American Journal of Operational Research

p-ISSN: 2324-6537    e-ISSN: 2324-6545

2015;  5(2): 21-28

doi:10.5923/j.ajor.20150502.01

Developing Efficient Ratio and Product Type Exponential Estimators of Population Mean under Two Phase Sampling for Stratification

Subhash Kumar Yadav 1, S. S. Mishra 1, Alok Kumar Shukla 2

1Department of Mathematics and Statistics (A Centre of Excellence), Dr. RML Avadh University, Faizabad, India

2Department of Statistics, D.A-V College, Kanpur, India

Correspondence to: S. S. Mishra , Department of Mathematics and Statistics (A Centre of Excellence), Dr. RML Avadh University, Faizabad, India.

Email:

Copyright © 2015 Scientific & Academic Publishing. All Rights Reserved.

Abstract

This manuscript deals with the estimation of population mean of the main variable Y under study using auxiliary information under two phase sampling for stratification. When the main variable under consideration is homogeneous and the mean of the auxiliary variable is not known, double sampling is used to estimate the population mean of the main variable. Stratified random sampling is used when the main variable under study is not homogeneous. In the present manuscript we have proposed the efficient ratio and product type exponential estimators under two phase (double) sampling scheme for stratification for estimating the population mean. The expressions for the biases and mean square errors of proposed estimators have been obtained up to the first order of approximation. The minimum mean square errors have also been obtained for the proposed estimators. A comparison has been made with the existing estimators of population mean in double sampling for stratification. An empirical study is carried out to meet out the theoretical findings.

Keywords: Auxiliary variable, Double sampling, Stratification, Bias, Mean square error

Cite this paper: Subhash Kumar Yadav , S. S. Mishra , Alok Kumar Shukla , Developing Efficient Ratio and Product Type Exponential Estimators of Population Mean under Two Phase Sampling for Stratification, American Journal of Operational Research, Vol. 5 No. 2, 2015, pp. 21-28. doi: 10.5923/j.ajor.20150502.01.

Article Outline

1. Introduction
2. Methods and Material
3. Proposed Estimators
4. Efficiency Conditions
5. Empirical Study
6. Results and Conclusions

1. Introduction

The proper use of auxiliary information increases the efficiencies of the estimators of the population parameters. This information is supplied by the auxiliary variable (X) and it is completely known to the experimenter. The auxiliary variable is highly (positively or negatively) correlated with the main variable (Y) under study. When X is highly positively correlated with that of Y, ratio type estimators are used to estimate the population parameters while product type estimators are used when X is highly negatively correlated with Y, otherwise regression estimators are used to estimate the population parameters.
Cochran [1] utilized the positively correlated auxiliary information and proposed the traditional ratio estimator of population mean of study variable. Robson [2] proposed the traditional product estimator of population mean using negatively correlated auxiliary information. Later many authors including Upadhyaya and Singh [3], Singh [4], Singh and Tailor [5], Singh et.al ([6], [7], [8]), Singh et al. [9], Kadilar and Cingi ([10], [11]), Tailor and Sharma [12], Yan and Tian [13], Yadav [14], Pandey et al. [15], Subramani and Kumarapandiyan [16], Solanki et al. [17], Onyeka [18], Jeelani et.al [19], Yadav and Kadilar [20] etc. used auxiliary information for improved estimation of population mean of the study variable.

2. Methods and Material

Let the finite population consist of N distinct and identifiable heterogeneous units for the characteristic under study. In this situation simple random sampling is not an appropriate technique for estimating population mean as it will have a very large variance. To overcome this problem we divide the whole population into relatively homogeneous groups known as strata. Let the whole population is divided into L strata of size with as strata weights. When these strata weights are not known, two phase sampling scheme for stratification is used. Following procedure has been followed for two phase sampling scheme for stratification as given by Tailor et.al [21],
(1) A sample of size is drawn at first phase using simple random sampling without replacement technique and the observations are taken on auxiliary variable X only.
(2) This sample of size is stratified into L strata based on auxiliary variable. Let be the number of units in hth stratum such that .
(3) From these units, a sample of size is drawn, where is the predetermined probability of selecting a sample of size from strata of size and it constitutes a sample of size . From this sample observations on both the variables Y and X are taken.
Following notations have been used in this manuscript,
- Size of the sample at second phase
- hth stratum weight for the second phase sample
- hth stratum mean for the study variable Y
- hth stratum mean for the auxiliary variable X
- Population mean square deviation for the study variable Y
- Population mean square deviation for the auxiliary variable X
- hth stratum mean square deviation for the study variable Y
- hth stratum mean square deviation for the auxiliary variable X
- Covariance between y and x in hth stratum
- Correlation coefficient between y and x in the hth stratum
- Unbiased estimator of population mean in second phase sampling
- Unbiased estimator of population mean in second phase sampling
- hth stratum mean at second phase sampling for the study variable Y
- hth stratum mean at second phase sampling for the auxiliary variable X
- hth stratum mean at first phase sampling for the auxiliary variable X
- First phase sampling fraction
Hansen et.al [22] proposed the classical combined ratio estimator for population mean under stratified random sampling as,
(2.1)
Ige and Tripathi [23] proposed the double sampling version estimator of (1.1) as,
(2.2)
The combined product estimator of population mean in stratified random sampling is defined as,
(2.3)
The double sampling version estimator of (1.3) is given by,
(2.4)
where is an auxiliary variable which is highly negatively correlated with main variable y under study and , have their usual meanings.
The biases and mean square errors of the estimators in (1.2) and (1.4) respectively are,
(2.5)
(2.6)
(2.7)
(2.8)
where and .
Singh et.al [24] suggested following exponential ratio and product type estimators in stratified random sampling based on Bahl and Tuteja [25] estimators of population mean under simple random sampling as,
(2.9)
(2.10)
Tailor et.al [21] proposed the double sampling estimators of the estimators in (2.9) and (2.10) respectively as,
(2.11)
(2.12)
The biases and mean square errors of above estimators, up to the first order of approximation are respectively,
(2.13)
(2.14)
(2.15)
(2.16)
where,

3. Proposed Estimators

Motivated by Tailor et.al [21] and Prasad [26], we proposed the following ratio and product type estimators as,
(3.1)
(3.2)
where and are suitably chosen constant to be determined such that the mean square errors of the estimators (3.1) and (3.2) are minimum respectively.
To study the large sample properties of the proposed estimators, we define
and such that and
Expressing in terms of , we have
On simplification, we have
After expansion and simplification, we get
(3.3)
Subtracting from both the sides and taking expectation both sides, we get bias of as,
(3.4)
Subtracting from both the of (3.3), squaring and simplifying up to the first order of approximation we have,
Expanding, simplifying and taking expectations on both the sides, we get the mean square error of , up to the first order of approximation as,
(3.5)
which is minimum for,
(3.6)
where,
And the minimum mean square error is,
(3.7)
Similarly we can get the bias and mean square error of the estimator in (3.2) up to the first order of approximation as,
(3.8)
where,
and such that and
(3.9)
which is minimum for,
(3.10)
where,
And the minimum mean square error is,
(3.11)

4. Efficiency Conditions

From (2.7) and (3.7) we have
(4.1)
From (2.8) and (3.7) we have that the proposed estimator is better than the estimator if
(4.2)
From (2.15) and (3.7) we have
(4.3)
The estimator is better than the estimator if
(4.4)
Note: Similar conditions are for the estimator .

5. Empirical Study

We examine the efficiency of the proposed estimator over other estimators. We consider published data sets whose statistics are given in Table1. The MSE and percent relative efficiency (PRE) values are given in Table 2. It is obviously seen that the proposed estimator are quite efficient than the other estimators for the given Population.
Table 1. Population (Source: Gujarati [27])
      (Thousands of wildcats),
      (per barrel price),
      (time)
     
Table 2. MSE and PRE Values of Estimators
     

6. Results and Conclusions

In this manuscript we proposed the improved ratio and product type exponential estimators of population mean under two phase sampling using the powerful kappa technique. As we observe from the table-2 that the proposed estimators and have lesser mean square errors than the other mentioned estimators of population mean under two phase sampling for stratification. It means the estimated values of the parameter (population mean) by these estimators are very close to the actual value of the parameter. The main aim of the study is to find the estimators which predict the values very close to the actual value of the parameter with minimum mean squared error. Here in the present study the major goal is obtained as the proposed estimators have the minimum mean square errors. Thus we infer that the proposed estimators should be preferred for the estimation of population mean in two phase sampling for stratification.

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