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Indian Journal of Pure & Applied Biosciences (IJPAB)
Year : 2020, Volume : 8, Issue : 6
First page : (523) Last page : (528)
Article doi: :

Modified Estimation Using Different Linear Combination

Shabnum Gul*, P.P. Singh and Nisar Ahmad Khan
J.S. University Shikohabad Firozabad (283135) U.P India
*Corresponding Author E-mail:
Received: 15.11.2020 | Revised: 17.12.2020 | Accepted: 26.12.2020 


The present study was taken under consideration in order to propose new modified estimators using different linear combinations for population mean using the auxiliary information of Mid-Range with coefficient of correlation, coefficient of variation and coefficient of skewness in order to achieve more precision in estimates than the already existing estimators. The properties associated with the proposed estimators are assessed by mean square error and bias and compared with the existing estimators. In the support of the theoretical proposed work we have given numerical illustration.

Keywords: Auxiliary information, Linear combinations estimators, Mean square error, Bias, Efficiency.

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Cite this article: Gul, S., Singh, P. P., & Khan, N. A. (2020). Modified Estimation Using Different Linear Combination, Ind. J. Pure App. Biosci. 8(6), 523-528. doi:


An important purpose of sampling theory is to make sampling more efficient. It attempts to develop methods of sample selection and of estimation that provides, at the lowest possible cost, estimates that are precise enough for our purpose. This principle of specified precision at minimum cost recurs repeatedly in the presentation of theory. Estimation theory is an important part of statistical studies, whereby, population parameters are obtained using sample statistics. In any survey work, the experimenter’s interest is to make use of methods that will improve precisions of estimates of the population parameters both at the design stage and estimation stage. These parameters can be totals, means or proportions of some desired characters. In sample surveys, auxiliary information is used at selection as well as estimation stages to improve the design as well as obtaining more efficient estimators. Increased precision can be obtained when study variable Y is highly correlated with auxiliary variable X. Usually, in a class of efficient estimators, the estimator with minimum variance or mean square error is regarded as the most efficient estimator. Linear combination estimators are good examples in this context. Cochran (1940) initiated the use of auxiliary information at estimation stage and proposed ratio estimator for population mean. It is well established fact that linear combination estimators provide better efficiency in comparison to simple mean estimator if the study variable and auxiliary variable are positively correlated.

If the correlation between the study variable and auxiliary variables negative, product estimator given by Robson (1957) is more efficient than simple mean estimator.
Further improvements are also achieved on the classical linear combination by introducing a large number of modified linear combinatiob with the use of known parameters like, coefficient of variation, coefficient of kurtosis, coefficient of skewness and population correlation coefficient. For more detailed discussion one may refer to  Kadilar and Cingi (2004, 2006), Koyuncu and Kadilar (2009), Murthy (1967), Prasad (1989), Rao (1991), Singh (2003), Singh and Tailor (2003, 2005), Singh et al (2004), Sisodia and Dwivedi (1981), Upadhyaya and Singh (1999) and Yan and Tian (2010).
Further, had taken initiative by proposed modified linear combinations estimator for estimating the population mean of the study variable by using the population median of the auxiliary variable.


The objective of the paper is to propose modified estimators for estimating the population mean by using their linear combinations with the correlation coefficient and the coefficient of skewness of the auxiliary variable.

Table 3: The mean squared errors and Bias of the existing and proposed estimators




T0; Kadilar and Cingi (2004)



T1;Yan and Tian (2010)



T2; Yan and Tian (2010)



T3; Kadilar and Cingi (2006)



T4; Kadilar and Cingi (2006)



T5; Kadilar and Cingi (2006)



T6; Kadilar and Cingi (2006)



Proposed 1



Proposed 2



Proposed 3





In this paper we have proposed modified estimators based on simple random sampling without replacement by using the auxiliary variable, under the situation when mid-range, coefficient of skewness and correlation coefficient is known. We found that the performances of our proposed estimators in terms of mean square error are more efficient than all other existing estimators in the literature. Hence we strongly recommend that our proposed estimators preferred over the existing estimators for use in practical application.


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