Trade-off between efficiency and variance estimation of spatially balanced augmented samples

In this paper, we construct three types of augmented samples, which are samples generated from two separate randomization events. The first type combines a simple random sample (SRS) with a spatially balanced sample (SBS) selected from the same finite population. The second type combines an SBS with an SRS. The third type combines two spatially balanced samples. The simple random sample is constructed without replacement and does not contain any ties. The spatially balanced samples are constructed using the properties of the Halton sequence. We provide the first and second order inclusion probabilities for the augmented samples. Next, using the inclusion probabilities of the augmented samples, we construct estimators for the mean and total of a finite population. The efficiency of the augmented samples varies between the efficiency of SRS and SBS samples. If the number of SRS observations in the augmented sample is large, the efficiency is closer to the efficiency of SRS. Otherwise, it is closer to the efficiency of SBS. We also provide estimators for the variances of the estimators of population total of augmented samples. The stability of these variance estimators depends on the proportion of SRS observations in the augmented samples. The larger number of SRS observations lead to stable variance estimators.