Improved Ratio Type Estimator Using Two Auxiliary Variables under Second Order Approximation Prayas Sharma and Rajesh Singh Department of Statistics, Banaras Hindu University Varanasi-221005, India
[email protected],
[email protected]
Abstract In this paper, we have proposed a new Ratio Type Estimator using auxiliary information on two auxiliary variables based on Simple random sampling without replacement (SRSWOR). The proposed estimator is found to be more efficient than the estimators constructed by Olkin (1958), Singh (1965), Lu (2010) and Singh and Kumar (2012) in terms of second order mean square error. Keywords: simple random sampling, population mean, study variable, auxiliary variable, ratio type estimator, product estimator, Bias and MSE. 1. Introduction In sampling survey, the use of auxiliary information is always useful in considerable reduction of the MSE of a ratio type estimator. Therefore, many authors suggested estimators using some known population parameters of an auxiliary variable. Hartley-Ross (1954), Quenouille’s (1956) and Olkin (1958) have considered the problem of estimating the mean of a survey variable when auxiliary variables are made available. Jhajj and Srivastava (1983), Singh et al.(1995),
Upadhyaya and Singh (1999), Singh and Tailor (2003), Kadilar and Cingi (2006), Khoshnevisan et al. (2007), Singh et al. (2007), Singh and Kumar (2011,), etc. suggested estimators in simple random sampling using auxiliary variable. Moreover, when two auxiliary variables are present Singh (1965,1967) and Perri(2007) suggested some ratio -cum -product type estimators.
Most of these authors discussed the
properties of estimators along with their first order bias and MSE. Hossain et al. (2006), Sharma et al. (2013a, b) studied the properties of some estimators under second order approximation. In this paper, we have suggested an estimator using auxiliary information in simple random sampling and compared with some existing estimators under second order of approximation when information on two auxiliary variables are available. Let U= (U1 ,U2 , U3, …..,Ui, ….UN ) denotes a finite population of distinct and identifiable units. For estimating the population mean Y of a study variable Y, let us consider X and Z are the two auxiliary variable that are correlated with study variable Y, taking the corresponding values of the units. Let a sample of size n be drawn from this population using simple random sampling without replacement (SRSWOR) and yi , xi and zi (i=1,2,…..n ) are the values of the study variable and auxiliary variables respectively for the i-th units of the sample. When the information on two auxiliary variables are available Singh (1965,1967) proposed some ratio-cum-product estimators in simple random sampling without replacement to estimate the population mean Y of the study variable y, generalized version of one of these estimators is given by,
X t 1 y x
1
Z z
2
(1.1)
where 1 and 2 are suitably chosen scalars such that the mean square error of t1 is minimum.
and y
1 n yi n i 1
, x
1 n 1 n x z and i zi , n i 1 n i 1
Olkin(1958) proposed an estimator t2 as-
X Z t 2 y 1 2 z x
(1.2)
where, 1 and 2 are the weights that satisfy the condition 1 + 2 =1 Lu (2010) proposed multivariate ratio estimator using information on two auxiliary variables as w X w 2 X2 t 3 y 1 1 w1x1 w 2 x 2
(1.3)
where, w 1 and w 2 are weights that satisfy the condition: w 1 + w 2 =1. Singh and Kumar (2012) proposed exponential ratio-cum-product estimator. Generalized form of this estimator is given by 1
X x Z z t 4 y exp exp X x Z z
2
where 1 and 2 are constants such that the MSE of estimator t4 is minimise. Theorem1.1 Let e 0
xX zZ yY , e1 and e 2 Y X Z
then E(e 0 ) E(e1 ) E(e 2 ) =0 and variances and co-variances are as follows: (i) V(e 0 ) E{(e 0 ) 2 }
L1C 200 V200 Y2
(ii) V(e1 ) E{(e1 ) 2 }
L1C 020 V020 X2
(iii) V(e1 ) E{(e 2 ) 2 }
L1C 002 V002 Z2
(iv) COV(e 0 , e1 ) E{(e 0 e1 )}
L1C110 V110 XY
(1.4)
(v) COV(e1 , e 2 ) E{(e1e 2 )}
L1C 011 V011 XZ
(vi) COV(e 0 , e 2 ) E{(e 0 e 2 )}
L1C101 V101 YZ
(vii)
(viii)
E{(e 0 e1 )} 2
E{(e 0 e 2 )} 2
L 2 C 210 V210 Y2X
L 2 C 201 V201 Y2Z
(ix) E{(e1 e 2 )}
L 2 C 021 V021 X2Z
(x) E{(e 0 e12 )}
L 2 C120 V120 YX 2
(xi) E{(e1e 22 )}
L 2 C 012 V012 XZ 2
2
(xii)
(xiii)
E{(e 0 e 22 )}
L 2 C102 V102 YZ 2
E{(e1 )} 3
(xiv)
E (e1 e 2 )
(xv)
E(e1e 32 )
(xvi)
E (e 0 e1 )
3
3
L 2 C 030 V030 X3
L 3 C 031 3L 4 C 020 C 011 V031 X3Z
L 3 C 013 3L 4 C 002 C 011 V013 XZ 3 L 3 C130 3L 4 C 020 C110 V130 YX 3
where, L1
L3
(N n) 1 , ( N 1) n
L2
( N n ))( N 2n ) 1 ( N 1)( N 2) n 2
( N n ))( N 2 N 6nN 6n 2 ) 1 N( N n ))( N n 1)( n 1) 1 , L4 3 ( N 1)( N 2)( N 3) ( N 1)( N 2)( N 3) n 3 n
Cpqr = (Xi - X) p (Yi - Y) q Z i Z N
And
r
i 1
This theorem will be used to obtain MSE expressions of estimators considered here. Proof of this theorem is straight forward by using SRSWOR ( see Sukhatme and Sukhatme (1970)). 2. First Order Biases and Mean Squared Errors The expression of the biases of the estimators t1, t2 t3 and t4 to the first order of approximation are respectively, written as
Bias ( t 1 ) Y 1 V110 2 V102 R 1 V020 S1 V002 1 2 V011
where , R 1
(2.1)
1 (1 1) ( 1) S1 2 2 . 2 2
Bias ( t 2 ) Y w 1 V110 w 2 V101 w 1 V020 w 2 V002 1 2 V011
(2.2)
Bias ( t 3 ) Y w 1 X1 V110 w 2 V101
2
where,
1 w 2 X 2 V 2
1
1
020
w 22 X 22 V002 2 w 1 w 1 X1 X 2 V011
(2.3)
1 w 1 X1 w 2 X 2
2 2 Bias ( t 4 ) Y 1 V110 2 V101 V020 1 1 2 2 V002 2 8 4 8 2 4
(2.4)
Expressions for the MSE of the estimators t1,t2, t3 and t4 to the first order of approximation are respectively, given by
MSE(t 1 ) Y 2 V200 1 V020 22 V002 21V110 2 2 V101 21 2 V011 2
(2.5)
The MSE of the estimator t1 is minimized for
y x y z xz V200 1* 2 1 xz V020
(2.6)
And
y z y x xz V200 1* 2 1 xz V002
(2.7)
where 1* and *2 are, respectively, partial regression coefficients of y on x and of y on z in simple random sampling.
MSE(t 2 ) Y 2 V200 1 V020 22 V002 21V110 2 2 V101 21 2 V011 2
(2.8)
The MSE of the estimator t2 is minimum for *1
V002 V101 V110 V012 V020 V002 2V012
and *2 1 *1
(2.9)
MSE ( t 3 ) Y 2 V200 2 2 ( w 12 X12 V020 w 22 X 22 V002 2 w 1 w 2 X1 X 2 V011 )
2 ( w 1 X1 V110 w 2 X 2 V101
(2.10)
Differentiating (2.10) with respect to w1 and w2 partially, we get the optimum values of w1 and w2 respectively as
X1V110 X 2 V101 X 22 V002 X1 X 2 V011 w X12 V020 X 22 V002 2X1 X 2 V011 * 1
and w *2 1 w 1*
(2.11)
For optimum value of w 1 w 1* and w 2 w *2 , MSE of the estimator t3 is minimum. 2 2 MSE( t 4 ) Y 2 V200 1 V020 2 V002 1 V110 2 V101 1 2 V011 4 4 2
(2.12)
On differentiating (2.12) with respect to 1 and 2 respectively, we get the optimum values of 1 and 2 as 1*
2V110 V002 V101 V011 2 V002 V020 V011
(2.13)
2V020 V101 V110 V011 2 V002 V020 V011
(2.14)
and *2
Estimators t1, t2 t3 and t4 at their respective optimum values attains MSE values which are equal to the MSE of regression estimator for two auxiliary variables.
3.
Proposed Estimator
When auxiliary information on two auxiliary variables are known, we propose an estimator t5 as 2 cX dx 1 z t 5 y k 1 k 2 2 c d X Z
(3.1)
where d and c are either real numbers or a function of the known parameters associated with auxiliary information . k1 and k2 are constants to be determined such that the MSE of estimator t5 is minimum under the condition that k1+k2 = 1 and 1 and 2 are integers and can take values -1, 0 and +1 . Expressing the estimator t 5 in terms of e’s we have
t 5 Y k1 (1 1e1 ) 1 k 2 2 (1 e 2 ) 2 where, 1
(3.2)
d . cd
We assume that 1e1 1so that (1 1e1 ) 1 are expandable. Expanding the right hand side of (3.2), and neglecting terms of e’s having power greater than two we have
t
5
Y Y k 111e 0 e1 k 2 2 e 0 e 2 k 1 M 1e12 k 2 N 1e 22 1 2 e1e 2
(3.3)
Taking expectation on both sides we get bias of estimator t5, to the first degree of approximation as Bias ( t 5 ) Y k 111 V110 k 2 2 V101 k 1 M 1 V020 k 2 N 1 V002 1 2 V011
(3.4)
where, M1
1 (1 1) 2 1 , 2
N1
2 ( 2 1) . 2
Squaring both sides of (3.3) and neglecting terms of e’s having power greater than two we have
( t 5 Y ) 2 Y e 02 k 12 12 12 e12 k 22 22 e 22 2k 111e 0 e1 2k 2 2 e 0 e 2 2k 1 k 2 1 2 1e1e 2
(3.5)
Taking expectation on both sides of (3.5) and using theorem 1.1, we get the MSE of t5 up to first degree of approximation as
MSE(t 5 ) Y 2 V200 k12 12 1 V020 k 22 22 V002 2k111V110 2k 2 2 V101 2k1k 2 1 2 1V011 (3.6) 2
Differentiating (3.6) with respect to k1 and k2 partially, equating them to zero and after simplification, we get the optimum values of k1 and k2 respectively, as
k 1*
11V110 22 V002 2 V102 1 2 1V011 12 12 V110 22 V002 21 2 1V011
,
k *2 1 k 1* .
(3.7)
Putting these values in (3.6) we get minimum MSE of estimator t5. The minimum MSE of the estimator t1, t2, t3, t4 and proposed estimator t5 is equal to the MSE of combined regression estimator based on two auxiliary variables, which motivated us to study the properties of estimators up to the second order of approximation. 4.
Second Order Biases and Mean Squared Errors
Expressing estimator ti’s (i=1,2,3,4,5) in terms of e’s (i=0,1), we get
t 1 Y1 e 0 1 e1
1
1 e1
2
(4.1)
Or
(t 1 Y) Y e 0 1e1 1e 0 e1 2 e 2 2 e 0 e 2 1 2 e1e 2 1 2 e 0 e1e 2 R1e1 R1e 0 e1 2
2
S1e 22 S1e 0 e 22 R 2 e13 R 2 e 0 e13 S 2 e 32 S 2 e 0 e 32 R 2 e13 2 R 1e 2 e12 1S1e1e 22
2 R 2 e13 e 2 1 R 1e1e 32
(4.2)
Squaring both sides and neglecting terms of e’s having power greater than four, we have
t
Y Y 2 e 02 12 e12 22 e 22 21e 0 e1 2 2 e 0 e 2 21 2 e1e 2 21e 02 e1 2 2 e 02 e 2 2
1
(2R 1 2 12 )e 0 e12 2S1e 0 e 22 2 12 2 e12 e 2 2S1 1e1e 22 2R 1 1e13 61 2 e 0 e1e 2 (12 2R 1 )e 02 e12 ( 22 2S1 )e 02 e 22 (12 22 2R 1S1 )e12 e 22 (412 2 6R 11 )e 0 e12 e 2 41 (S1 22 )e 0 e1e 22 41 2 e 02 e1e 2 2(R 2 21 R 1 )e 0 e13 2(S 2 2 2 S1 )e 0 e 32
21 (S 2 2 S1 )e1e 22 2 2 (R 2 1 R 1 )e12 e 2 (R 12 21 R 2 )e14 (S12 2 2S 2 )e 42 (4.3)
Taking expectations, and using theorem 1.1 we get the MSE of the estimator t 1 up to the second order of approximation as MSE 2 ( t 1 ) Y 2 V200 12 V020 22 V002 21 V110 2 2 V101 21 2 V011 21 V210 2 2 V201
(2R 1 212 )V120 2S1 V102 212 2 V021 2S11 V012 2R 11 V030 61 2 V111
( 12 2R 1 )V220 ( 22 2S1 )V202 ( 12 22 2R 1S1 )V022 (4 12 2 6M 1 1 )V121 4 1 (S1 22 )V112 4 1 2 V211 2(R 2 21 R 1 )V130 2(S 2 2 2 S1 )V103
21 (S 2 2 S1 )V012 2 2 (R 2 1 R 1 )V021 (R 12 21 R 2 )V040 (S12 2 2 S 2 )V004 (4.4)
where,
R1
1 (1 1) , 2
R2
1 (1 1)(1 2) , 6
R3
1 (1 1)(1 2)(1 3) , 24
S1
2 ( 2 1) , 2
S2
2 ( 2 1)( 2 2) , 6
S3
2 ( 2 1)( 2 2)( 2 3) . 24
Similarly, MSE expression of estimator t2 is given by
MSE 2 ( t 2 ) Y 2 V200 21 V020 V220 3V040 2V120 2V030 4V130
22 V002 V202 3V004 2V102 2V003 4V103
21 V110 V210 V120 V220 V130 2 2 V101 V201 V202 21 2 V011 2V111 V012 2V112 V013 V211 V021 2V121 V022 V031 (4.5)
MSE expression of estimator t3 is given by
MSE 2 ( t 3 ) Y 2 V200 w 12 X1 2 2 (V020 V220 2V120 ) 2A 1 2 (V120 V220 )
w 22 X 2 2 2 (V002 V202 2V120 ) 2A 1 2 (V120 V202 )
2w 1 w 2 X1 X 2 2 2 (V211 2V111 ) 2A1 2 (V111 V211
2w 13 w 2 X13 X 2 A12 4 4A 2 4 V031 2w 1 w 32 X1 X 23 A 12 4 4A 2 4 V013 2 w 1 X1 V110 V210 2 w 2 X 2 V101 V201
3w 12 w 2 X12 X 2 2A 2 3 V121 4A 1 3 V121 2A 1 3 V012 3w 1 X1 w 22 X 22 2A 2 3 V112 4A 1 3 V112 2A 1 3 V012
w 14 X14 A 14 4 2A 2 4 V040 w 42 X 24 A 14 4 2A 2 4 V002
w 13 X13 2A 2 3 V130 4A 1 3 V130 2A 1 3 V030 w 32 X 23 2A 2 3 V103 4A 1 3 V103 2A 1 3 V003
6 w 12 X12 w 22 X 22 A 12 4 2A 2 4 V022
(4.6)
MSE expression of estimator t4 is given by
MSE 2 ( t 4 ) Y V200 2
12 22 12 V102 22 V020 V002 1V110 1V210 M N 4 4 4 2 4
3 1 3 1 V021 M 12 2 V012 N 221 1 2 V111 1 2 V011 1MV030 2 NV003 2 2 4 2
V V220 12 V202 22 V022 12 22 1 2 Q MN 130 O 21M M N 2 4 2 4 8 2 4
V103 P 2 2 N V031 1Q O SM V013 2 R 1P SN 4 8 8
V V121 V Q 212 2 2 M 112 2 221 21 N R 040 21O M 2 4 4 16
V004 V 2 2 P N 2 211 12 2 S 16 2
where,
2 M 1 1 2
2 , N 2 2 , 2
3 P 22 2 , 6
3 O 12 1 6
2 2 Q 1 2 1 2 , R 1 2 1 2 2 2
MSE of Proposed estimator t5 up to second order of approximation
(4.7)
Expanding right hand side of equation (3.2) and neglecting terms of e’s having power greater than four, we get
t 5 Y Y e 0 k 1 11e1 M 1e12 M 2 e13 M 3 e14 k 2 2 e 2 N 1e 22 N 2 e 32 N 3 e 42
k 1 11e 0 e1 M 1e 0 e12 M 2 e 0 e13 k 2 2 e 0 e 2 N 1e 0 e 22 N 2 e 0 e 32
(4.8)
Squaring both sides of (4.8) and neglecting terms of e’s having power greater than four, we have
t
Y Y 2 e 02 k12 12 12 (e12 e12 e 22 2e 0 e12 ) 211 (M1e14 2M1e13 ) M12 e14 2
5
+ k 22 22 (e 22 e 02 e 22 2e 0 e 22 ) 2 2 ( N 1e 0 e 32 N 1e12 N 2 e 42 ) N 12 e 42
2k 1 11 (e 0 e1 e 02 e1 ) M 1 (e 0 e12 e 02 e12 ) M 2 e 0 e13 2k 2 2 (e 0 e 2 e 02 e 2 ) N 1 (e 0 e 22 e 02 e 22 ) N 2 e 0 e 32
2k 1 k 2 1 2 1 (e1e 2 2e 0 e1e 2 e 02 e1e 2 ) 11 ( N 1e1e 22 N 2 e1e 32 )
(4.9)
Taking expectations on both sides of (4.9) and using theorem1.1, we get the MSE of estimator t 5 ,up to the second order of approximation as
MSE 2 ( t 5 ) Y 2 V200 k 12 12 12 (V020 V022 2V120 ) 211 (M 1 V040 2M 1 V030 ) M 12 V040
+ k 22 22 (V002 V202 2V102 ) 2 2 ( N 1 V103 N 1 V030 N 2 V004 ) N 12 V004 2k 1 11 (V110 V210 ) M1 (V120 V220 ) M 2 V130 2k 2 2 (V101 V201 ) N1 (V102 V202 ) N 2 V103 2k 1 k 2 1 2 1 (V011 2V111 V211 ) 11 ( N1 V012 N 2 V013 )
(4.10)
5.
Numerical Illustration
For a natural population data, we have calculated the mean square error’s of the estimator’s and compared MSE’s of the estimator’s under first and second order of approximations. Data Set The data for the empirical analysis are taken from Book, “An Introduction to Multivariate Statistical Analysis”, page no. 58, 2nd Edition By T.W. Anderson. The population consist of 25 persons with Y= Head length of second son, X= Head length of first son and Z= Head breadth of first son. The following values are obtained from Raw data given on page no. 58. Y 183 .84 , X 185 .72 , Z 151.12 , N 25, n 7 and
V020=0.000244833,V200=0.000306792,V020=0.000284171,V110=0.00020986,V011=0.000193753, V101= 0.000189972, V111= -0.000059732, V210 = -0.0000004582, V201= -0.0000002.77, V021= -0.0000002775, V102= -0.0000002354, V120= -0.00000036455, V012=0.00000025179, V202=0.000013,V003=0.000001544,V031=0.3893411 ,V013=0.380025, V030= -0.00001363,V103=0.00001215,V040=0.0000214,V220=0.000015 V022=0.001624,V121=0.0000115,V211=0.0000122,V004=0.00001384,V112=0.000000085 Table 5.1: MSE’s of the estimators ti (i=1,2,3,4,5)
Estimators
MSE First order
Second order
t1
4.508
16156.644
t2
4.508
27204.321
t3
4.508
17679.890
t4
4.508
20928.689
t5
4.508
275.926*
*for 1 = 2 =1 This table shows the comparison of estimators on the basis of MSE because y cannot be extended up to second order of approximation therefore, we are unable to calculate PRE for second order approximation.
6.
Conclusion In the Table 5.1 the MSE’s of the estimators t1, t2 , t3, t4 and t5 are written under first order
and second order of approximations. It has been observed that for all the estimators, the mean squared error increases for second order. Observing second order MSE’s we conclude that the estimator t5 is best estimator among the estimators considered here for the given data set. Acknowledgement: The authors are thankful to the learned referee’s for their valuable suggestions leading to the improvement of contents and presentation of the original manuscript. REFERENCES 1. Anderson, T. W. (1984). An introduction to multivariate statistical analysis (2nd ed.). New York: Wiley. 2. Hartley, H.O. and Ross,A. (1954): Unbiased ratio estimators. Nature, 174, 270-271. 3. Hossain, M.I., Rahman, M.I. and Tareq, M.(2006) : Second order biases and mean squared errors of some estimators using auxiliary variable. SSRN.
4. Jhajj H.S and Srivastva S.K., (1983): A class of PPS estimators of population mean using auxiliary information. Jour. Indian Soc. Agril. Statist. 35, 57-61. 5. Kadilar, C.; Cingi, H., (2006): Improvement in Estimating the Population Mean in Simple Random Sampling, Applied Mathematics Letters, 19, 1, 75-79. 6. Khoshnevisan, M., Singh, R., Chauhan, P., Sawan, N., and Smarandache, F. (2007). A general family of estimators for estimating population mean using known value of some population parameter(s), Far East Journal of Theoretical Statistics 22 181–191.
7. Lu,J. Yan,Z. Ding,C. Hong,Z., (2010): 2010 International Conference On Computer and Communication Technologies in Agriculture Engineering (CCTAE), 3, 136-139. 8. Olkin, I. (1958) Multivariate ratio estimation for finite populations, Biometrika, 45, 154– 165. 9. Perri, P. F. (2007). Improved ratio-cum-product type estimators. Statist. Trans. 8:51–69. 10. Quenouille, M. H. (1956): Notes on bias in estimation, Biometrika, 43, 353-360. 11. Sharma, P., Singh, R. ,Jong, Min-kim. (2013a): Study of Some Improved Ratio Type Estimators using information on auxiliary attributes under second order approximation. Journal of Scientific Research, vol.57, 138-146. 12. Sharma, P., Singh, R.,Verma, H. (2013b): Some Exponential Ratio-Product Type Estimators using information on Auxiliary Attributes under Second Order Approximation. International Journal of Statistics and Economics, Vol. 12, issue no. 3. 13. Singh, H.P. and Tailor, R. (2003). Use of known correlation coefficient in estimating the finite population mean. Statistics in Transition 6 555-560. 14. Singh, M. P. (1965). On the estimation of ratio and product of the population parameters. Sankhya B 27:231-328. 15. Singh, M. P. (1967). Ratio cum product method of estimation. Metrika 12:34–42. 16. Singh, R. and Kumar, M. (2011): A note on transformations on auxiliary variable in survey sampling. MASA, 6:1, 17-19. 17. Singh, R. and Kumar, M. (2012 Improved Estimators of Population Mean Using Two Auxiliary Variables in Stratified Random Sampling Pak.j.stat.oper.res. Vol.VIII No.1 pp65-72. 18. Singh, R., Cauhan, P., Sawan, N., and Smarandache, F. (2007). Auxiliary Information and A Priori Values in Construction of Improved Estimators. Renaissance High Press.
19. Singh S, Mangat N.S and Mahajan P.K., (1995): General Class of Estimators. Jour. Indian Soc. Agril. Statist.47(2), 129-133. 20. Srivastava, S.K. (1967) : An estimator using auxiliary information in sample surveys. Cal. Stat. Ass. Bull. 15:127-134.
21. Sukhatme, P.V. and Sukhatme, B.V. (1970): Sampling theory of surveys with applications. Iowa State University Press, Ames, U.S.A. 22. Upadhyaya, L. N. and Singh, H. P. (1999): Use of transformed auxiliary variable in estimating the finite population mean. Biom. Jour., 41, 627-636.
