This paper suggests some estimators for population mean of the study variable in simple random sampling and two-phase sampling using information on an auxiliary variable under second order approximation. Bahl and Tuteja (1991) and Singh et al. (2008) proposed some efficient estimators and studied the properties of the estimators to the first order of approximation. In this paper, we have tried to find out the second order biases and mean square errors of these estimators using information on auxiliary variable based on simple random sampling and two-phase sampling. Finally, an empirical study is carried out to judge the merits of the estimators over others under first and second order of approximation.

1. Introduction

Let U = (U₁, U₂, U₃, …, U_N) denote a finite population of distinct and identifiable units. For the estimation of population mean of a study variable Y, let us consider X to be the auxiliary variable that is correlated with study variable Y, taking the corresponding values of the units. Let a sample of size n be drawn from this population using simple random sampling without replacement (SRSWOR) and y_i, x_i (i = 1,2, …, n) are the values of the study variable and auxiliary variable, respectively, for the ith units of the sample.

In sampling theory the use of suitable auxiliary information results in considerable reduction in MSE of the ratio estimators. Many authors including Singh and Tailor [1], Kadilar and Cingi [2], Singh et al. [3], and Singh and Kumar [4] suggested estimators using some known population parameters of an auxiliary variable in simple random sampling. These authors studied the properties of the estimators to the first order of approximation. But sometimes it is important to know the behavior of the estimators to the second order of approximation because up to the first order of approximation the behavior of the estimators is almost the same, while the properties for second order change drastically. Hossain et al. [5] and Sharma and Singh [6, 7] studied the properties of some estimators to the second order approximation. Sharma et al. [8, 9] also studied the properties of some estimators under second order of approximation using information on auxiliary attributes. In this paper we have studied properties of some exponential estimators under second order of approximation in simple random sampling and two-phase sampling using information on an auxiliary variable.

2. Some Estimators in Simple Random Sampling

For estimating the population mean

of Y, the exponential ratio estimator t_1S is given by

()

where

and

(the notation is used to represents for simple random sampling).

The classical exponential product type estimator is given by

()

Following Srivastava [10] an estimator t_3S is defined as

()

where α is a constant suitably chosen by minimizing MSE of t_3S. For α = 1, t_3S is the same as conventional exponential ratio estimator, whereas, for α = −1, it becomes conventional exponential product type estimator.

Again for estimating the population mean

of Y, Singh et al. [11] defined an estimator t_4S as

()

where θ is the constant and suitably chosen by minimizing mean square error of the estimator t_4S.

3. Notations Used

Let us define, and , and then E(e₀) = E(e₁) = 0.

For obtaining the bias and MSE the following lemmas will be used.

Lemma 1. Consider

()

Lemma 2. Consider

()

Lemma 3. Consider

()

where

()

For proof of these lemmas see P. V. Sukhatme and B. V. Sukhatme [12].

4. Biases and Mean Squared Errors to the First Order of Approximation

Bias and MSE of the estimators t_1S, t_2S, and t_3S are, respectively, written as

()

By minimizing MSE(t_3S), the optimum value of α is obtained as α_o = 2C₁₁/C₂₀. By putting this optimum value of α in (13) and (14) we get the minimum value for bias and MSE of the estimator t_3S.

The bias and MSE of estimator t_4S are given, respectively, as

()

By minimizing MSE(t_4S), the optimum value of θ is obtained as θ_o = (C₁₁/C₂₀) + (1/2). By putting this optimum value of θ in (15) we get the minimum value for bias and MSE of the estimator t_3S. We found that for the optimum cases the biases of the estimators t_3S and t_4S are different but the MSE expressions of estimators t_3S and t_4S are similar to the first order of approximation. It is also analyzed that the MSEs of the estimators t_3S and t_4S are always less than the MSEs of the estimators t_1S and t_2S. This prompted us to study the estimators t_3S and t_4S under second order approximation.

5. Second Order Biases and Mean Squared Errors

Expressing estimator t_i’s (i = 1,2, 3,4) in terms of e’s (i = 0,1), we get

()

The Bias of the estimators t_1S is

()

Using (17), we have

()

The biases and MSE’s of the estimators t_2S, t_3S, and t_4S to second order of approximation, respectively, as

()

The optimum value of α we get by minimizing MSE₂(t_3S). But theoretically the determination of the optimum value of α is very difficult; we have calculated the optimum value by using numerical techniques. Similarly, the optimum value of θ which minimizes the MSE of the estimator t_4S is obtained by using numerical techniques.

6. Empirical Study

For a natural population data, we calculate the biases and the mean squared errors of the estimators and compare biases and MSEs of the estimators under first and second order of approximation.

6.1. Data Set

The data is taken from 1981, Utter Pradesh District Census Handbook, Aligarh. The population consists of 340 villages under koil police station, with Y being number of agricultural labour in 1981 and X being area of the villages (in acre) in 1981. The following values are obtained:

()

Table 1 exhibits the biases and MSEs of the estimators t_1S, t_2S, t_3S, and t_4S which are written under first order and second order of approximation. The estimator t_2S is exponential product estimator and it is proposed for the case of negative correlation; therefore, the bias and mean squared error for estimator t_2S are greater than the other estimators considered here. For ratio estimators, it is observed that the biases and the mean squared errors increased for second order. Estimators t_3S and t_4S have the same mean squared errors for the first order but the mean squared errors of t_3S for the second order are less than t_4S. So, on the basis of the given dataset, we conclude that the estimator t_3S is best followed by the estimator t_4S among the estimators considered here.

Table 1. Bias and MSE of estimators.

Estimators	Bias		MSE
Estimators	First order	Second order	First order	Second order
t_1S	0.062981	0.062546	39.231071	39.348971
t_2S	0.053623	0.052483	72.256693	73.329202
t_3S	0.057063	0.056678	39.218063	39.131656
t_4S	0.062513	0.062043	39.218073	39.334929

7. Two-Phase Sampling

In the case when population mean of the auxiliary character is not known in advance, we go for two-phase (double) sampling. The two-phase sampling can be powerful and cost-effective (economical) procedure for finding the infallible estimate for first phase sample for the unknown parameters of the auxiliary character x and hence plays an eminent role in survey sampling; for instance, see Hidiroglou and Sarndal [13].

Considering SRSWOR (simple random sampling without replacement) design in each phase, the two-phase sampling scheme is as follows:

(i)
the first phase sample () of a fixed size n^′ is drawn to measure only x in order to formulate a good estimate of a population mean ;
(ii)
given , the second phase sample s_n () of a fixed size n is drawn to measure y only.

Let

, and

The estimators t_1S considered in Section 2 can be defined in two-phase sampling as

()

The classical exponential product type estimator in two-phase sampling is given by

()

The estimator t_3S in two-phase sampling is defined as

()

where α_d is a constant suitably chosen by minimizing MSE of t_3d. For α_d = 1, t_3d is the same as conventional exponential ratio estimator, whereas, for α_d = −1, it becomes conventional exponential product type estimator.

The estimator t_4S in two-phase sampling is defined as

()

where θ_d is a constant and is suitably chosen by minimizing mean square error of the estimator t_4d.

8. Notations under Two-Phase Sampling

Notations defined in Section 3 can be written in two-phase sampling for SRSWOR as follows.

Lemma 4. Consider

()

Lemma 5. Consider

()

Lemma 6. Consider

()

where

()

Proof of these lemmas is straight forward by using SRSWOR (see P. V. Sukhatme and B. V. Sukhatme [12]).

9. First Order Biases and Mean Squared Errors in Two-Phase Sampling

The bias and MSEs of the estimators t_1d, t_2d, and t_3d in two-phase sampling, respectively, are

()

By minimizing MSE(t_3d), the optimum value of α_d is obtained as

. By putting this optimum value of α_d in (41) and (42), we get the minimum value for bias and MSE of the estimator t_3d.

The expressions for the bias and MSE of t_4d to the first order of approximation are given below

()

By minimizing MSE(t_4d), the optimum value of θ is obtained as

. By putting this optimum value of θ_d in (43) we get the minimum value for bias and MSE of the estimator t_3d. We analyzed that our study should be extended to the second order of approximation as earlier.

10. Second Order Biases and Mean Squared Errors in Two-Phase Sampling

Expressing estimator t_id (i = 1,2, 3,4) in terms of e’s (i = 0,1), we get

()

The bias of the estimator t_1d to the second order of approximation is

()

Using (45) we get MSE of t_1d up to second order of approximation as

()

The biases and MSE’s of the estimators t_2d, t_3d, and t_4d to the second order of approximation are, respectively, as follows:

()

We can get the optimum value of α_d and θ_d by minimizing MSE₂(t_3d) and MSE₂(t_4d), respectively. But theoretically the determination of the optimum values of α_d and θ_d is very difficult; therefore, we have calculated the optimum values by using numerical techniques.

11. Empirical Study

For a natural population data set considered in Section 6, we calculate the biases and the mean squared errors of the estimators and compare the biases and MSE’s of the estimators under first and second order of approximations.

Table 2 exhibits the biases and MSE’s of the estimators t_1d, t_2d, t_3d, and t_4d which are written under first order and second order of approximation for two-phase sampling. The estimator t_2d is exponential product estimator and it is considered in case of negative correlation. So the bias and mean squared error for this estimator is more than the other estimators considered here. For the classical exponential ratio estimator in two-phase sampling, it is observed that the biases and the mean squared errors increased for second order. The estimators t_3d and t_4d have the same mean squared error for the first order but the mean squared error of t_3d is less than t_4d for the second order. So, on the basis of the given dataset we conclude that the estimator t_3d is best followed by the estimator t_4d in tow phase sampling, among the estimators considered here.

Table 2. Bias and MSE of estimators under two phase sampling.

Estimators	Bias		MSE
Estimators	First order	Second order	First order	Second order
t_1d	0.075448919	0.0753124	158.5067369	158.90969
t_2d	0.049751632	0.0474159	221.9313664	222.50215
t_3d	0.061931	0.0620135	158.320	158.3868
t_4d	0.07413	0.0739541	158.320	158.99492

12. Conclusion

In this paper we have studied the Bahl and Tuteja [14] exponential ratio and exponential product type estimators and Singh et al. [11] estimators under first order and second order of approximation in simple random sampling and two-phase sampling. It is observed that up to the first order of approximation both estimators are equally efficient in the sense of mean squared error but when we consider the second order approximation the estimator t_3S (t_3d in two-phase sampling) is best followed by the estimator t_4S (t_4d in two-phase sampling). Theoretical results are also supported through two natural population datasets.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

Authors wish to thank the editor Chin-Shang Li and two anonymous referees for their helpful comments that aided in improving this paper.

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Efficient Estimators Using Auxiliary Variable under Second Order Approximation in Simple Random Sampling and Two-Phase Sampling

Abstract

1. Introduction

2. Some Estimators in Simple Random Sampling

3. Notations Used

4. Biases and Mean Squared Errors to the First Order of Approximation

5. Second Order Biases and Mean Squared Errors

6. Empirical Study

6.1. Data Set

7. Two-Phase Sampling

8. Notations under Two-Phase Sampling

9. First Order Biases and Mean Squared Errors in Two-Phase Sampling

10. Second Order Biases and Mean Squared Errors in Two-Phase Sampling

11. Empirical Study

12. Conclusion

Conflict of Interests

Acknowledgments

References

References

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Efficient Estimators Using Auxiliary Variable under Second Order Approximation in Simple Random Sampling and Two-Phase Sampling

Abstract

1. Introduction

2. Some Estimators in Simple Random Sampling

3. Notations Used

4. Biases and Mean Squared Errors to the First Order of Approximation

5. Second Order Biases and Mean Squared Errors

6. Empirical Study

6.1. Data Set

7. Two-Phase Sampling

8. Notations under Two-Phase Sampling

9. First Order Biases and Mean Squared Errors in Two-Phase Sampling

10. Second Order Biases and Mean Squared Errors in Two-Phase Sampling

11. Empirical Study

12. Conclusion

Conflict of Interests

Acknowledgments

References

References

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