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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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References

[1] Cressie, N. A. C. (1993). Statistics for Spatial Data. John Wiley, New York.CrossRefGoogle Scholar
[2] Daley, D. J. and Vere-Jones, D. (1988). Introduction to the Theory of Point Processes. Springer, Berlin.Google Scholar
[3] Hall, P. (1979). On the invariance principle for U-statistics. Stoch. Proc. Appl. 9, 163–174.CrossRefGoogle Scholar
[4] Kozintsev, B. and Kedem, B. (2000). Generation of ‘similar’ images from a given discrete image. J. Comput. Graph. Statist. 9, 286–302.Google Scholar
[5] Mase, S. (1982). Properties of fourth-order strong mixing rates and its application to random set theory. J. Multivar. Anal. 12, 549–561.CrossRefGoogle Scholar
[6] Matheron, G. (1975). Random Sets and Integral Geometry. John Wiley, New York.Google Scholar
[7] Mattfeldt, T., Vogel, U., Gottfried, H.-W. and Frey, H. (1993). Second-order stereology of prostatic adenocarcinoma and normal prostatic tissue. Acta Stereologica 12, 203–208.Google Scholar
[8] Molchanov, I. (1997). Statistics of the Boolean Model for Practitioners and Mathematicians. John Wiley, New York.Google Scholar
[9] Picka, J. D. (1997). Variance-reducing modifications for estimators of dependence in random sets. , University of Chicago.Google Scholar
[10] Picka, J. D. (1998). Measures of dependence structure for random sets. Tech. Rept 88, National Institute of Statistical Sciences, NC.Google Scholar
[11] Picka, J. D., Jaiswal, S. S., Igusa, T., Karr, A. F. and Shah, S. P. (2000). Quantitative description of coarse aggregate volume fraction gradients. Cement, Concrete, and Aggregates 22, 151–159.Google Scholar
[12] Ripley, B. D. (1976). The second-order analysis of stationary point processes. J. Appl. Prob. 13, 255–266.CrossRefGoogle Scholar
[13] Ripley, B. D. (1988). Statistical Inference for Spatial Processes. John Wiley, New York.CrossRefGoogle Scholar
[14] Stein, M. L. (1993). Asymptotically optimal estimation for the reduced second moment measure of point processes. Biometrika 80, 443–449.CrossRefGoogle Scholar
[15] Stein, M. L. (1995). An approach to asymptotic inference for spatial point processes. Statist. Sinica 5, 221–234.Google Scholar
[16] Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and its Applications, 2nd edn. John Wiley, New York.Google Scholar
[17] Stoyan, D. and Stoyan, H. (2000). Improving ratio estimators of second order point process charateristics. Scand. J. Statist. 27, 619–634.CrossRefGoogle Scholar