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 in-tended 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. Sandal. 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. Resume: Plusieurs ajustements des estimateurs classiques ont ete proposes dans le cadre d'echantillonage de Poisson afin de compenser l'augmentation de la variance due a la taille aleatoire de l'echantillon. Toutefois, aucun de ces ajustements n'ont ete consideres dans le cadre des inventaires forestiers extensifs. Cet article analyse les performances de divers estima-teurs dans le contexte d'inventaires forestiers a une phase et deux degres, dans lesquels les arbres sont echantillonnes, au ni-veau de la placette, par cercles concentriques ou comptage angulaire pour le premier degre et par un tirage de Poisson pour le deuxiee degre. L'estimateur originel est la somme de deux composantes: la premiee est la moyenne d'estimateurs Hor-witz--ompson bases sur des approximations simples du volume n'utilisant que le diamere et l'espee de tous les arbres du premier degre dans chaque placette d'inventaire, et la deuxieme est la moyenne d'estimateurs Horwitz--ompson utili-sant les differences entre approximations simples et approximations affinees requerant des mesures supplementaires de dia-mere et de hauteur sur les arbres du second degre dans chaque placette. Cet estimateur permet d'obtenir des estimations sans biais, mee si le modee approximatif est incorrect, ce qui est particulieement utile pour les estimations locales. Les trois autres estimateurs s'inspirent d'ajustements proposes dans la litterature par L.R. Grosenbaugh et C.E. Sandal. Il s'a-vee que ces ajustements introduisent un biais negligeable mais que les performances, en terme de variance, de l'estimateur originel sont aussi bonnes, voire meilleures, que celles des nouveaux estimateurs.
Introduction In forest inventory and other fields of application of sampling techniques, costs to measure the response variable [Y.sub.i] are high. A commonly used approach, when obtaining [Y.sub.i] be-comes prohibitively [...]