Variance estimation for adaptive cluster sampling with a single primary unit and the partially systematic adaptive cluster sampling

Abstract

Two topics are investigated in this dissertation. The first concerns variance estimation when a single primary sampling unit is selected. Two new bias variance estimators, based on splitting the initial sample into sub-samples and regarding the initial sample as a stratified sample, are proposed. The results of this study indicated that both new variance estimators are underestimated. The first variance estimator is not preferable when the number of sub-samples is two because its relative bias is too large to be useful. Increasing the number of sub-samples made its relative bias decrease. When the sub-sample size and stratum size equal two, the second variance estimator is more efficient than the first in terms of minimum relative bias and mean squared error. However, both new variance estimators are less efficient than nonadaptive variance estimators in systematic sampling in terms of relative bias in some case. This may result from ignoring the correlation terms between sub-samples or between units in stratum. In addition, the percentage of intervals containing the true population mean for both new variance estimators is less than ninety-five percent. For the second topic, the design of partially systematic adaptive cluster sampling, in which the initial sample selected by sampling without replacement of units, without replacement of networks, and without replacement of clusters, is studied. The results of this study indicated that all three sampling procedures can provide unbiased estimators of the population mean and its variance. An unbiased estimator of the population mean based on a selection without replacement of clusters is the most efficient in terms of minimum variance, while an unbiased estimator of the population mean obtained by sampling without replacement of units is the least efficient. The efficiency comparison between all three estimators proposed for partially systematic adaptive cluster sampling and a modified Raj type estimator proposed by Raj (1956) indicated that the proposed estimators are more efficient than the modified Raj type estimator. However, the percentage of intervals containing the true population mean for the proposed estimators is less than ninety-five percent. This may have been caused by the distribution of proposed estimators not being asymptotic to normal distribution.