Comparison of estimators in one-phase two-stage Poisson sampling in forest inventories

In the context of Poisson sampling, numerous adjustments to classical estimators have been proposed that are intended to compensate for inflated variance due to random sample size. However, such adjustments have never been applied to extensive forest inventories. This work investigates the performances of four estimators for the timber volume in one-phase two-stage forest inventories, where trees in the first stage are selected, at the plot level, by concentric circles or angle-count methods and a subset thereof are selected by Poisson sampling for further measurements to get a better estimation. The original two-stage estimator is the sum of two components: the first is the mean of Horwitz-Thompson estimators using simple volume approximations, based on diameter and species alone, of all first-stage trees in each inventory plot, and the second is the mean of Horwitz-Thompson estimators based on the differences between the simple volume approximations and refined volume determinations based on further diameter and height measurements on the second-stage trees within each inventory plot. This two-stage estimator is particularly useful because it provides unbiased estimates even if the simple prediction model is not correct, which is particularly important for small area estimation. The other three estimators rely on adjustments of the second component of the original estimator that are adapted from estimators proposed in the literature by L. R. Grosenbaugh and C.-E. Sarndal. It turns out that these adjustments introduce a negligible bias and that the original simple estimator performs just as well or even better than the new estimators with respect to the variance.