The Rao-blackwell Theorem in Stereology and Some Counterexamples

A version of the Rao-Blackwell theorem is shown to apply to most, but not all, stereological sampling designs. Estimators based on random test grids typically have larger variance than quadrat estimators; random s-dimensional samples are worse than random r-dimensional samples for s < r. Furthermore, the standard stereological ratio estimators of diierent dimensions are canonically related to each other by the Rao-Blackwell process. However there are realistic cases where sampling with a lower-dimensional probe increases eeciency. For example, estimators based on (conditionally) non-randomised test point grids may have smaller variance than quadrat estimators. Relative eeciency is related to issues in geostatis-tics and the theory of wide-sense stationary random elds.