Journal of Statistical Theory and Applications

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Volume 19, Issue 4, December 2020, Pages 534 - 539

An Unbiased Estimator of Finite Population Mean Using Auxiliary Information

Authors
B. Mahanty1, *, ORCID, G. Mishra2
1P.G. Department of Statistics, Utkal University, Bhubaneswar-751004, India
2P.G. Department of Statistics, Utkal University, Bhubaneswar-751004, India
*Corresponding author. Email: bulu.mahanty@gmail.com
Corresponding Author
B. Mahanty
Received 20 July 2020, Accepted 25 December 2020, Available Online 4 January 2021.
DOI
10.2991/jsta.d.201228.001How to use a DOI?
Keywords
Simple random sampling; Auxiliary variable; Unbiased estimator; Optimality; Efficiency
Abstract

In this paper, an unbiased estimator is constructed by using a linear combination of an estimator of study variable and mean per unit estimator of an auxiliary variable under simple random sampling without replacement scheme. The efficiency of the estimator under optimality compared with the mean per unit estimator, an almost unbiased ratio estimator, an unbiased product estimator, and a regression estimator both theoretically and with the numerical illustration.

Copyright
© 2021 The Authors. Published by Atlantis Press B.V.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

In the survey sampling method, it is a common practice to utilize auxiliary information, which is frequently acknowledged to the higher precision of the estimators of population parameters. The classical ratio estimator, product estimator, and regression estimator are good examples in this context. When the correlation between the study variable (y) and auxiliary variable (x) is highly positively correlated, the ratio method of estimation is quite effective. Similarly when there is an existence of a high negative correlation between y and x then the product method of estimation is effectively used.

The property of unbiasedness is one of the important properties of an estimator. So it is desirable to construct unbiased estimators simultaneously keeping in mind its efficiency.

Tin [1] suggested an almost unbiased ratio estimator estimate the population means. Tin showed that this estimator is an almost unbiased and more efficient than the conventional ratio estimator suggested by Cochran [2]. Robson [3] suggested an estimator when there exists a negative correlation between study variable and auxiliary variable and is known as product estimator. Further, Robson constructed an unbiased estimator to estimate the population means by subtracting estimated bias from the product estimator.

In this paper, the unbiased estimator is suggested and its efficiency is compared with the mean per unit estimator, with an almost unbiased ratio estimator, suggested by Tin [1], unbiased product estimator suggested by Robson [3] and regression estimator by Watson [4].

2. UNBIASED ESTIMATORS

Let’s consider a finite population U=U1,U2,U3,...,UN size N. Let (y, x) be the study and auxiliary variables respectively. Now we consider yi,xi, i=1,2,3,…,n. denotes a sample of size “n” on the characteristics y and x have drawn from the population U with SRSWOR.

We denote Y¯ and X¯ are the population means of y and x respectively.

Let y¯=1n∑i=1nyi and x¯=1n∑i=1nxi are the sample mean of y and x respectively.

Further, we denote sy2=1n−1∑i=1nyi−y¯2, sx2=1n−1∑i=1nxi−x¯2 , and sxy=1n−1∑i=1nxi−x¯yi−y¯

Tin [1] suggested an almost unbiased ratio type estimator and it is given by

tTin=y¯X¯x¯1+θsyxx¯y¯−sx2x¯2(1)
where, θ=1n−1N
VtTin=θY¯2Cy2+Cx2−2Cyx Considering up to O1n.(2)

Robson [3] suggested an unbiased product type estimator and it is given by

tRob=y¯x¯X¯−θsyxX¯.(3)
VtRob=θY¯2Cy2+Cx2+2Cyx Considering up to O1n.(4)

Further, the regression estimator which is biased but in most cases, it is more precise is given by

tlr=y¯+bX¯−x¯(5)
Vtlr=θY¯2Cy21−ρ2(6)
where, b=syxsx2, the regression coefficient of y on x.

2.1. Proposed Unbiased Estimator

Considering the linear combination of Robson’s an unbiased estimator and the mean per unit estimator of auxiliary variable x, we construct an unbiased estimator, given by

tMU=λ0y¯x¯X¯−θsxyX¯+λ1x¯(7)
where, λ0 and λ1 are two suitable chosen constants such that estimator tMU is unbiased.

Hence, we put an unbiased condition

EtMU=Eλ0tRob+λ1x¯=Y¯
⇒λ0EtRob+λ1E(x¯)=Y¯
⇒λ0Y¯+λ1X¯=Y¯
⇒λ0−1Y¯+λ1X¯=0(8)

Now, we derive the variance of the proposed unbiased estimator by considering the condition. The value of λ0 and λ1, which makes the estimator unbiased.

3. VARIANCE OF THE PROPOSED ESTIMATOR tMU

The variance of the estimator tMU is given as

VtMU=λ02VtRob+λ12V(x¯)+2λ0λ1Covx¯,tRob=λ′S2λ(9)
where, λ′=λ0λ1, S2=S00S01S10S11 , and λ=λ0λ1

Further, S00=Vy¯Rob, S10=S01=Covx¯,tRob, and S11=V(x¯)

S00=VtRob=θY¯2Cy2+Cx2+2Cyx, Considering up to O1n

S10=S01=CovtRob,x¯=θY¯X¯Cyx+Cx2, Considering up to O1n

S11=V(x¯)=θX¯2Cx2, Considering up to O1n

The value of λ0 and λ1 which minimize the V(tMU) given in (9) and subject to the condition of unbiasedness of tMU given in (7).

The optimum value of λ is obtained as

λ=Y¯S2−1Q2Q2′S2−1Q2(10)
where, S2−1 is the inverse of the matrix of S2, i.e.,
S2−1=1S00S11−S102S11−S10−S01S00
Q2=Y¯X¯
where, Q2′ is the transpose of Q2 i.e.,
Q′2=Y¯X¯

Hence, after the expansion

λ=Y¯Y¯S11−X¯S01−Y¯S10+X¯S00Y¯2S11−2X¯Y¯S10+X¯2S00(11)

From the Equation (11) we get the optimum values of λ0 and λ1

λ0=Y¯Y¯S11−X¯S10Y¯2S11−2X¯Y¯S10+X¯2S00=R2V(x¯)−RCov(x¯,tRob)V(tRob)+R2V(x¯)−2RCov(x¯,tRob)(12)
λ1=Y¯X¯S00−Y¯S10Y¯2S11−2X¯Y¯S10+X¯2S00=R.VtRob−RCovx¯,tRobVtRob+R2V(x¯)−2RCovx¯,tRob(13)
where, R=Y¯X¯

The optimum variance is obtained by substituting the optimum value of λ0 and λ1

VtMUopt=Y¯2Q′S2−1Q=Y¯2S00S11−S102Y¯2S11−2X¯Y¯S10+X¯2S00=θY¯2Cx21−ρ2(14)

4. COMPARISON OF EFFICIENCY

  1. Comparison of tMU under optimality with mean per unit estimator y¯.

    The variance of mean per unit estimator is

    V(y¯)=θY¯2Cy2(15)

    Comparing the variance of suggested unbiased estimator tMU under optimum value from the Equation (14) with the variance of Mean per unit estimator from Equation (15) we have,

    V(y¯)−VtMUopt=θY¯2Cy2−θY¯2Cx21−ρ2=θY¯2Cy2−Cx21−ρ2(16)

    Hence, the proposed estimator tMU under optimality more efficient than the mean per unit estimator if

    Cy2>Cx21−ρ2.(17)

  2. Comparison of tMU under optimality with an almost unbiased estimator due to Tin tTin.

    From the above Equations (2) and (14)

    MSEtTin−VtMU=θY¯2Cy2+Cx2−2ρCyCx−θY¯2Cx21−ρ2=θY¯2Cy2+ρ2Cx2−2ρCyCx=θY¯2Cy−ρCx2>0(18)

    Since, Cy−ρCx2 always positive the proposed estimator tMU is more efficient than the Tin estimator tTin.

  3. Comparison of tMU under optimality with an unbiased product estimator due to Robson tRob.

    From the above Equations (4) and (14)

    MSEtRob−VtMU=θY¯2Cy2+Cx2+2ρCyCx−θY¯2Cx21−ρ2=θY¯2Cy2+ρ2Cx2+2ρCyCx=θY¯2Cy+ρCx2>0(19)

    Since, Cy+ρCx2 always positive the proposed estimator tMU is more efficient than the unbiased product estimator tRob.

  4. Comparison of tMU under optimality with regression estimator tlr

    From the above Equations (6) and (14)

    Now,

    MSEtlr−VtMU=θY¯2Cy21−ρ2−θY¯2Cx21−ρ2=θY¯21−ρ2Cy2−Cx2>0(20)
    when, Cy2 is greater than, Cx2, tMU is more efficient than tlr

5. NUMERICALLY ILLUSTRATIONS

To study the performance of the estimators, y¯ (mean per unit estimator), tTin, tRob, tlr, tMU numerically, we consider seven natural populations with positive correlation and seven natural population with negative correlation described in the Tables 1 and 2. The comparison is based on the computation of the variance of different estimators.

Pop.n No. Sources Popn size (N) Y x Cy Cx ρ
1 Daniel and Cross [5], p. 474 25 Paired Serum Dry Blood Spot Specimens 0.5177 0.4358 0.95
2 Gujarati [6], p. 598 10 Income (thousands of dollars) No. of Families Owning a House 0.3096 0.1811 0.36
3 Gupta [7], p. 451 11 Fathers Height (cm) Sons Height (cm) 0.0014 0.0015 0.55
4 Armitage and Berry [8], p. 161 17 Birth Weight Increase in Weight 0.6076 0.2104 0.64
5 Sukhatme and Sukhatme [9], p. 166 20 No. of Banana Bunches No. of Banana Pits 0.0605 0.0431 0.75
6 Daniel and Cross [5], pp. 455–456 20 Age (years) Bilirubin Levels (mg/dl) 0.4518 0.0483 0.46
7 Cochran [10], p. 186 21 Number of Family Members Number of Cars 0.5231 0.1156 0.97
Table 1

Description of the population (Correlation coefficient is positive).

Pop.n No. Sources Popn Size (N) y x Cy Cx ρ
1 Singh and Mangat [11], p. 71 10 Number of Tube wells Samples of Villages 0.7412 0.3981 −0.19
2 Singh and Mangat [11], p. 227 36 Samples Student OGPA Number of hours per Week Devoted to TV Viewing 0.2852 0.0190 −0.58
3 Singh and Mangat [11], p. 195 37 Collected in respect of OGPA Said nonacademic Activities 0.1648 0.0443 −0.36
4 Armitage and Berry [8], p. 161 32 Increase in Weight after 70–100 days Birth Weight, 0.1069 0.0256 −0.68
5 Maddala [12], p. 316 16 Veal (Consumption per capita LB) Veal (Price per pound) 0.0519 0.0097 −0.68
6 Maddala [12], p. 316 16 Lamb (Consumption per capita LB) Lamb (Price per pound) 0.011 0.0105 −0.75
7 Gujarati [6], p. 598 10 Hypothetical data on Income 1000$ No. of Families at Income 0.344 0.1423 −0.43
Table 2

Description of the population (Correlation coefficient is negative).

Remarks: Table 3, which shows that the variance of the proposed estimator tMU is the least when, Cy2 is greater than Cx2, followed by the regression estimator.

Pop.n No. y¯ tTin tRob tlr tMU
1 137.4757 13.5067 492.9409 13.3607 11.2491
2 18.954 19.5847 40.4995 16.4883 9.6457
3 0.9545 0.8289 2.9216 0.6572 0.6340
4 3.2559 1.9426 6.8246 1.9350 0.6701
5 9842.1 4470.5 4438.3 6232.4 3128.54
6 4.4792 3.5936 6.3234 3.5075 0.3753
7 0.3111 0.0949 0.6648 0.0158 0.0035
Table 3

MSE of different estimators (sample size n = 4) (Correlation coefficient is positive).

Remarks: Table 4, which shows that the variance of the estimator tMU is the least when, Cy2 is greater than Cx2, followed by the regression estimator.

Pop.n No. y¯ tTin tRob tlr tMU
1 2800.20 5087.90 3521.06 2698.20 1449.48
2 2.9360 4.0209 2.2425 1.9258 0.1283
3 8.0612 13.2546 7.2073 7.0078 1.8862
4 115.5169 221.0861 65.4413 60.9492 14.6398
5 0.568 0.3396 0.3412 0.7866 0.0568
6 10.657 5.4721 5.4356 8.3346 4.8318
7 18.954 37.359 16.2325 15.3967 6.3700
Table 4

MSE of different estimators (sample size n = 4) (Correlation coefficient is negative).

6. CONCLUSION

In this paper, an almost unbiased estimator tMU perform better than the estimators mean per unit estimator (y¯), Tin estimator (tTin), Robson estimator (tRob), and regression estimator (tlr) both theoretically and numerically when the coefficient variation of y is greater than the coefficient variation of x.

CONFLICTS OF INTEREST

We the author(s) declare(s) that there is no conflict of interest.

AUTHORS' CONTRIBUTIONS

B. Mahanty developed the theoretical formalism, performed the analytic calculations and performed the numerical illustrations. Both B. Mahanty and G. Mishra. authors contributed to the final version of the manuscript. G.Mishra. supervised the manuscript.

Funding Statement

The author(s) received no financial support to carry out this research in the manuscript.

ACKNOWLEDGMENTS

The authors are indebted to two anonymous referees and the Editor for their productive comments and suggestions, which led to improve the presentation of this manuscript.

REFERENCES

5.W.W. Daniel and L.C. Cross, Biostatistics: Basic Concepts and Methodology for the Health Sciences, International Student Version, tenth, Wiley India Pvt. Ltd., New Delhi, India, 2014.
6.D.N. Gujarati, Basics Econometrics, fourth, McGraw-Hill, New Delhi, India, 2003.
7.S.P. Gupta, Elementary Statistical Methods, Sultan and Chand & Sons, New Delhi, India, 2008.
9.P.V. Sukhatme and B.V. Sukhatme, Sampling Theory of Surveys with Applications, first, Piyush Publications, Delhi, India, 1997.
10.W.G. Cochran, Sampling Techniques, third, John Wiley and Sons, Inc., New York, NY, USA, 1977.
12.G.S. Maddala, Introduction to Econometrics, second, Macmillan Publishing Company, New York, NY, USA, 1992.
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Journal
Journal of Statistical Theory and Applications
Volume-Issue
19 - 4
Pages
534 - 539
Publication Date
2021/01/04
ISSN (Online)
2214-1766
ISSN (Print)
1538-7887
DOI
10.2991/jsta.d.201228.001How to use a DOI?
Copyright
© 2021 The Authors. Published by Atlantis Press B.V.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

risenwbib
TY  - JOUR
AU  - B. Mahanty
AU  - G. Mishra
PY  - 2021
DA  - 2021/01/04
TI  - An Unbiased Estimator of Finite Population Mean Using Auxiliary Information
JO  - Journal of Statistical Theory and Applications
SP  - 534
EP  - 539
VL  - 19
IS  - 4
SN  - 2214-1766
UR  - https://doi.org/10.2991/jsta.d.201228.001
DO  - 10.2991/jsta.d.201228.001
ID  - Mahanty2021
ER  -