modified linear regression estimator is assessed with that of simple random sampling without replacement (SRSWOR) sample mean, ratio estimator, some of the ...
Research Article Jambulingam Subramani Department of Statistics, Ramanujan School of Mathematical Sciences, Pondicherry University, R. V. Nagar, Kalapet, Puducherry-605014. E-mail Id: drjsubramani@yahoo. co.in
A Modified Approach in Linear Regression Estimator in Simple Random Sampling Abstract The present article deals with the estimation of finite population mean of the study variable by introducing a modified form of linear regression estimator. The proposed modified linear regression estimator is assessed with that of simple random sampling without replacement (SRSWOR) sample mean, ratio estimator, some of the modified linear regression estimators and linear regression estimator for certain natural population available in the literature. Further the optimum value of mean squared error of the proposed estimator is also obtained.
Keywords: Bias, Linear regression estimator, Mean squared error, Natural population, Simple random sampling.
1. Introduction It is well known that ratio and linear regression estimators are used to improve the efficiency of the estimator based on SRSWOR whenever there exists positive correlation between the study variable and the auxiliary variable such that takes values
and
which is measured on the population
. Further improvements on ratio and regression estimators are achieved by addition of known parameters of the auxiliary variable and are known as modified ratio and modified linear regression estimators. For a detailed discussion on the ratio estimator, linear regression estimators and their modifications, the readers are referred to see the following articles: Al-Jararha and AlHaj Ebrahem [1], Banerjie and Tiwari [2], Bhushan et al. [3], Misra et al. [8], Sen [16], Singh [10], Singh and Tailor [11, 12], Singh et al. [14], Sisodia and Dwivedi [15], and Subramani [17, 18]. The other important and related works are due to Kadilar and Cingi [5, 6], Koyuncu and Kadilar [7], Yan and Tian [24] and Subramani and Kumarapandiyan [19-21], Tailor et al. [22], Upadhyaya and Singh [23] and the references cited therein. Let denote a random sample of size selected from the population of size . In the case of SRSWOR, the sample mean is used to estimate the population mean . The variance of the estimator
is given below: (1)
How to cite this article: Subramani J. A Modified Approach in Linear Regression Estimator in Simple Random Sampling. J Adv Res Appl Math Stat 2016; 1(2): 1-7.
The ratio estimator for estimating the population mean defined as
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of the study variable
is
(2)
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Where
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is the estimate of
. The bias
(13)
and mean squared error of to the first degree of approximation are given below:
(14)
)
(3) (4)
The usual linear regression estimator and its variance are given as (5) (6) where
is the sample regression coefficient of
on
.
) and kurtosis (
)
where
1.2.
Kadilar and Cingi (2006) Estimators
Using the linear combinations of correlation coefficient ( ), coefficient of variation ( ) and kurtosis ( ) of the auxiliary variable, Kadilar and Cingi [6] have proposed some more modified linear regression-type ratio estimators. The modified linear regression-type ratio estimators and the mean squared errors are given below: (15)
1.1. Kadilar and Cingi (2004) Estimators When the coefficient of variation (
of the auxiliary variable are known, Kadilar and Cingi [5] have proposed several modified regression-type ratio estimators. The estimators and the mean squared errors are given below:
(16) where (17)
(7) (18) (8) where where (19) (9) (20) (10) where where (21) (11) (22) (12) where where
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1.3. Yan and Tian (2010) Estimators Yan and Tian [24] have proposed two modified linear regression-type ratio estimators using the coefficient of skewness ( ) and kurtosis ( ) of the auxiliary variable. The estimators and the mean squared errors are as follows: (23)
2. Proposed Modified Estimator
Linear
Regression
In this section, a modified form of linear regression estimator for the estimation of finite population mean of the study variable has been proposed. The proposed estimator together with the bias and mean squared error are given below: 28)
(24) (29) where (30) (25)
The optimum value of
is obtained by differentiating
Eq. (30) with respect to
and equating it to zero. That
(26) is, where Since the expressions differ only by the value of , the mean squared error of
to
can be written as (27) The In this article, an attempt is made to introduce a modified form of linear regression estimator and is discussed in the upcoming section.
can also be written as given below so as
to get the optimum value.
Consider
(31) The optimum value of mean squared error is obtained by substituting the value of
3
in Eq. (31)
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J. Adv. Res. Appl. Math. Stat. 2016; 1(2)
After a little algebra, the optimum value of
is obtained as given below:
3. Numerical Comparison for Certain Natural Populations In order to assess the performance of the proposed modified linear regression estimator with that of the SRSWOR sample mean, ratio estimator and modified ratio estimators
to
, six natural populations have
been considered. The populations 1 and 2 are taken from Murthy (1967, page 228) [9], population 3 is taken from Murthy (1967, page 422) [9], population 4 is taken from Murthy (1967, page 178) [9], population 5 is taken from Cochran (1977, page 325) [4] and population 6 is taken from Koyuncu and Kadilar [7]. The population parameters obtained for the above data are given below:
Population-1: Murthy (1967, page 228) [9] X-Fixed Capital and Y-Output for 80 factories in a region N = 80 ρ = 0.9413 = 0.7507
n = 20
= 51.8264
= 11.2646
= 18.3569
= 0.3542
= 8.4563
= −0.06339
= 1.05
Population-2: Murthy (1967, page 228) [9] X-Data on number of workers and Y-Output for 80 factories in a region N = 80 ρ = 0.9150 = 0.9484
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n = 20
= 51.8264
= 18.3569
= 0.3542
= 1.3005
= 0.6978
= 2.8513 = 2.7042
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Population-3: Murthy (1967, page 422) [9] Number of cattle for 24 sample villages X-Census and Y-Survey N = 24
n = 12
ρ = 0.9589 = 0.9293
= 568.5833
= 506.1174
= 0.8901
= 2.2559
= 1.6651
= 568.25 =528.0501
Population-4: Murthy (1967, page 178) [9] X-Geographical area and Y-Area under winter paddy N = 108
n = 16
ρ = 0.7896 = 0.6903
= 172.3704
= 134.3567
= 0.7795
=1.6307
= 1.3612
= 461.3981 = 318.5022
Population-5: Cochran (1977, page 325) [4] X-Number of rooms and Y-Number of persons N = 10
n=4
ρ = 0.6515 = 0.1281
= 101.1
= 14.6523
= 0.1449
=−0.3814
= 0.5764
= 58.8 = 7.5339
Population-6: Koyuncu and Kadilar (2009) [7] X-Number of students and Y-Number of teachers N = 923
n = 180
ρ = 0.9543 = 1.8645
= 436.4345
= 749.9394
= 1.7183
=18.7208
= 3.9365
= 11440.498 = 21331.131
The mean squared error of the proposed modified linear discussed above are computed and are given in Table 1. regression estimator and the existing estimators Table 1.Mean Squared Error of the Existing Estimators and Proposed Modified Linear Regression Estimator Estimators Mean Squared Errors Populations 1 2 3 4 5 6 12.6366 12.6366 10673.1176 961.0871 32.2035 2515.1677 SRSWOR sample mean
5
Ratio estimator
18.9791
41.3163
936.5000
370.7382
20.2765
267.6515
Kadilar and Cingi [5]
58.2026
92.6563
12491.1400
1115.6570
43.7044
3185.9900
Kadilar and Cingi [5]
51.3313
53.0736
12453.1900
1113.4060
43.5951
3185.0250
Kadilar and Cingi [5]
58.8469
44.7874
12399.3300
1110.3570
44.0341
3176.3220
Kadilar and Cingi [5]
59.0633
43.3674
12392.3900
1107.9970
46.4609
3180.7980
Kadilar and Cingi [6]
49.7853
53.9825
12451.9800
1113.0830
43.1558
3185.4960
Kadilar and Cingi [6]
47.4010
52.6365
12449.0100
1111.9330
39.8564
3185.7250
Kadilar and Cingi [6]
50.9448
50.7876
12451.5600
1112.8080
43.5369
3184.9780
Kadilar and Cingi [6]
58.8875
42.4051
12395.4200
1108.9540
44.2132
3175.8600
Yan and Tian [24]
48.9356
60.5325
12423.2700
1111.2290
43.2181
3183.9530
Yan and Tian [24]
58.8160
35.1887
12435.8700
1111.7580
44.2806
3183.5290
(Linear regression estimator)
1.4399
2.0569
859.2285
361.9124
18.5273
224.6250
(Proposed estimator)
1.4386
2.0542
855.3437
354.3171
18.4760
224.1717
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From Table 1, it is observed that the optimum mean squared error of the proposed modified linear regression estimator is less than the variance of SRSWOR sample mean, ratio estimator, linear
regression estimator and some of the modified linear regression estimators discussed in this article. From this, the following inequality holds good:
The percentage relative efficienciesof the proposed estimator with respect to the existing estimators are obtained by the formula given: Estimator
Table 2.PRE of the Proposed Modified Linear Regression Estimator PRE (e, p) Populations 1 2 3 4 5 878.40 615.16 1247.82 271.25 174.30
6 1121.98
1319.28
2011.31
109.49
104.63
109.75
119.40
4045.78
4510.58
1460.36
314.88
236.55
1421.23
3568.14
2583.66
1455.93
314.24
235.96
1420.80
4090.57
2180.28
1449.63
313.38
238.33
1416.91
4105.61
2111.16
1448.82
312.71
251.47
1418.91
3460.68
2627.91
1455.79
314.15
233.58
1421.01
3294.94
2562.38
1455.44
313.82
215.72
1421.11
3541.28
2472.38
1455.74
314.07
235.64
1420.78
4093.39
2064.31
1449.17
312.98
239.30
1416.71
3401.61
2946.77
1452.43
313.63
233.91
1420.32
4088.42
1713.01
1453.90
313.77
239.67
1420.13
100.09
100.13
100.45
102.14
100.28
100.20
From Table 2, it is further observed the following. The PRE is ranging from:
174.30 to 1247.82 in case of SRSWOR sample mean 104.63 to 2011.31 in case of Ratio estimator 215.72 to 4093.39 in case of modified linear regression estimators and 100.09 to 102.14 in case of linear regression estimator
From this, it is clear that the proposed estimator performs better than the SRSWOR sample mean, ratio estimator, modified ratio estimator and linear regression estimator since the PREs of proposed estimator with respect to the existing estimators are more than 100.
4. Conclusion The present article deals with the estimation of finite population mean of the study variable by introducing
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the new modified linear regression estimator. The mean squared error of the proposed estimator is derived and is compared with that of SRSWOR sample mean, ratioestimator, modified ratio estimator and linear regression estimators by numerically. The computed PREs reveal that the proposed estimator outperforms all the existing estimators including the linear regression estimator.
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Journal of Reliability and Statistical Studies 2008; 1(1): 67-73. Cochran WG. Sampling Techniques. 3rd Edn. Wiley Eastern Limited, 1977. Kadilar C, Cingi H. Ratio estimators in simple random sampling. Applied Mathematics and Computation 2004; 151: 893-902. Kadilar C, Cingi H. An improvement in estimating the population mean by using the correlation coefficient. Hacettepe Journal of Mathematics and Statistics 2006; 35(1): 103-109. Koyuncu N, Kadilar C. Efficient Estimators for the Population mean. Hacettepe Journal of Mathematics and Statistics 2009; 38(2): 217-25. Misra GC, Shukla AK, Yadav SK. A comparison of regression methods for improved estimation in sampling. Journal of Reliability and Statistical Studies 2009; 2(2): 85-90. Murthy MN. Sampling theory and methods. Calcutta, India: Statistical Publishing Society, 1967. Singh GN. On the improvement of product method of estimation in sample surveys. Journal of the Indian Society of Agricultural Statistics 2003; 56(3): 265-67. Singh HP, Tailor R. Use of known correlation coefficient in estimating the finite population means. Statistics in Transition 2003; 6(4): 555-60. Singh HP, Tailor R. Estimation of finite population mean with known coefficient of variation of an auxiliary character. STATISTICA 2005; anno LXV, n.3: 301-13. Singh HP, Tailor R, Kakran MS. An improved estimator of population mean using power transformation. Journal of the Indian Society of Agricultural Statistics 2004; 58(2): 223-30. Singh R, Malik S, Chaudhary MK et al. A general family of ratio-type estimators in systematic sampling. Journal of Reliability and Statistical
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