Two-stage, model-assisted estimation using remotely sensed auxiliary data

The utility of remotely sensed auxiliary data for increasing the precision of sample-based inventory estimates of population parameters is well-established. To this end, the model-assisted estimators with remotely sensed auxiliary data are particularly effective for use with continuous dependent variables. The model-assisted estimators take somewhat different forms, depending on the sampling design used to collect the data. The forms are well-documented for simple random sampling but considerably less so for variations of two-stage sampling designs. The objectives of the study were three-fold: (1) to derive the more commonly used Cochran (1977)-like notation for two-stage, model-assisted regression estimators for population means from the Särndal et al. (1992) estimators for population totals, (2) to assess the unbiasedness of the two-stage, model-assisted regression estimators of the population mean and the corresponding variance estimators of the estimate of the population mean, and (3) to compare the precision of estimates of the mean for different combinations of PSU size and first- and second-stage sample sizes. The conclusions were that the two-stage, model-assisted estimators of Särndal et al. (1992) could be much more simply expressed using the familiar Cochran notation and that neither the estimators of the mean nor the estimators of the variance of the estimate of the mean exhibited any indications of bias other than possibly for small sample sizes. For equal total sample sizes, standard errors were smaller for combinations of larger first-stage and smaller second-stage sample sizes and slightly smaller for a larger PSU size.