Surprising optimal estimators for the area fraction.

For a random closed set X and a compact observation window W the mean coverage fraction of W can be estimated by Smeasuring the area of W covered by X. Jensen and Gundersen, and Baddeley and Cruz-Orive described cases where a point counting estimator is more efficient than area measurement. We give two other examples, where at first glance unnatural estimators are not only better than the area measurement but by Grenanderʹs Theorem have minimal variance. Whittleʹs Theorem is used to show that the point counting estimator in the original Jensen-Gundersen paradox is optimal for large randomly translated discs.