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Variance reducing modifications for estimators of standardized moments of random sets

Part of: Inference from stochastic processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  19 February 2016

Jeffrey D. Picka
Jeffrey D. Picka*
Affiliation:
University of Maryland
*
∗ Postal address: Department of Mathematics, University of Maryland, College Park, MD 20742, USA. Email address: jdp@math.umd.edu
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Abstract

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.

Keywords

Random setgerm-grain modelscumulantsedge effects

MSC classification

Primary: 62M30: Spatial processes
Secondary: 60D05: Geometric probability and stochastic geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 32 , Issue 3 , September 2000 , pp. 682 - 700
Copyright
Copyright © Applied Probability Trust 2000 

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