1. Introduction

Ratio estimation involves the use of known population totals for auxiliary variables to improve the weighting from sample values to population estimates of interest. It operates by comparing the sample estimate for an auxiliary variable with the known population total for the same variable on the frame. The ratio of the sample estimate of the auxiliary variable to its population total on the frame is used to adjust the sample estimate for the variable of interest. The ratio weights are given by Xx (where X is the known population total for the auxiliary variable and x is the corresponding estimate of the total based on all responding units in the sample). These weights assume that the population for the variable of interest will be estimated by the sample equally as well (or poorly) as the population total for the auxiliary variable is estimated by the sample.

Consider a finite population U = {U₁, U₂, ….U_N}of N distinct and identifiable units. Let Y be a study variable with value Y_i measured on Ui, i = 1, 2, 3,………., N giving a vector Y= {Y₁, Y₂…, Y_N}. The problem is to estimate the population mean with some desirable properties on the basis of a random sample selected from the population U. The simplest estimator of population mean is the sample mean obtained by using simple random sampling without replacement, when there is no additional information on the auxiliary variable available. Sometimes in sample surveys, along with the study variable Y, information on

auxiliary variable x correlated with Y is also collected. This information on auxiliary variable x, may be utilized to obtain a more efficient estimator of the population mean.

The two broad categories of estimators using auxiliary information are the ratio method and product method of estimation.

Among those who have worked on these estimators are Sisodia and Dwivedi[12], Pandey and Dubey[7], Singh, Taylor[10], Adewara and Singh[2]. The study is motivated by the success recorded by J. Subramani and G. Kumarapandiyan[8] on their respective works on “Modified ratio estimators using known median and coefficient of Kurtosis” and “Efficiency of some modified ratio and product Estimators using known value of some population parameters”.

This work is focused on improving the efficiency of some Ratio and Product estimators in the literature by obtaining the optimal values of coefficient of variation of the auxiliary variable Cx in the MSE and then substitute back with some re-arrangement to obtain an improved estimate of the estimator and their respective MSE.

2. Literature Review

Sen., A.R.[9] presented an historical development of the ratio method of estimation starting from the year 1662. Auxiliary information is any information closely related to the study variable. The use of auxiliary information usually leads to the sampling strategy with higher efficiency compared to those in which no auxiliary information is used. Higher precision can also be achieved by using the auxiliary information for the dual purposes of selection and the estimation procedure.

It is important to note that proper use of the knowledge of auxiliary information may result in appreciable gain in precision of the estimates. But indiscriminate use of auxiliary information might not provide the desired precision and in some extreme cases might even lead to loss in precision.

Many authors since the discovery of ratio method of estimation have come up with various degree of modification of the conventional ratio estimations for better performance. These among others include Sisodia and Dwivedi[12], Singh and Tailor[10], Pandey and Dubey[7], Adewara and Singh[10] among others.

3. Methodology

3.1. Existing Modified Ratio and Product Estimators

Suppose a pairs (x, y) (i=1, 2, ….., n) observations are taken on n units sampled form N population units using simple random sampling without replacement scheme, are the population means for the auxiliary variable (x) and variable of interest (y) and are the sample means based on the sample drawn.

Khoshnevison et al[5] defined their family of estimators as:

where , b are either real numbers or a functions of the known parameters of the auxiliary variable x such as standard derivation , coefficient of variation, C_x, Skewness , Kurtosis and correlation coefficient ρ.

i. When α=1 a=1, b= 0, g=0, we have the usual ratio estimator, with

(3.1)

ii. When α=0 a=1, b= 0, g=1, we have the usual ratio estimator, with

(3.2)

iii. When α=1 a=1, b= 0, g=-1, we have the usual product estimator, with

(3.3)

iv. When α=1 a=1, b= C_x, g=1, Sisodia and Dwivedi[12] ratio estimator with

(3.4)

v. When α=1 a=1, b=C_x, g=1, we have Pandey and Dubey[7] product estimator with with

(3.5)

vi. When α=1 a=1, b=, g=1, we have Singh, Taylor[10] ratio estimator as with

(3.6)

vii. When α=1 a=-1, b=ρ ,g=1, we have Singh, Taylor[10] product estimator with

(3.7)

It should be noted that there are other ratio and product estimators from the family mentioned above, but here attention is focused on those estimators that uses the coefficient of variation Cx and correlation coefficient. Conventionally, for ratio estimators to hold, and also for product estimators to hold, .

3.2. Proposed Ratio and Product Estimators

The proposed optimal estimators were obtained by minimizing the MSE(.) from each of the existing estimators with respect to C_x and equate to zero. The solution for C_x is then substituted back to initial MSE(.) to get the modified optimal estimators and corresponding MSE(*.) as follows in equations 3.8 to 3.12 below:

i.

(3.8)

ii.

(3.9)

iii.

(3.10)

iv.

v.

(3.11)

vi.

(3.12)

3.3. Data Used

Following Adewara et. al.[2], data sets from two populations are used:

Population I: Kadilar and Cingi1[3] with the parameters:

Population II: Maddala (1977) with the parameters:

Since conventionally, for ratio estimators to hold, and also for product estimators to hold, .

4. Results and Discussions

The results obtained from the application of the two data sets to the Mean square error of the conventional and proposed ratio and product estimators in equations 3.8 to 3.12 are shown tables 1 and 2 below:

It can be observed from table 1 that the MSE for the proposed estimators are less that each of the existing ratio estimators.

Also it can be observed from table 2 that the MSE for the proposed estimators are less that each of the existing product estimators.

Table 1. Mean Square Errors (MSE) of the estimates of Existing and Proposed Estimators (Ratio Estimators) with Population I

Table 2. Mean Square Errors (MSE) of the estimates of Existing and Proposed Estimators (Product Estimators) with Population II

5. Conclusions

In this study, we have proposed an optimal estimators for the ratio estimators and product estimators by minimizing the coefficient of variation of the auxiliary variable x. Using the Kadilar and Cingi1[3] and Maddala[6] data for the proposed estimators, a gain of a least 10% precision was achieved over the existing estimators for both the ratio and product estimators in the literature.

Therefore, the proposed optimal estimators is uniformly better than all other estimators and thus most preferred over the existing modified ratio and product estimators for the use in practical applications for certain population with peculiar characteristics.