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The Rao-Blackwell theorem in stereology and some counterexamples. (English) Zbl 0818.60010

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.

MSC:

60D05 Geometric probability and stochastic geometry
62D05 Sampling theory, sample surveys
62M30 Inference from spatial processes
62B05 Sufficient statistics and fields
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