A new improved difference-cum-exponential ratio type estimator in systematic sampling using two auxiliary variables

This article is published under the Creative Commons CC-BY-ND License (http://creativecommons.org/licenses/by-nd/4.0/). This license permits use, distribution and reproduction, commercial and non-commercial, provided that the original work is properly cited and is not changed in anyway. Abstract: A difference-cum-exponential ratio type estimator was proposed for estimating the finite population mean using two auxiliary variables under systematic sampling. Expressions for the biases and mean square errors (MSEs) were derived up to first order of approximation. It was observed that the proposed estimator is more efficient than the usual sample mean estimator, traditional ratio estimator, exponential-ratio estimator and many other recently proposed difference type estimators in terms of MSEs. Four datasets were used for efficiency comparisons.


INTRODUCTION
The use of auxiliary information in sample survey in an appropriate way may increase the precision of estimators by taking advantages of correlation between the study variable and the auxiliary variable. The ratio, product, exponential and regression estimators have frequently been used by many researchers in different forms either at the estimation stage or at designing stage or at both stages. In daily life information on single as well as two auxiliary variables are commonly used to enhance the precision of estimators. For example: (i) let y be the yield of a particular crop based on the area of a crop (x) and the amount of water utilisation (z); (ii) let y be the electricity consumed in the households (HHDs) based on income of the HHDs (x) and size of the HHDs (z) and (iii) let y be the average grades of the undergraduate students based on the number of study hours used by students (x) and intelligence coefficient (IQ) level of the students (z). In all these examples, we are interested to use the auxiliary information in parallel to the study variable. Madow and Madow (1944), Cochran (1946) and Gautschi (1957) were among the earlier contributors in the area of systematic sampling. Other authors who have contributed in this area include: Swain (1964), Kushwaha and Singh (1989), Banarasi et al. (1993), Singh and Singh (1998), Singh et al. (2011), Singh and Jatwa (2012), Tailor et al. (2013), Khan and Singh (2015), Khan (2016), Pal and Singh (2017), Kocyigit and Cingi (2017), Riaz et al. (2017), Tailor and Mishra (2018), Qureshi et al. (2018) and Mishra et al. (2018). The main objective of this study was to construct a new estimator by taking advantage of two auxiliary variables to improve the efficiency of the estimator in systematic sampling. ij ij

METHODOLOGY
x z ( 1, 2,3,..., ; 1, 2,3,..., ) i k j n = = be the observed values of the study variable ( ) y and the auxiliary variables ( , ) x z , respectively for the j th unit in the i th possible the intra-class correlation coefficients between pairs of units within the same systematic sample for y, x and ( , ) y z, respectively. Let We also define the following error terms.
Now we discuss some of the existing estimators.
The usual sample mean estimator is given by The variance of 0 Y is given by Single auxiliary variable (i) The traditional ratio estimator is given by The bias and MSE of ˆR Y to first order of approximation are respectively given by The usual exponential ratio type estimator is given by The bias and MSE of ˆR E Y to first order of approximation are respectively given by ... (7) and 2 2 2 iii) The usual difference estimator is given by where d is the constant.
where ( , ) R α δ ∈ and ( 1, 2,3) i q i = are constants and a and b are the functions of known population parameters, which may be the population mean, population coefficient of variation, population coefficient of skewness and population coefficient kurtosis of the auxiliary variable x.
which is equal to the variance of the linear regression estimator.
(v) Riaz et al. (2017) suggested the following class of estimators in systematic sampling: where ( 1, 2) i t i = are constants and γ is the scalar quantity.
For 1 γ = , the above estimator becomes: The bias of ˆR DS Y to first order of approximation is given by The MSE of ˆR DS Y is given by 2 2 2 2 2 2 2 1 1 1

Two auxiliary variables
(i) The traditional ratio-ratio estimator using two auxiliary variables is given by The bias and MSE of ˆR R Y to first order of approximation are respectively given by Tailor et al. (2013) suggested the ratio-product estimator which is given by

March 2020
Journal of the National Science Foundation of Sri Lanka 48(1) The bias and MSE of ˆR P Y to first order of approximation are given by (iv) The exponential ratio-ratio type estimator is given by The bias and MSE of ˆR RE Y to first order of approximation are given by (24) (v) Tailor and Mishra (2018) suggested the exponential ratio-product type estimator, which is given by The bias and MSE of ˆR PE Y to first order of approximation are given by 2 2 .. (28) where 1 d and 2 d are constants.
is the multiple correlation coefficient of y on x and z.
Solving equation (33), the correct expressions of bias and MSE of estimator ˆM SS Y to first order of approximation are: The correct optimum value of

Proposed estimator
Motivated by Khan (2016) and Riaz et al. (2017), the following difference-cum-exponential ratio type estimator is proposed for the population mean under systematic sampling: ... (40) where ( 1, 2, 3) i A i = are constants whose values are to be determined.

The bias of Pr
Y to first order of approximation is given by ( ) The MSE of Pr Y to first order of approximation is given by From equation (42), the optimum values of ( 1, 2, 3) Substituting the optimum values of ( 1, 2, 3) (42)

Numerical examples
We use the following 4 datasets from different sources.
Data 1: [Source: Mukhopadhyay (1998)] Let y = output of factories, x = number of workers and z = fixed capital.  The PRE values for different estimators based on the above datasets are given in Table 1. show poor performances for some datasets, i.e. (Data 1, Data 2), Data 2, (Data 1, Data 2, Data 4), (Data 1, Data 3), (Data 1, Data 2, Data 4) and Data 3, respectively. The performance of difference estimator (ˆD Y ) and Kocyigit and Cingi (2017) estimator (ˆK C Y ) are equal when using single auxiliary variable. The improvement in Riaz et al. (2017) estimator (ˆR DS Y ) is more than (ˆD Y ) and (ˆK C Y ) estimators. Also difference estimators (ˆD D Y ,ˆQ KH Y ) are equally efficient but (ˆD D Y ) is preferable because of unbiasedness for using two auxiliary variables.

CONCLUSION
A difference-cum-exponential ratio type estimator is proposed for estimating the finite population mean under systematic sampling using single and two auxiliary variables. Some estimators, when using two auxiliary variables do not perform better than the estimators using single auxiliary variable. The proposed estimator Pr ( ) Y is more efficient than the usual sample mean estimator