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In this paper, we have proposed three classes of mixture ratio estimators for estimating population mean by using information on auxiliary variables and attributes simultaneously in two-phase sampling under full, partial and no information cases and analyzed the properties of the estimators. A simulated study was carried out to compare the performance of the proposed estimators with the existing estimators of finite population mean. It has been found that the mixture ratio estimator in full information case using multiple auxiliary variables and attributes is more efficient than mean per unit, ratio estimator using one auxiliary variable and one attribute, ratio estimator using multiple auxiliary variable and multiple auxiliary attributes and mixture ratio estimators in both partial and no information case in two-phase sampling. A mixture ratio estimator in partial information case is more efficient than mixture ratio estimators in no information case.

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

Olkin [

The concept of double sampling was first proposed by Neyman [

Jhajj, Sharma and Grover [

Hanif, Haq and Shahbaz [

In this paper, we will extend the mixture ratio estimator proposed by Moeen, Shahbaz and Hanif [

Consider a population of N units. Let Y be the variable for which we want to estimate the population mean and

Further, let

where

Consider a sample of size n drawn by simple random sampling without replacement from a population of size

In defining the attributes we assume complete dichotomy so that,

Let

attributes

where

Let

Also

We shall take

Similarly,

The coefficient of variation and correlation coefficient are given by

Then for simple random sampling without replacement for both first and second phases we write by using phase wise operation of expectations as:

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

The sample mean

while its variance is given,

Let

information on one auxiliary variables is available for population (full information case) is:

The mean square error of

where

The ratio estimator by Haq [

The optimum values of unknown constants are,

The mean square error of

In order to have an estimate of the population mean

The MSE of

where

The ratio estimators by Hanif, Haq and Shahbaz [

The optimum values of unknown constants are,

The MSE of the

In general these estimators have a bias of order

If we estimate a study variable when information on all auxiliary variables is available from population, it is utilized in the form of their means. By taking the advantage of mixture ratio estimators technique for two-phase sampling, a generalized estimator for estimating population mean of study variable

Using (1.0), (1.3) and (1.4) in (3.0) and ignoring the second and higher terms for each expansion of product and after simplification, we write,

The mean squared error of

We differentiate the Equation (3.2) partially with respect to

Using normal equation that is used to find the optimum values given (3.2) we can write,

Taking expectation and using (1.6) in (3.5), we get,

Substituting the optimum (3.3) and (3.4) in (3.6), we get,

Or

Or

Using (1.8), we get,

In this case suppose we have no information on all s and t auxiliary variables but only for r and g auxiliary variables from population. Considering mixture ratio technique of estimating technique, the population mean of study variable

(3.11)

Using (1.0), (1.3) and (1.4) in (3.1) and ignoring the second and higher terms for each expansion of product and after simplification, we write,

Mean squared error of

We differentiate the Equation (3.13) with respect to

Using normal equation that are used to find the optimum values given (3.13) we can write

Taking expectation and using (1.6) in (3.15), we get,

Using the optimum value (3.14) in (3.16), we get,

Or

Or

Or

Or

Using (1.8) in (3.21), we get,

Simplifying (3.22) we get,

If we estimate a study variable when information on all auxiliary variables is unavailable from population, it is utilized in the form of their means. By taking the advantage of mixture ratio technique for two-phase sampling, a generalized estimator for estimating population mean of study variable

Using (1.0), (1.3) and (1.4) in (3.24) and ignoring the second and higher terms for each expansion of product and after simplification, we write,

Mean squared error of

We differentiate the Equation (3.27) partially with respect to

Using normal equation that are used to find the optimum values given (3.26) we can write,

Taking expectation and using (1.6) in (3.29), we get

Or

Or

Or

Or

Using (1.8) in (3.34), we get,

Simplifying (3.35), we get,

These mixture ratio estimators using multiple auxiliary variables and attributes in two-phase sampling are biased. However, these biases are negligible for large samples that is

We carried out data simulation experiments to compare the performance of mixture ratio estimators using multiple auxiliary variables and attributes in two-phase sampling with ratio estimator using one auxiliary variable and one auxiliary attribute or ratio estimator using multiple auxiliary variable or multiple auxiliary attributes in two-phase sampling estimators for finite population.

All the results were obtained after carrying out two hundred simulations and taking their average.

1) Study variable

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

Correlation coefficients

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

Correlation coefficients

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.

The proposed mixture ratio estimator under full information case is recommended for estimating the finite population mean since it is the most efficient estimator compared to mean per unit, ratio estimator using one auxiliary variable, ratio estimator using one auxiliary attribute, ratio estimator using multiple auxiliary variable and ratio estimator using multiple auxiliary attributes in two-phase sampling. In case some auxiliary variables or attributes are unknown, we recommend mixture ratio estimator under partial information case since it is more

Population | Percent relative efficiency of full and partial to no information | Percent relative efficiency of full to partial in formation case | ||||
---|---|---|---|---|---|---|

Estimator | ||||||

Relative percent efficiency | 100 | 139 | 172 | 100 | 127 |

Population | Percent relative efficiency of full and partial to no information | Percent relative efficiency of full to partial in formation case | ||||
---|---|---|---|---|---|---|

Estimator | ||||||

Relative percent efficiency | 100 | 139 | 172 | 100 | 127 |

efficient than the mixture ratio estimator under no information case and if all are unknown, we recommend the mixture ratio estimator under no information case to estimate finite population mean.