Baddeley, A. J.; Cruz-Orive, L. M. The Rao-Blackwell theorem in stereology and some counterexamples. (English) Zbl 0818.60010 Adv. Appl. Probab. 27, No. 1, 2-19 (1995). Summary: 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 different 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 efficiency. For example, estimators based on (coditionally) non-randomised test point grids may have smaller variance than quadrat estimators. Relative efficiency is related to issues in geostatics and the theory of wide-sense stationary random fields. A uniform minimum variance unbiased estimator typically does not exist in our context. Cited in 1 ReviewCited in 3 Documents MSC: 60D05 Geometric probability and stochastic geometry 62D05 Sampling theory, sample surveys 62M30 Inference from spatial processes 62B05 Sufficient statistics and fields Keywords:Cartier’s formula; point counting; random closed sets; sampling design; spatial statistics; stochastic geometry; wide-sense stationary random fields; Rao-Blackwell theorem PDFBibTeX XMLCite \textit{A. J. Baddeley} and \textit{L. M. Cruz-Orive}, Adv. Appl. Probab. 27, No. 1, 2--19 (1995; Zbl 0818.60010) Full Text: DOI Link