In the statistical analysis of random sets, it is useful to have simple statistics that can be used to describe the realizations of these sets. The cumulants and several other standardized moments such as the correlation and second cumulant can be used for this purpose, but their estimators can be excessively 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 found through use of a specialized asymptotic regime. Simulation results establish that the modifications are highly effective at reducing estimator standard 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 that are not used in second moment estimation.
Journal of Applied Probability and Advances in Applied Probability have for four decades provided a forum for original research and reviews in applied probability, mapping the development of probability theory and its applications to physical, biological, medical, social and technological problems. Their wide readership includes leading researchers in the many fields in which stochastic models are used, including operations research, telecommunications, computer engineering, epidemiology, financial mathematics, information systems and traffic management. Advances includes a section dedicated to stochastic geometry and its statistical applications.
The Applied Probability Trust is a non-profit publishing foundation established in 1964 to promote study and research in the mathematical sciences. Its titles Journal of Applied Probability and Advances in Applied Probability were the first in the subject. The regular publications of the Trust also include The Mathematical Scientist, and the student mathematical magazine Mathematical Spectrum. The Trust publishes occasional special volumes on applied probability and related subjects.
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