In this paper, an initial sample of the adaptive cluster sampling design is considered in terms of primary and secondary units. The primary unit contains units that are called secondary units and all these secondary units are arranged in systematic order. Although in some situations in field work it is necessary to use a single primary, the variance estimator of the estimate of the population mean or total in this design is not available now. Two new variance estimators of the estimate of the population mean were found based on the method of splitting the sample. The preliminary study in terms of bias, MSE and percentage of confidence interval containing the population mean for each variance estimator was carried out with a small population. Adaptive cluster sampling, a design with primary and secondary units, was proposed by Thompson  and is a suitable design for rare and cluster populations, especially for a biological population. In this design a primary unit contains units that are called secondary units. All of these secondary units are arranged in systematic order. A primary unit, sometimes called a systematic sample, is selected at random.
[...] However, the bias, MSE and the proportion of confidence intervals that contain the population mean of these new variance estimators in this paper are preliminarily studied. Further study is needed the point of increasing q and p. Moreover, to investigate properties of these estimators, Monte-Carlo method can be used so that various situations can be compared. The other issue such as the “partially systematic sampling” and will be studied in the future. Acknowledgements I would like to thank Assistant Prof Dr Dryver for his suggestions. References Thompson, S. K Adaptive cluster sampling: Designs with primary and Secondary units. Biometrics. 47: 1103-1105. [...]
[...] Table Sample means of all possible samples for adaptive cluster sampling with a single primary unit from the population in figure 2B ( q = 2 Sample Unit labels: j ) Yij yk xk network number yk xk ˆ μ ˆ Table All possible samples for q = p = 2 and the calculation of v1 (μ1 ) . Method I Primary unit Sub-sample ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ˆ μ ˆ v1 (μ1 ) CI Lower - Upper ( ˆ Table All possible samples for q=2 and p=2 and the calculation of v1 (μ1 ) (continued). [...]
[...] Zinger, A Variance Estimation in Partially Systematic Sampling. Journal of the American Statistical Association. 75: 206-211. Appendix A ˆ E (v1 (μ Consider ˆ μ−μ = 1 p ˆ (μ t p t ˆ ˆ μ + μ μ) ˆ (μ μ = p 1 ˆ ˆ ˆ (μt μ + (μ μ + 2 p t ) (μˆ μˆ + μˆ μ )(μˆ p p t t t t p p t t t t ˆ ˆ μ + μ μ = p 1 ˆ ˆ ˆ (μt μ + (μ μ + p 2 t ) (μˆ μ )(μˆ p t t μ ( Q (μˆ t p t ˆ ˆ μ μ ) = 0 ) p 2 ˆ p μ ) = ˆ ˆ (μt μ t p p2 p + ˆ (μ t t t p ˆ μ t μ ) p2 (Subtract with ˆ (μ μ ) p q q 2 ˆ q μ ) = q p ˆ (μt μˆ t p ( p q q (μˆ t t t p p t ˆ μ t μ ) p q (Multiply with q ˆ (μ t t t p p p p q ˆ Since v1 (μ ) = q ˆ ˆ (μt μ t p q q ˆ ˆ μ So v1 (μ ) = q q t ˆ μ t μ ) p ( p p q 2 q 1 t ˆ E (μ μ ) E (v1 (μ = ˆ q q 2 (μˆ t t p ˆ μ t μ ) p t ˆ ˆ ˆ Since E (μ ) = μ so E (μ μ ) = V (μ ) q q q ˆ E (v1 (μ = q (μ ) q ˆ s ˆ (μ t t t p p ˆ μ t μ ) ˆ V (μ ) p t Appendix B ˆ E (v2 (μ ˆ ˆ Q μ can be rewritten in terms of U t as follows μ = n 2 U 2t + U 2t 1 2 n2 t p t p t , where p = n ˆ Consider μ μ = 1 n 2 U 2t U 2t ˆ ˆ +μ 2 t p ˆ (μ μ 1 U 2t + U 2t ˆ ˆ μ + μ μ . [...]
[...] Therefore, these two methods of estimation are of interest and will be adaptive cluster sampling with a single primary unit. In this paper, two methods of variance estimation will be proposed based on the above two methods. The properties of these estimators that will be considered are bias, MSE, and the proportion of confidence interval that contain the population mean Methodology In conventional systematic sampling from a finite population of size N = nq , where n is a sample size, q = N n is the sampling interval. [...]
[...] Table Sample means of all possible samples for adaptive cluster sampling with a single primary unit from the population in figure 2A ( q = 3 Sample Unit labels: j ) Yij yk xk network number yk xk ˆ μ ˆ ˆ Table All possible samples for q = 3 and p = 2 and the calculation of v1 (μ1 ) and v2 (μ1 ) . Method I Primary unit 1 ( Subsample ˆ μ Method II Upper 2.926 na na ˆ v2 (μ1 ) ˆ v1 (μ1 ) 95% CI Lower - 0.675 na na - CI Lower na - 0.148 Upper na - ( 3 ( Expectation 2.250 Bias 0.000 MSE of CI contains the true mean Note: na means is not available ˆ For q = table 3 shows the 2 possible sample mean μ = 2.417 and 2.083 and there are two possible values of that is, p = When p = it is impossible to obtain the second method variance since m are not be equal to 2. [...]
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