Advances in Research

21(9): 76-79, 2020; Article no.AIR.59985 ISSN: 2348-0394, NLM ID: 101666096

Ameliorated Ratio Estimator of Population Using New Linear Combination

T. A. Raja1* and S. Maqbool2

1Division of Agricultural and Economics, FoA, SKUAST-K, Wadura, Sopore, India. 2Division of AGB, FVSC & AH, SKUAST-K, Shuhama, India.

Authors’ contributions

This work was carried out in collaboration between both authors. Author TAR designed the study, performed the statistical analysis, wrote the protocol and wrote the first draft of the manuscript. Author SM managed the analyses of the study and the literature searches. Both authors read and approved the final manuscript.

Article Information

DOI: 10.9734/AIR/2020/v21i930235 Editor(s): (1) Dr. Carlos Humberto Martins, State University of Maringá, Brazil. Reviewers: (1) Somayeh Ahmadi Gooraji, Tehran University of Medical Sciences (TUMS), Iran. (2) Ahmed Sami Nori, University of Mosul, Iraq. Complete Peer review History: http://www.sdiarticle4.com/review-history/59985

Received 05 June 2020 Original Research Article Accepted 10 August 2020 Published 22 August 2020

ABSTRACT

We propose a new modified ratio estimator of population mean of the main variable using the linear combination of known values of Co-efficient of and Tri-Mean of the auxiliary variable. Mean Square Error (MSE) and bias of the proposed estimator is calculated and compared with the existing estimator. The comparison is demonstrated numerically which shows that the proposed estimator performs better than the existing estimators.

Keywords: Estimator; simple random ; auxiliary variable; mean square error; bias; co-efficient of Kurtosis; tri-mean.

1. INTRODUCTION utilized to obtain a more efficient estimator of the population mean. Ratio method of estimation The information on study variable is studied using auxiliary information is an attempt in this with the information on auxiliary variable . This direction. The method is utilized under the information on auxiliary variable , may be condition only if the variables are correlated. ______

*Corresponding author: E-mail: [email protected];

Raja and Maqbool; AIR, 21(9): 76-79, 2020; Article no.AIR.59985

when the population parameters of the auxiliary = (2.1) variable X such as Population Mean, Coefficient ̅ of variation, Kurtosis, , Correlation coefficient, etc., are known, a number of = (2.2) ̅ estimators such as ratio, product and estimators and their modifications are ∑ ( ) where, = is Co-efficient of available in literature and are performing better ()() than the simple random sample mean under Skewness of auxiliary variable and = certain conditions. Among these estimators the () ∑ ( ) () − is Co-efficient of ratio estimator and its modifications have widely ()() ()() attracted many researchers for the estimation of kurtosis. To the first degree of approximation the the mean of the study variable (see for example bias and mean square error are given below: Kadilar and Cingi [1,2,3], Cingi and Kadilar [4], () Subramani [5]). Further improvements are = ( − ) (2.3) achieved by introducing a large number of modified ratio estimators with the use of known () = ( + − 2 ) (2.4) , kurtosis, skewness, median, coefficient of correlation, Subramani and Kumarpandiyan [6,7,8]. Subzar et al. [9] had where, = taken initiative by proposed modified ratio estimator for estimating the population mean of the study variable by using the population deciles and and correlation coefficient of the auxiliary () variable. = ( − ) (2.5)

These points have motivated the authors to () = ( + − 2 ) (2.6) propose modified ratio estimators using the above linear combinations. The classical ratio estimator for population mean is defined as: where, = = = , (1.1) ̅ 3. THE SUGGESTED MODIFIED ESTIMATOR where = = is the estimate of = = ̅ We propose the modified ratio estimator using and it is assumed that the population mean of the linear combination of Co-efficient of Kurtosis the auxiliary variate is known. Hence is the and Tri-Mean (TM) of the auxiliary variable. This sample mean of the variate of the interest and ̅ measure is the weighted average of the is the sample mean of the auxiliary variate. The population median and two quartiles. For more bias and the mean square error of to the first detailed properties of tri-mean see Wang et al. degree of approximation is given below: [11].

( ≅ − 2 (1.2) ) = (3.1) ̅

() ≅ + − 2 (1.3) () ∑() () where, = − , Co- ()() ()() where, , are the co-efficient of variation and efficient of kurtosis and (TM) which is the weighted average of the population median and is the co-efficient of correlation. two quartiles and is defined as

TM=(Q +2Q +Q )/4, 2. YAN AND TIAN [10] ESTIMATOR 1 2 3

The bias of upto the first degree of Yan and Tian [10] have proposed two modified ratio estimators using the linear combinations of approximation is given by; Co-efficient of skewness and Co-efficient of () kurtosis. = ( − ), (3.2)

77

Raja and Maqbool; AIR, 21(9): 76-79, 2020; Article no.AIR.59985

Table 1. Parameters and constants of population

T 80 20 51.8264 11.2646 0.9413 18.3569 0.3542 8.4542 0.750 2.866 1.05 9.318

Table 2. Constants, Bias and MSE’s of estimators

Estimator Constants Bias MSE 0.968 0.554 881.97 Yan and Tian [10] 0.804 0.316 539.35 Yan and Tian [10] Subramani [5] 0.798 0.352 527.82 Subzar et al. [9] 0.875 0.302 486.32 0.776 0.281 396.62 ( Proposed Estimator)

MSE of the above estimator can be found using 6. CONCLUSION Taylor series method as : A new estimator has been proposed using the () = + − 2 ; linear combination of Co -efficient of Kurtosis and (3.3) Tri-mean of the auxiliary variable. Numerically it has also been proved that the estimator produces smallest bias as well as MSE value as where, = compared to the already existing estimators. We conclude that the proposed estimator is proficient 4. COMPARISON OF and more efficient than the existing estimators and be used in practical applications in We compare the efficiencies for which the estimating finite population mean. estimator given in (3.1) is more efficient than COMPETING INTERESTS the existing estimator;

Authors have declared that no competing < if interests exist. ( ) < ; 1,2( (2010) (4.1) REFERENCES

5. NUMERICAL ILLUSTRATION 1. Kadilar C, Cingi H. Ratio estimators in simple random sampling. Applied Data from Murthy [12] page 228 in which fixed Mathematics and Computation. capital is denoted by X (auxiliary variable) and 2004;151:893-902. output of 80 factories are denoted by Y (study 2. Kadilar C, Cingi H. An improvement in variable) have been utilized for the purpose. The estimating the population mean by using population parameters are given in Table 1 and the correlation co-efficient. Hacettepe Constants, Bias and MSE’s of estimators are Journal of Mathematics and Statistics. given in Table 2: 2006a;35(1):103-109. 3. Kadilar C, Cingi H. Improvement in % Relative Efficiency= MSE estimating the population mean in simple (Existing)/MSE(Proposed) * 100= 222.37% random sampling. Applied Mathematics Letters. 2006b;19:75-79. It is evident from the Table 2 that the proposed 4. Cingi H, Kadilar C. Advances in sampling estimator has smallest bias as well as MSE value theory- ratio method of estimation. as compared to the already existing estimators. Bentham Science Publishers; 2009. We also calculate the value of the condition 5. Subramani J. Generalized modified ratio ( ) given in (4.1) < ; 1,2 i.e., 0.941 < estimator for estimation of finite population mean. Journal of Modern Applied 1.408, which also is satisfied. Statistical Methods. 2013;12(2):121-155.

78

Raja and Maqbool; AIR, 21(9): 76-79, 2020; Article no.AIR.59985

6. Subramani J, Kumarapandiyan G. 9. Subzar M, Raja AT, Maqbool S, Nazir Estimation of population mean using co- Nageena. New alternative to ratio efficient of variation and median of an estimator of population mean. Inter. J. auxiliary variable. International Journal of of Agri. and Stat. Sci. 2016;12(1):221- Probability and Statistics. 2012a;1(4):111– 225. 118. 10. Yan Z, Tian B. Ratio method to the mean 7. Subramani J, Kumarapandiyan G. estimation using co-efficient of skewness Estimation of population mean using of auxiliary variable. ICICA: Part II. known median and co-efficient of skewness. 2010;103–110. American Journal of Mathematics and 11. Wang T, Li Y, Cui H. On weighted Statistics. 2012b;2(5):101–107. randomly trimmed . Journal of 8. Subramani J, Kumarapandiyan G. Modified Systems Science and Complexity. ratio estimators using known median and 2007;20(1):47-65. co-efficient of kurtosis. American Journal of 12. Murthy MN. Sampling theory and methods. Mathematics and Statistics. 2012c;2(4): Calcutta Statistical Publishing House, 95–100. India; 1967. ______© 2020 Raja and Maqbool; This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Peer-review history: The peer review history for this paper can be accessed here: http://www.sdiarticle4.com/review-history/59985

79