In this paper, we have proposed a class of ratio-cum-product estimator for estimating population mean of study variable for single phase sampling using multi-auxiliary attributes. The expressions for mean square error are derived. An empirical study is given to compare the performance of the estimator with existing estimators. It has been found that the ratio-cum-product estimator using multiple auxiliary attributes is more efficient than mean per unit, product and ratio estimators using one auxiliary attribute, and Product and Ratio estimators using multiple auxiliary attributes in single phase sampling.
The use of supplementary information is widely in sampling theory. Auxiliary variables are commonly used in sample survey practices to obtain improved designs and to achieve higher precision in the estimates of some population parameters such as the mean or the total of a study variable. The concept of ratio estimation was in- troduced in sample survey by Cochran [
Shabbir and Gupta [
Hanif, Haq and Shahbaz [
Let
interest and
complete dichotomy so that;
Let
sessing attribute
attributes
Further, let
where
We assume that
Let
Then for simple random sampling without replacement for first phase sampling, we write by using phase wise operation of expectations as:
, and (1.2)
where
respectively.
The bi-serial correlation coefficient between study variable and auxiliary attributes is given by
If the information proportion
gested as
are to be bounded set in
If A is a square matrix, its inverse can be written using adjoint matrix as,
The following notations will be used in deriving the mean square errors of proposed estimator.
(1.5)
The sample mean
while variance of
In order to have an estimate of the population mean
The MSE of
The optimums value are
The ratio and product estimators by Hanif, Haq and Shahbaz [
The MSE of the
If we estimate a study variable when information on all auxiliary attributes is available from population, it is utilized in the form of their means. By taking the advantage of ratio-cum-product technique for single phase sampling, a genera- lized estimator for estimating population mean of study variable Y with the use of multi-auxiliary attributes is given by:
Simplifying (3.0) and substituting (1.0) we get,
Using (1.0) and (1.1) and ignoring the second and higher terms for each expansion of product and after simpli- fication, we write,
The mean squared error of ratio-cum-product estimator is given by,
We differentiate Equation (3.3) partially with respect to
Using normal equations that are used to find the optimum values of
Using (1.3) in (3.6), we get,
Using the optimum value
Or
Or
Or
These ratio-cum-product estimators using multiple auxiliary attribute in single phase sampling is biased. However, these biases are negligible for moderate and large samples. It’s easily shown that the ratio-cum- product estimator is consistent estimator using multiple auxiliary variables since it is a linear combinations
. Relative efficiency of existing and proposed estimator with respect to mean per unit estimator for single phase sampling.
Estimators | Percent relative efficiency with respect with the mean per unit | |
---|---|---|
. Relative efficiency of existing and proposed estimator with respect to mean per unit estimator for single phase sampling. | 100 | |
. Relative efficiency of existing and proposed estimator with respect to mean per unit estimator for single phase sampling. | 161 | |
. Relative efficiency of existing and proposed estimator with respect to mean per unit estimator for single phase sampling. | 128 | |
. Relative efficiency of existing and proposed estimator with respect to mean per unit estimator for single phase sampling. | 202 | |
. Relative efficiency of existing and proposed estimator with respect to mean per unit estimator for single phase sampling. | 189 | |
. Relative efficiency of existing and proposed estimator with respect to mean per unit estimator for single phase sampling. | 230 | |
of consistent estimators it follows that it is also consistent.
In this section, we carried out some data simulation experiments to compare the performance of ratio-cum product estimator with already existing estimators of finite population mean that uses one or multiple auxiliary attributes. The data for the empirical study is a normally distributed simulated population with the following va- riables
N = 300, n = 45, mean = 45, standard deviation = 5
In order to evaluate the efficiency gain we could achieve by using the proposed estimators, we calculated the variance of mean per unit and the mean squared error of all estimators we considered. We calculated percent relative efficiency of each estimator in relation to variance of mean per unit. We then compared the percent rela- tive efficiency of each estimator, the estimator with the highest percent relative efficiency is considered to be the most efficient compared to other estimators. The efficiency was calculated using the following formulae.
According to