Variance reducing modifications for estimators of standardized moments of random sets

In the statistical analysis of random sets, it is useful to have simple sta tistics that can be used to describe the realizations of these sets. The cu mulants and several other standardized moments such as the correlation and second cumulant can be used for this purpose, but their estimators can be e xcessively variable if the most straightforward estimation strategy is used . Through exploitation of similarities between this estimation problem and a similar one for a point process statistic, two modifications are proposed . Analytical results concerning the effects of these modifications are foun d through use of a specialized asymptotic regime. Simulation results establ ish that the modifications are highly effective at reducing estimator stand ard deviations for Boolean models. The results suggest that the reductions in variance result from a balanced use of information in the estimation of the first and second moments, through eliminating the use of observations t hat are not used in second moment estimation.