terms of e’s, we have
t
ª ° e § e · 1 ½° ° e § e · 1 ½°º S 2y 1 e 0 «D exp® 1 ¨1 1 ¸ ¾ (1 D) exp® 2 ¨1 2 ¸ ¾» 2 ¹ °¿ 2 ¹ °¿» °¯ 2 © °¯ 2 © «¬ ¼
(3.2)
Expanding the right hand side of (3.2) and retaining terms up to second power of e’s, we have ª § e e e2 e e e2 t # S 2y «1 e 0 2 2 0 2 D¨¨ 1 1 2 8 2 8 «¬ © 2
· §e e2 ¸¸ D¨¨ 2 2 8 ¹ © 2
§ e §e e 2 ·º e2 · e 0 D¨¨ 1 1 ¸¸ De 0 ¨¨ 2 2 ¸¸ » 8 ¹ ¼» 8 ¹ © 2 © 2
· ¸¸ ¹
(3.3)
Taking expectations of both sides of (3.3) and then subtracting S 2y from both sides, we get the bias of the estimator t, up to the first order of approximation, as
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S 2y ÂŞ D w 040 1 1 D w 004 1 1 D w 202 1
n ÂŤÂŹ 8 8 2
B( t )
D w 220 1 º 2 Ÿ
(3.4)
From (3.4), we have
t S # S 2 y
2 y
De1 1 D Âş ÂŞ ÂŤe 0 2 2 e 2 Âť Âź ÂŹ
(3.5)
Squaring both the sides of (3.5) and then taking expectation, we get MSE of the estimator t, up to the first order of approximation, as
MSE( t ) #
2 S 4y ÂŞ D2 w 040 1 1 D w 004 1
ÂŤ w 400 1 n ÂŹ 4 4
D w 220 1 1 D w 202 1
D 1 D
w 022 1 º 2 Ÿ
(3.6)
Minimization of (3.6) with respect to D yields its optimum value as
D
^w 004 2(w 220 w 202 ) w 022 6` w 040 w 004 2w 022 4
D 0 (say)
(3.7)
Substitution of D 0 from (3.7) into (3.6) gives minimum value of MSE of t. 4.
Proposed estimators in two-phase sampling
In certain practical situations when S 2x is not known a priori, the technique of twophase or double sampling is used. This scheme requires collection of information on x and z the first phase sample sâ&#x20AC;&#x2122; of size nâ&#x20AC;&#x2122; (nâ&#x20AC;&#x2122;<N) and on y for the second phase sample s of size n (n<nâ&#x20AC;&#x2122;) from the first phase sample. The estimators t1, t2 and t in two-phase sampling will take the following form, respectively
t 1d
ª s' 2x s 2x º s 2y exp  2 2   s' x s x Ÿ
(4.1)
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t 2d
ª s' 2 s 2 º s 2z exp« 2z z2 » ¬ s ' z s z ¼
(4.2)
ª  s' 2 s 2 ½  s ' 2 s 2 ½ º s 2y «k exp ® 2x x2 ¾ (1 k ) exp ® 2z z2 ¾» ¯ s' x s x ¿ ¯ s' z s z ¿¼» ¬«
td
(4.3)
To obtain the bias and MSE of t1d, t2d, td, we write
S2x 1 e1 , s' 2x
s 2y
S 2y 1 e 0 , s 2x
s 2z
S2z 1 e 2 , s' 2z S2z 1 e' 2
1 n' x i x ' 2 , s 2z n ' 1 ¦ i 1
where s' 2x
x'
1 n' ¦ x i , zc n' i 1
S2x 1 e'1
1 n' z i z ' 2 n ' 1 ¦ i 1
1 n' ¦ zi n' i 1
Also, E(eâ&#x20AC;&#x2122;1) = E(eâ&#x20AC;&#x2122;2) = 0,
1 w 040 1 , E(e' 22 ) n'
E(e'12 )
E(e'1 e' 2 )
1 w 004 1 , n
1 w 220 1
n'
Expressing t1d, t2d, and td in terms of eâ&#x20AC;&#x2122;s and following the procedure explained in section 2 and section3 we get the MSE of these estimators, respectively asª1 1§1 1 · MSE ( t 1d ) # S 4y « w 400 1 ¨ ¸ w 040 1
4 © n n' ¹ ¬n º § 1 1· ¨ ¸ w 220 1 » n ' n © ¹ ¼
(4.4)
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ª1 1§1 1 · MSE ( t 2d ) # S 4y « w 400 1 ¨ ¸ w 004 1
n 4 © n n' ¹ ¬ º § 1 1· ¨ ¸ w 202 1 » © n' n ¹ ¼
ª1 k2 MSE( t d ) # S 4y « w 400 1 4 ¬n
(4.5)
§¨ 1 1 ·¸ w
k 2 1 §1 1· ¨ ¸ w 040 1 4 © n n' ¹
©n
n' ¹
004
1
§1 1 · § 1 1· k ¨ ¸ w 220 1 k 1 ¨ ¸ w 202 1
© n n' ¹ © n' n ¹
k ( k 1) § 1 1 · ¨ ¸ w 022 1
2 © n' n ¹
(4.6)
Minimization of (4.6) with respect to k yields its optimum value as
k
^w 004 2(w 220 1) w 022 6` w 040 w 004 2w 022 4
k 0 (say)
(4.7)
Substitution of k0 from (4.7) to (4.6) gives minimum value of MSE of td.
5.
Empirical Study
To illustrate the performance of various estimators of S 2y , we consider the data given in Murthy(1967, p.-226). The variates are: y: output, x: number of workers, z: fixed capital, N=80, nâ&#x20AC;&#x2122;=25, n=10.
w 400
2.2667 , w 040
3.65 , w 004
2.8664 , w 220
2.3377 , w 202
2.2208 , w 400
3.14
The percent relative efficiency (PRE) of various estimators of S 2y with respect to conventional estimator s 2y has been computed and displayed in table 5.1.
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Table 5.1 : PRE of s 2y , t1, t2 and min. MSE (t) with respect to s 2y Estimator
PRE(., s 2y )
s 2y
100
t1
214.35
t2
42.90
t
215.47
In table 5.2 PRE of various estimators of s 2y in two-phase sampling with respect to S 2y are displayed. Table 5.2 : PRE of s 2y , t1d, t2d and min.MSE (td) with respect to s 2y
6.
Estimator
PRE (., s 2y )
s 2y
100
t1d
1470.76
t2d
513.86
td
1472.77
Conclusion
From table 5.1 and 5.2, we infer that the proposed estimators t performs better than conventional estimator s 2y and other mentioned estimators.
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References
Isaki, C. T. (1983): Variance estimation using auxiliary information. Journal of American Statistical Association. Bahl, S. and Tuteja, R.K. (1991): Ratio and Product type exponential estimator, Information and Optimization sciences, Vol. XII, I, 159-163. Kadilar,C. and Cingi,H. (2006) : Improvement in estimating the population mean in simple random sampling. Applied Mathematics Letters 19 (2006), 75â&#x20AC;&#x201C;79. Singh, R. Chauhan, P. Sawan, N. and Smarandache, F. (2007): Auxiliary information and a priori values in construction of improved estimators. Renaissance high press, USA. Murthy, M.N.(1967): Sampling Theory and Methods. Statistical Publishing Society, Calcutta, India.
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