1. Introduction

The system of supplementary population information is modeled to rejig the entire spectrum of estimation process in the theory and practice of sample surveys. Presently, it is widely used by theorist and practitioners of estimation knowledge of population parameters of the given population. Availability of various approaches to construct improved efficient estimators for population parameters is not only the necessary condition but training of mind to them is also sufficient. Why do we use the theory of prediction is a quintessential aspect of enchantment for the researchers engaged in this field of investigation. The answer to this question of attraction of the method lies in the fact that it provides us general framework for statistical inference on the character of finite population, vide Cochran (1977).

In the theory of survey sampling, it is unanimously accepted that the suitable use of auxiliary information improves the efficiency of the estimators of the parameters of the population under consideration for the main characteristic (y) under study. The auxiliary variable (x) is the variable about which we have full information and which is highly positively or negatively correlated with the main variable under study. When the auxiliary variable is highly positively correlated with the main variable, then the ratio method of estimation is used in which the ratio type estimators are used for the estimation of the parameters. On the other hand product type estimators under product method of estimation are used for the estimation of the population parameters when the auxiliary variable is highly negatively correlated with the main variable under study; vide Murthy (1967) and Das (1988).

Many authors used auxiliary information for improved estimation of population mean through ratio and product type estimators for the main characteristic under discussion. The latest references can be made of Yan and Tian (2010), Yadav (2011), Pandey et al. (2011), Nayak and Sahoo (2012), Subramani and Kumarapandiyan (2012), Solanki et al. (2012), Onyeka (2012), Jeelani et al. (2013), Saini (2013), Yadav and Kadilar (2013), Singh et al. (2014) etc.. Optimal search for solution by using computational methods have been widely used in boiling down the numerical complexity including inventory populations in supply chain management etc for attaining efficient estimators of performance measures of inventory system vide for examples Mishra and Singh (2012, 2012, 2013), Mishra and Mishra (2013) and Yadav et al.(2012, 2014).

The fresh attempt has been made to present optimal search for efficient estimator of finite population mean using auxiliary information in which most efficient ratio and product type exponential estimators have been developed. Optimal search continues unless data computation provides us better efficiencies as compared to previously existing estimators. In the proposed estimator, kappa technique has been used which seeks to introduce a constant and further the optimal value is obtained by minimizing the mean square error of the estimator. Next, after putting this optimal value, minimum value of mean square error is obtained. Lastly, result discussion and conclusion are given in the last section.

2. Material and Methods

Let the finite population consists of distinct and identifiable units. Let the main variable under study is denoted by and the auxiliary variable by . Thus denote the i^th observations for the main and auxiliary variables respectively. Thus we have

, the population mean of the main variable to be estimated.

Further let denote the set of all possible sample of fixed size from the population . Let denote the effective sample size that is the number of distinct elements in and denote the collection of all those units of which are not in . We denote,

.

In predictive estimation a specific model is considered to predict the non sampled values of the population. The prediction theory is based on the model-based theory. It considers a general framework for statistical inferences to be drawn for the parameters of the population in question. The general prediction theory considers different ratio, product and regression type estimators of population parameters as the predictive estimators i.e, predictors of the unobserved units of the population under some specific model. Several authors used ratio, product and regression type estimators as predictors of population parameters in predictive estimation theory. In predictive estimation theory it is well known that the use of ratio and product type estimators as predictors of the parameters under consideration of the unobserved units of the population result in the corresponding estimators of the population parameters for the whole population.

For a specific sample , the population mean can be written as,

(2.1)

The sample mean of the sample of size (i.e. ) in simple random sampling is,

that is .

Thus, the population mean in (1) may be rewritten as,

(2.2)

The suitable estimator for using (2.2) can be considered as,

(2.3)

where is the predictor of the population mean of unobserved units of the population.

Srivastava (1983) considered the following estimators as the predictors of as,

,

,

where,

,

and .

He has shown that whenever these estimators are used as predictive estimators of , then the estimator given in (2.3) results in corresponding classical estimators,

and respectively.

However, he has shown that for the product predictive estimator, of , the estimator does not result in the classical product estimator of population mean .

Thus, for , we have

(2.4)

The biases and the mean square errors of the estimators and , up to the first order of approximations respectively are,

(2.5)

(2.6)

where,

, , , ,, ,, and .

Singh et al. (2014) suggested two ratio and product type exponential estimators of population mean using Bahl and Tuteja (1991) ratio and product types exponential estimators of population mean as the predictive estimators of respectively as,

(2.7)

(2.8)

The biases and the mean square errors of above estimators up to the first order of approximations respectively are,

(2.9)

(2.10)

3. Proposed Estimators

Motivated by Singh et al. (2014) and Prasad (1989), we attempt to propose the following efficient ratio and product type exponential estimator in predictive estimation as,

(3.1)

(3.2)

where and are the constants known as kappas to be determined such that the mean square errors of the estimators and are minimum respectively.

To study the large sample properties of the proposed estimators and , we define,

and such that and up to the first order of approximation,

, and .

Expressing (3.1) in terms of ’s and simplifying, we have

(3.3)

Subtracting on both sides of (3.3), simplifying and taking expectation both sides, we get the bias of as,

Putting the values of different expectations, we get

(3.4)

Squaring on both sides of (3.3) after subtracting , expanding, simplifying up to the first order of approximation and taking expectation on both sides, we have the mean square error of as,

which is minimum for

where

And the minimum mean square error of is,

(3.5)

Similarly, the bias and mean square error of , up to the first order of approximation respectively are,

(3.6)

which is minimum for

where

And the minimum mean square error of is,

(3.7)

4. Conditions of Optimal Search and Efficiency Comparison

Using (2.5) and (3.5), we fairly get

(4.1)

In view of (2.6) and (3.5), we have

(4.2)

Further using (2.9) and (3.5), we find

(4.3)

Finally, in the light of (2.10) and (3.5), we happen to obtain

(4.4)

Above conditions are conditions of under which the proposed estimator performs better than the above mentioned estimators of population mean.

Remark: Similar conditions are also drawn for the estimator to be more efficient than the above mentioned estimators.

5. Empirical Study

An empirical study has been carried out numerically to show the usefulness of suggested methodology in this paper. To verify the theoretical findings of the proposed estimator over the estimators and of population mean in predictive estimation, we have taken the following three populations in table 1.

Table 1. The data description

The following table2 represents the percentage relative efficiency (PRE) of different mentioned estimators with respect to mean per unit estimator of population mean in predictive estimation approach.

Table 2. The PREs of different estimators with respect to

6. Results and Conclusions

In the paper, we have succeeded in developing the improved efficient estimators for positively and negatively correlated data for estimating population mean in predictive estimation approach. Upon drawing the observations from the theoretical discussion of section-4 and the results in table-2, it is evident that the proposed estimators and are better than the Srivastava (1983) estimators , and the Singh et.al (2014) estimators as they have lesser mean square error than all these estimators. Therefore the proposed estimators and should be preferred for the estimation of population mean in predictive estimation approach for positively and negatively correlated data respectively.