Estimators Proposed by Geometric Mean, Harmonic Mean and Quadratic Mean

In the studies in literature up to date arithmetic population mean of auxiliary variable is used to obtain the proportional estimators. In this paper geometric mean, harmonic mean and quadratic mean is used as well as arithmetic population mean. Using geometric mean, harmonic mean and quadratic mean do not affect the variance of ratio estimator ( ). However new approaches are obtained for the estimation and variance of the dependent variable when these means are used as well as arithmetic population mean. In the application, the mean number of teaching staff of the departments in Ondokuz Mayıs University is estimated by auxiliary variable which is the number of students in the departments. In addition the variances of proportional estimation method are obtained and interpreted by using population arithmetic mean, geometric mean, harmonic and quadratic mean.


Introduction
One of the basic estimation method, using auxiliary variable to estimate the population mean, in simple random sampling is the proportional estimator method. In this study the proportional estimators, used to estimate population mean, and the variances of these estimators are given. In all studies regarding the proportional estimators, arithmetic mean is chosen as the population mean of the auxiliary variable. In this paper geometric mean, harmonic mean and quadratic mean are used as the population mean as well as arithmetic mean. Using these means as well as arithmetic will not change the variance of proportional estimator . However by using these means and under assumption > 0, ( = 1, 2, … , ), new approaches are obtained for the estimation of the dependent variable and its variance. Geometric mean, harmonic mean and quadratic mean are investigated because the population mean is fixed. The aim is to minimize the variance for the estimators. In the application part the data is obtained from the departments of Ondokuz Mayıs University. The number of teaching staff at the departments are estimated via the number of student which is considered as the auxiliary variable. Some other means of the population also calculated and variances are calculated for the proportional estimation method approaches. There are several new estimators in the literature proposed for population mean and population sum derived from known proportional estimation. " Cochran (1977) investigated the use of information obtained from the auxiliary variable [1]." "Dayveh et al. (2003) proposed two estimators for the estimation of population mean using two auxiliary variable and generalized these estimators for more than two auxiliary variables [2]". "Sign and Tailor (2003) proposed new estimators by using correlation coefficient information [3]". " Ray and Singh (1981) developed a new proportional estimator using regression coefficient [4]". On the other hand " Kadılar and Çıngı (2004) investigated a special case of this estimator [5]". " Kadılar and Çıngı (2005) developed a new estimator, for population mean in two variable regression estimator, using the two auxiliary variable multiplicative estimator proposed by Dayveh et al. [6]". "İsaki (1983) proposed a proportional estimator for the population variance [10]" and "by developing this estimator Prasad and Singh (1992) proposed new estimators for the population variance [11]". " Garcia and Cebrian (1996) proposed a new estimator for the population variance by using auxiliary variable and generalized this estimator for more than one auxiliary variable by a iteration method [12]". Proceeding from here the following estimators are proposed. " Beale (1962) and Tin (1965) used variation coefficients of dependent variable and independent variable as auxiliary variable to propose new proportional estimators for population mean [13,14]". Additionally "Zaman et al. (2014), investigated mean square errors of some ratio estimators and means via Taylor Series approach and made comparisons [15]". Proceeding from here the following estimators are proposed.

Proportional Estimators
Suppose that a sample with a size is chosen out of a population by random simple sampling method. It is assumed that for the values of y and x the change of the proportion ⁄ is very small and the correlation between the variables is denoted by a line passing from the origin point. Then the sample means of the variables are as follows: The population means of the variables are: If ( , ) > 0 then the proportional estimation of the population mean of variable is obtained as, and proportion of the variables is The estimator of is given as follows = Substituting equation (7) in equation (5), we have

Proportional Estimator and New Estimators Proposed by Other Means
New estimators are given by using geometric mean, harmonic mean and quadratic mean instead of arithmetic population mean in equation (8) under condition > 0( = 1, 2, … , ).

Estimator Proposed by Geometric Mean
Definition 1. The geometric mean of a population is given as, where . , 4 , ⋯ , are values of auxiliary variable of the population.
Replacing equation (10) instead of in equation (8) we have following estimator ! "67 = GM Expected value of this estimator is

Estimator Proposed by Harmonic Mean
Definition 2. The harmonic mean of a population is given as, > (21)

Estimator Proposed by Quadratic Mean
Definition 3. The quadratic mean of a population is given as, where . , 4 , 5 , are values of auxiliary variable of the population.

The Comparison of the Estimators
The following inequalities are given for the comparison of arithmetic mean against other means "Shahbazov, 2005, [8]" where . , 4 , 5 , are values of auxiliary variable of the population.

Comparison of the Means
Using inequalities (28), the following inequalities are obtained: (29)

The Comparison of the Variances
Similarly using inequalities (28), the inequalities for the variances of estimators are given below: In equation (30) the inequality &'( 9 ! " : F &'( 9 ! "EB : is not a desirable situation therefore the estimator ! "GH is ignored. In this manner, using the following inequality According to these results, if the population values of the auxiliary variable are known and positive then the estimators ! "JH or ! "KH can be proposed. An actual data application, presented in section 4, supports these results.

Application
The actual data needed for application is obtained from the actively teaching departments of Ondokuz Mayıs University. The number of teaching staff in the university is denoted by the variable y and the number of students in the university is denoted by the variable x. When the correlation between these two variables is positive the variable, considered by auxiliary variable, can be estimated. Each of the department in the university is population unit and the sample size estimation is calculated by equation ( In equation (33) when R 0.05 , T 0.30 the sample equals to 14. Arithmetic mean, geometric mean and harmonic mean of the number of students are given in Table 1 respectively. Data of the chosen sample is obtained as in Table 2 by using random numbers table. The estimated values and estimated variances of the proposed new estimators are calculated by Table 2 and given in Table 3. In Table 3, we notice that the variances of estimators, obtained by using geometric and harmonic means, are smaller.

Conclusions
In this paper new mean estimators are proposed by the population geometric mean, population harmonic mean and population quadratic mean of the auxiliary variable. The expected values and variances of these proposed new estimators theoretically obtained. The estimators proposed by the quadratic mean are ignored since the variances of these estimators are greater than or equal to the variances of the estimators obtained by population arithmetic mean. However the estimators proposed by geometric mean and harmonic mean are analyzed since the variances of these estimators are smaller than or equal to the variances of estimators obtained by population arithmetic mean. When the obtained results are discussed it is seen that the variance of the estimator, proposed by harmonic mean, is the smallest. The aim of this paper, assuming that the population values of the auxiliary variable are positive, is to propose new estimators and to find their variances by using inequalities concerning the means. Finally an application is done using actual data to show that the numerical results tend to theoretically obtained results.