Enhancing the Mean Ratio Estimators for Estimating Population Mean Using Non-Conventional Location Parameters

Conventional measures of location are commonly used to develop ratio estimators. However, in this article, we attempt to use some non-conventional location measures. We have incorporated tri-mean, Hodges-Lehmann, and mid-range of the auxiliary variable for this purpose. To enhance the efficiency of the proposed mean ratio estimators, population correlation coefficient, coefficient of variation and the linear combinations of auxiliary variable have also been exploited. The properties associated with the proposed estimators are evaluated through bias and mean square errors. We also provide an empirical study for illustration and verification.


Introduction
A mechanism, in statistical analysis, in which a predetermined number of observations are taken from a statistical population is called sampling.Sampling plays a vital role in all kinds of disciplines.The purpose is to reduce the cost and/or the amount of work that it would take to survey the entire target population.The variable of interest or the variable about which we want to draw some inference is called a study variable.
In survey research, there are situations in which when the information is available on every unit in the population.If a variable, that is known for every unit of the population, is not a variable of direct interest but instead employed to improve the sampling plan or to enhance estimation of the variables of interest, then it is called an auxiliary variable.The auxiliary information is commonly associated with the use of ratio, product and regression estimation methods and to improve the efficiency of the estimators in survey sampling.The ratio estimator is most effective for estimating population mean when there is a linear relationship between study variable and auxiliary variable and they have a positive correlation.
Consider a finite population U = {U 1 , U 2 , . . ., U N } of N distinct and identifiable units.Let Y be the study variable with value Y i measured on U i , i = 1, 2, . . ., N giving a vector Y = {Y 1 , Y 2 , . . ., Y N }.The objective is to estimate population mean Y = 1 N N i=1 Y i , on the basis of a random sample.When the population parameters of the auxiliary variable X such as population mean, coefficient of variation, co-efficient of kurtosis, co-efficient of skewness, median, quartiles, population correlation coefficient, deciles etc., are known then a number of estimators available in the literature which performs better than the usual simple random mean under certain conditions.Below we provide here the complete list of notations that are used in this paper: Mean squared error of the estimator Mid-range Tri-mean

Quartile Deviation
Subscript i For existing estimators j For proposed estimators Greek ρ

Coefficient of correlation
Coeff. of skewness of auxiliary variable

Coeff. of kurtosis of auxiliary variable
Proposed modified ratio estimator of Y Based on the above mentioned notations, the mean ratio estimator for estimating the population mean Y of the study variable Y is defined as where The bias, constant and the mean square error of the mean ratio estimator is given by: The mean ratio estimator given in ( 1) is used to improve the precision of the estimate of the population mean in comparison with the sample mean estimator whenever a positive correlation exists between the study variable and the auxiliary variable.Moreover, we mode improvements by introducing a large number of modified ratio estimators with the use of the known coefficient of variation, coefficient of kurtosis, co-efficient of skewness, deciles etc. Cochran (1940) suggested a classical ratio type estimator for the estimation of a finite population mean using one auxiliary variable under a simple random sampling scheme.Murthy (1967) proposed a product type estimator to estimate the population mean or total of study variable y by using auxiliary information when the coefficient of a correlation is negative.Rao (1991) introduced a difference type estimator that outperforms conventional ratio and linear regression estimators.Singh & Tailor (2003) proposed a family of estimators using known values of some parameters by using SRSWOR to estimate the population mean of the study variable.Singh, Tailor, Tailor & Kakran (2004) and Sisodia & Dwivedi (2012) used a coefficient of variation of the auxiliary variate.Upadhyaya & Singh (1999) derived ratio type estimators using a coefficient of variation and coefficient of kurtosis of the auxiliary variate.
The organization of the rest of the article is as follows: Section 2 provides a description of the existing estimators.The structure of the proposed mean ratio estimator and the efficiency comparison of the proposed estimator with the existing estimator are presented in Section 3. Section 4 consists of an empirical study of proposed estimators.Finally, Section 5 summarizes the findings of this study.Kadilar & Cingi (2004) suggested the following ratio estimators for the population mean Y in simple random sampling using some auxiliary information.

Existing Modified Ratio Estimators
The biases, constants and the mean squared errors of estimators of Kadilar & Cingi (2004) are given by: They showed that the above ratio estimators are more efficient than traditional ratio estimators to estimate the population mean.Kadilar & Cingi (2006) has developed some modified ratio estimators using the correlation coefficient, which are shown below: The biases, constants and the mean squared errors in Kadilar & Cingi (2006) are given by: These five estimators proved to be more efficient than all the existing ratio estimators, as is evident from their application to some natural populations.
Yan & Tian (2010) proposed the following two modified ratio estimators using coefficient of skewness and kurtosis; The biases, constants and the mean squared errors for Yan & Tian (2010) are given by: The above-defined estimators that use the skewness and kurtosis coefficient respectively provide better estimates of the population mean in comparison with traditional ratio estimators.Subramani & Kumarapandiyan (2012a, 2012b, 2012c) have introduced estimators with the use of population median, skewness, kurtosis and coefficient of variation for auxiliary information in simple random sampling to estimate population mean.
The biases, constants and the mean squared errors for Subramani & Kumarapandiyan (2012a, 2012b, 2012c) are given by; Jeelani, Maqbool & Mir (2013) suggested an estimator with the use of the coefficient of skweness and quartile deviation of auxiliary information in simple random sampling to estimate population mean.
The bias, constant and the mean square error for Jeelani et al. ( 2013) is as follows;

The Proposed Modified Ratio Estimators
In this section, we propose different modified ratio type estimators using the population tri-mean, mid-range, Hodges-Lehmann, coefficient of variation and population correlation coefficient.The midrange is defined as M R = , where X (1) and X (N ) are the lowest and highest order statistics in a population of size N .It is highly sensitive to outliers as its design structure is based on only extreme values of data (cf. Ferrell (1953) for more details).We also include the measure based on the median of the pairwise Walsh averages which is defined as: The main advantage of the HL is that it is robust against outliers.For more properties of HL see Hettmansperger & McKean (1988).The HL is also known as the Hodges-Lehmann estimator.The next measure included in this study is the trimean, which is the weighted average of the population median and two quartiles.It is defined as: T M = Q1+2Q2+Q3 4 , where Q p (p = 1, 2, 3) denote one of the three quartiles in a population.For detailed properties of trimean (T M ) see Wang, Li & Cui (2007) and Nazir, Riaz, Ronald & Abbas (2013).
The proposed estimators are given below: The biases, constants and MSEs of proposed estimators are given below.

Efficiency Comparisons
In this section, the condition under which the proposed modified ratio estimators will have minimum mean square error compared to usual ratio estimators and existing modified ratio estimators to estimate the finite population mean have been derived algebraically.

Comparison with Usual Mean Ratio Estimator
From the expression of proposed and usual ratio estimator MSEs, we have derived the conditions for which the proposed estimators are more efficient than the ratio estimator as: x − 2RρS x S y , where j = 1, 2, . . ., 9.

Comparison with Existing Modified Ratio Estimators
From the expression of proposed and existing MSEs, we have derived the conditions for which the proposed estimators are more efficient than the existing modified ratio estimators as: where j = 1, 2, . . ., 9 and i = 1, 2, . . ., 17.

Empirical Study
The performance of the proposed modified mean ratio estimators and the existing modified ratio estimators are evaluated by using 5 natural populations.The population 1 and 2 are taken from Singh & Chaudhary (1986) page 177, population 3 is taken from Cochran (1940) page 152 and population 4 and population 5 are taken from Murthy (1967) page 228.The characteristics of the five populations are given below in Table 1, whereas the constants and the biases are given in Tables 2-5.The MSEs of the existing and proposed modified ratio estimators are given in Tables 6 and 7.The percentage relative efficiencies (PREs) of the proposed estimators (p), with respect to the existing estimators (e), are computed as and are given in Tables 8-12.
Tables 2-5 reveals that the constants and bias for the proposed ratio estimators are smaller compared to the existing ratio estimators.It is evident in Table 6 and 7 that the MSEs of the proposed ratio estimators with regard to the existing ones are much smaller, which indicates that proposed modified mean ratio estimators are more efficient.It can be seen that the proposed ratio estimators perform well in comparison with the existing estimators in terms of PREs (cf.Tables 8-12).Existing Proposed

Conclusion
Sampling plays a vital role in all kinds of disciplines.The availability of auxiliary information enhances the efficiency of the estimators.Mean ratio estimators have been proposed using known values of population tri-mean, mid-range, Hodges-Lehmann estimator, coefficient of variation and population correlation coefficient by using the study variable and auxiliary variable information.It is observed that the mean squared errors of the suggested estimators based on the tri-mean, mid-range, Hodges-Lehmann, coefficient of variation and population correlation coefficient of the auxiliary variable are smaller than those for the existing modified ratio estimators for all the five known populations considered in the numerical study.Also, it is pertinent to note that the parameters such as the mean, coefficient of skewness and coefficient of kurtosis are affected by the extreme values in the population, whereas the tri-mean, mid-range, Hodges-Lehmann are robust to extreme values.Hence, the modified ratio estimators proposed in this study may be used for better and more stable results, and are preferred to the existing modified ratio estimators for practical applications.Moreover, empirical studies reveal that the bias and mean square error for the proposed estimators are lower than that of the existing methods in terms of various natural populations.For the given populations alone, the proposed estimators perform better than the exiting estimators.

Table 1 :
Characteristics of the Populations.

Table 2 :
The constants of existing ratio estimators.

Table 3 :
The biases of existing ratio estimators.

Table 4 :
The constants of proposed ratio estimators.

Table 5 :
The biases of proposed ratio estimators.

Table 6 :
Mean square errors of existing ratio estimators.

Table 7 :
Mean square errors of proposed modified ratio estimators.

Table 8 :
Percentage relative efficiency of existing estimators with respect to the proposed estimators of population 1.

Table 9 :
Percentage relative efficiency of existing estimators with respect to the proposed estimators of population 2.

Table 10 :
Percentage relative efficiency of existing estimators with respect to the proposed estimators of population 3.

Table 11 :
Percentage relative efficiency of existing estimators with respect to the proposed estimators of population 4.

Table 12 :
Percentage relative efficiency of existing estimators with respect to the proposed estimators of population 5.