The use of auxiliary information at both selection and estimation stages to increase the efficiency of estimators has been employed with several improvements at both selection and estimation stages since the work of [1]. Some estimation method that uses auxiliary variables includes ratio, product and regression estimators. Although, the first two methods give rise to biased estimators, the bias can be reduced by increasing the sample size. It is well known that to estimate any parameter, a suitable estimator is the corresponding statistic. Thus for estimating population mean, sample mean is the most appropriate estimator. Although it is unbiased, it has a large amount of variation. Therefore we seek an estimator which may be biased but has smaller man squared error as compared to sample mean. This is achieved through the use of an auxiliary variable that has strong positive or negative correlation with the study variable. When there is strong positive correlation between the study variable and the auxiliary variable and the line of regression passes through origin, then the [1] estimator is used for improved estimation of population mean. Estimator by [2] and [3] estimator are used when there is strong negative correlation. The regression type estimators are used for the improved estimation of population mean when the line of regression does not pass through the origin. This prompts the need for an appropriate transformation of the auxiliary variable to estimate the population total or mean of the study variable which may be asymptotically equal or close to the mean square error of the linear regression estimator. In light of the above shortcomings, several authors have studied to improve the existing classical ratio and product estimators to increase efficiency and also give better options in decision making. Eq.4,5,6,7,9, are amongst authors who have contributed in one way or the other to the modification of the classical ratio and product estimator for the population mean of the study variable under simple random sampling without replacement (SRSWOR) scheme with extensions to other sampling techniques.

NOTATIONS AND SOME EXISTING ESTIMATORS

Let U = {U1,UN} be a finite population of size N and let (yi,xi) be the values of the study variable Y and an auxiliary variable X on the ith unit Ui, I = 1,N. let a sample of size n be drawn   from this population  using    a   simple used    in     this      research    are     defined    below.

Population mean of the auxiliary variable:

Population mean of the study variable:

Sample mean of the auxiliary variable:

Sample mean of the study variable: 

Coefficient of variation of the auxiliary variable:

:

Coefficient of variation of the study variable:

Coefficient of variation between the study variable and the auxiliary variable:

And

is the sampling fraction:

Table 1: Some Existing Estimators with Their Mean Square Error (MSE)

 Population mean of the auxiliary variable:

Population mean of the study variable:

Sample mean of the auxiliary variable:

Sample mean of the study variable:

Proposed Estimator

The proposed estimator is a linear combination of Eq. 1, 2 and 3 denoted by:

                                                                               (1)

Bias and Mean Square Error of (1)

Let  and   

, , , , where

To obtain the Bias of the proposed estimator, we Substitute and in [1]

                                            (2)

                                                                 (3)

Using Taylor’s series expansion up to second order approximation and assuming higher orders are negligible, we obtain:

(4)

The bias of the proposed estimator to its first order approximation is obtained from Eq.4, as follows:

Bias () =

(5)

The expression for the MSE of the proposed estimator is also obtained from Eq. 4 as follows:

(6)

Optimality conditions for the proposed estimator 

To get the optimal value of α that will minimize the MSE, The partial derivative of Eq. 6 is taken with respect to α equated to zero and the value of α is obtained.

(7)

Substituting the value of α in Eq. 6, we obtained the optimal MSE () as:

(8)

From Eq. 8, it is observed that the optimum MSE of the proposed estimator is the same as the MSE of the linear regression estimator. 

Efficiency Comparison   

The efficiency comparisons in this study are done using the mean square error of the proposed estimators and that of three existing estimators: 

If

  if 

if

if 

 if 

 if 

Empirical Study
In this section, we consider four (4) real data sets to numerically evaluate the performances of the proposed and the existing estimators considered here.

Population 1
[Source: Cochran (1977), pp. 196] Let y be the peach production in bushels in an orchard and x be the number of peach trees in the orchard in North Carolina in June 1946. The summary statistics for this data set are: N = 256, n = 100, (Y ) ̅= 56.47, X ̅= 44.45, C_y = 1.42, C_x= 1.40, ρ_yx  = 0.887.

Population 2
[Source: Murthy (1977), pp. 228] Let y be the output and x be the number of workers. The summary statistics for this data set are: N = 80, n = 10, (Y ) ̅ = 51.8264, X ̅ = 2.8513, C_y = 0.3542, C_x = 0.9484, ρ_yx  = 0.915.
 

Population 3

[Source: Das (1988)] Let y be the number of agricultural labourers for 1971 and x be the number of agricultural labourers for 1961. The summary statistics for this data set are: N = 278, n = 25, = 39.068, = 25.111, = 1.4451, = 1.6198,  = 0.7213. 

Population 4

[Source: Steel, Torrie & Dickey (1960), pp. 282] Let y be the log of lef burn in sacs and x be the chlorine percentage. The summary statistics for this data set are: N = 30, n = 6, = 0.6860, = 0.8077, = 0.7001, = 0.7493,  = −0.4996. 

From the table above, it is shown that the proposed estimator   has higher percentage relative efficiency than those estimators provided in this article.

The theoretical and empirical studies carried out reveals that the proposed estimators is better than those estimators considered in this article because the conditions are 
satisfied and has higher percentage relative efficiency, also for optimum value of α the proposed estimators is equally efficient as the linear regression estimator. Hence the proposed estimator is recommended for its practical use for estimating population mean when the auxiliary information is available in survey sampling

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