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In this paper, we have proposed an estimator of finite population mean using a new regression type estimator with two auxiliary variables for single-phase sampling and investigated its finite sample properties. An empirical study has been carried out to compare the performance of the proposed estimator with the existing estimators that utilize auxiliary variables for finite population mean. It has been found that the new regression type estimator with two auxiliary variables for to be more efficient than mean per unit, ratio and product estimator and exponential ratio and exponential product estimators and exponential ratio-product estimator.

The history of using auxiliary information in survey sampling is as old as history of the survey sampling. The work of Neyman [

Olkin [

Singh [

Bahl and Tuteja [

In this paper, we will extend the modified regression estimator proposed by Hanif, Hamad and Shahbaz [

Let us consider a finite population

Let

cient of variation of study variable and the auxiliary variables respectively. Where the variances and covariance are given by,

The correlation coefficients between study variable and auxiliary variables are given by;

Let

sume that

The sampling error can also be written as,

Then for simple random sampling without replacement for both single-phase, we write by using phase wise operation of expectations as:

The following notations will be used in deriving the mean square errors of proposed estimator.

It is obtained by taking a sample of size n from N using simple random sampling without replacement.

Its variance is given by,

Classical ratio estimator by Cochran [

The mean squared error of the estimator

Classical regression estimator by Watson [

Mean squared error of estimator

Classical product estimator by Murthy [

The mean squared error of the estimator

Singh and Espejo [

The mean squared error of the estimator

Bahl and Tuteja [

The mean squared error of

The exponential ratio-product estimator proposed by Singh and Espejo [

The mean squared error is given by,

In general these estimators have a bias of order

is usually unimportant in samples of moderate and large sizes.

In this paper, we have extended the modified regression estimator by Hanif, Hamad and Shahbaz [

If we estimate a study variable when information on all auxiliary variables is available from the population, it is utilized in the form of their means. A new regression type estimator using two auxiliary variables for single variables is proposed as:

Substituting (1.3) equation in (3.0) we get,

Ignoring the second and higher terms for each expansion of product and after simplification we can write

Expanding the exponential in (3.2) and ignoring the second and higher terms for each expansion we get,

Simplifying (3.3) we get,

Expanding (3.4) and ignoring the second and higher terms we get,

The mean squared error of

Squaring the right sides of (3.6) and taking expectation, we get,

Differentiating (4.7) with respect to

Using normal equations that are used to find the optimum values of

Taking expectation in (3.10) we get,

Taking expectation and using (1.4) in (3.11) we get

Substituting the optimum value (3.8) and (3.9) in (3.12), we get

Simplifying (3.13) we get

Or

Or

We can also rewrite (3.16) as,

Using (1.6) in (3.17) we get

where

The regression-cum-exponential ratio-product estimator using multiple auxiliary variables in single-phase sampling is biased. However, this bias is negligible for moderate large samples. It is easily shown that the new regression type estimator with two auxiliary variables for single-phase is consistent estimator using two auxiliary variables since it is a linear combination of consistent estimators it follows that it’s also consistent.

We carried out some data simulation experiments to compare the performance of the new regression type estimator with mean per unit, ratio and product estimator using one auxiliary variable, ratio-product estimator, exponential ratio estimator, exponential product estimator and exponential ratio-product estimators in single-phase sampling for finite population.

1) Simulated population

1) Study variable

ii) For ratio estimator the auxiliary variable is strongly positively correlated with the study variable and the line passes through the origin.

iii) For regression estimator the auxiliary variable was strongly positively correlated with the study variable and the regression line does not pass through the origin.

Auxiliary variable

iv) For product estimator the auxiliary variable was strongly negatively correlated with the study variable.

Auxiliary variable

2) Natural population by Johnson (1996)

Lists estimates of the percentage of body fat determined by underwater weighing and various body circumference measurements for 252 men and data set was used to illustrate multiple regression techniques.

i) Body fat

ii) For ratio estimator the auxiliary variable (simulated) is strongly positively correlated with the study variable (body fat).

iii) For regression estimator the auxiliary variable (hips circumference) was strongly positively correlated with the study variable (body fat).

Auxiliary variable

iv) For product estimator the auxiliary variable (simulated) was strongly negatively correlated (body fat) with the study variable.

Auxiliary variable

In order to evaluate the efficiency gain we could achieve by using the proposed estimators, we have calculated the variance of mean per unit and the mean squared error of all estimators we have considered. We have then calculated percent relative efficiency of each estimator in relation to variance of mean per unit. We have then compared the percent relative efficiency of each estimator, the estimator with the highest percent relative efficiency is considered to be the more efficient than the other estimators. The percent relative efficiency is calculated using the following formulae.

efficient compared to mean per unit, ratio and product estimator using one auxiliary variables, ratio-product estimator, exponential ratio estimator, exponential product estimator and exponential ratio-product estimator estimators for population mean since it has the highest percent relative efficiency.

The proposed new regression type estimator with two auxiliary variables for single-phase sampling is recommended for estimating the finite population mean since it is the most efficient estimator compared to mean per unit, ratio and product estimator using one auxiliary variables, ratio-product estimator, exponential ratio estimator, exponential product estimator and exponential ratio-product estimator in term of efficiency in single-phase sampling.