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Statistics of the Boolean model for practitioners and mathematicians. (English) Zbl 0878.62068

Wiley Series in Probability and Mathematical Statistics. Chichester: Wiley. vi, 162 p. (1997).
From the author’s introduction: This book discusses statistical models of stationary random sets, estimation of parameters and fitting appropriate stochastic models. The presentation begins with the definition of the Boolean model of random closed sets. This model emerges from the Poisson point process if the points are replaced by sets of ‘full’ dimension. The overlappings that arise constitute the main problem in statistical estimation theory and statistical inference. In fact, all model parameters can be classified as either aggregate or individual parameters. The first are determined directly by the visible set, while the latter are not directly observable and can be related to aggregate parameters through mathematical equations. Note that the estimation of aggregate parameters is similar to the estimation of time-average characteristics for time series. Then equations relating individual and aggregate parameters and the corresponding estimators are discussed and applied to statistical estimation of parameters. The corresponding problems in time series statistics are fitting an autoregression scheme or estimating the trend. After this, the parametric approach and more sophisticated sampling schemes are considered. Finally, some testing problems are briefly discussed.
The material given in this book is divided into two levels. The first consists of basic ideas, definitions, explanations, simple properties, descriptions of algorithms, and recipes of how to implement them. It deals only with the planar case and does not go deep into the mathematical background, although corresponding references are usually mentioned. For this level intuitive understanding will be mostly enough. The other part of the presentation (given as notes) assumes general dimension of the space, and contains some mathematical background, and occasionally proofs. It is always possible to skip the second part, if only implementations are of interest.
This book has emerged from a manuscript written for applied scientists and explaining how to do statistics of the Boolean model. This material will be used in the first level of presentation. However, the existence of the interesting mathematical background and the necessity of further developments were the reasons to add the second (mathematical) part aimed at statisticians and probabilists. It is necessary to stress that statistics of the Boolean model is by no means complete. New ideas must appear to compare estimators and to extend for random sets (and Boolean models) most of the well-known methods of mathematical statistics.
Contents: 1) Introduction, 2) The Boolean model; 3) Estimation of aggregate parameters; 4) Estimation of functional aggregate parameters; 5) Estimation of numerical individual parameter; 6) Estimation of set-valued individual parameters; 7) Individual parameters: distributions; 8) Other sampling schemes; 9) Testing the Boolean model assumption; 10) An example; 11) Concluding remarks.
Reviewer: V.Schmidt (Ulm)

MSC:

62M99 Inference from stochastic processes
62-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistics
60-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory
60D05 Geometric probability and stochastic geometry
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