# Variance estimation for adaptive cluster sampling with a single primary unit

- Abstract
- Introduction
- Methodology
- Variance estimation: The proposed variance estimators
- Examples
- Conclusion
- References

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 [1] 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 ? ? ? Table All possible samples for q = p = 2 and the calculation of v1 (?1 ) . Method I Primary unit Sub-sample ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ? ? ? v1 (?1 ) CI Lower - Upper ( ? Table All possible samples for q=2 and p=2 and the calculation of v1 (?1 ) (continued). [...]

[...] Starting with the gray area in figure 1A which is a secondary unit within the selected primary unit, its neighborhood is a set of units in the North, South, East and West direction as shown in figure 1B. The set of units that is shaded with the gray color in figure 1C is called a network. Edge units and all units within the dash lines are a cluster as shown in figure 1D. { } A:Selected primary unit Neighborhood of unit Network Cluster Figure An example of a selected sample from adaptive clustering sampling with a single primary unit. In real work, an initial sample with a single primary unit is necessary in some situations. [...]