Asymptotic properties of estimators for parameters of the Boolean model

This paper considers estimators of parameters of the Boolean model which are obtained by means of the method of intensities. For an estimator of the intensity of the point process of germ points the asymptotic normality is proved and the corresponding variance is given. The theory is based on a study of second-order characteristics of the point process of lower-positive tangent points of the Boolean model. An estimator of the distribution of a typical grain is also discussed.

[1]  Dietrich Stoyan,et al.  Stereology for pores in wheat bread: statistical analyses for the Boolean model by serial sections , 1991 .

[2]  L. Heinrich,et al.  Asymptotic gaussianity of some estimators for reduced factorial moment measures and product densities of stationary poisson cluster processes , 1988 .

[3]  Thomas Fiksel,et al.  Edge-corrected density estimators for point processes , 1988 .

[4]  E. Jolivet,et al.  Upper bound of the speed of convergence of moment density Estimators for stationary point processes , 1984 .

[5]  Wolfgang Weil,et al.  Densities for stationary random sets and point processes , 1984, Advances in Applied Probability.

[6]  B. Ripley,et al.  Introduction to the Theory of Coverage Processes. , 1989 .

[7]  Wolfgang Weil,et al.  Expectation formulas and isoperimetric properties for non‐isotropic Boolean models , 1988 .

[8]  D. Stoyan,et al.  Stochastic Geometry and Its Applications , 1989 .

[9]  I. Molchanov.  Handling with spatial censored observations in statistics of Boolean models of random sets , 1992 .

[10]  H. Wendrock,et al.  Estimation variances for estimators of product densities and pair correlation functions of planar point processes , 1993 .

[11]  Noel A Cressie,et al.  Statistics for Spatial Data. , 1992 .

[12]  D. Stoyan,et al.  Formulas for stationary planar fibre processes I - general theory , 1980 .

[13]  E. CastroPeter,et al.  Infinitely Divisible Point Processes , 1982 .

[14]  Jean Serra,et al.  Image Analysis and Mathematical Morphology , 1983 .

[15]  L. Heinrich.  Asymptotic properties of minimum contrast estimators for parameters of boolean models , 1993 .

[16]  ASYMPTOTIC PROPERTIES OF STEREOLOGICAL ESTIMATORS OF VOLUME FRACTION FOR STATIONARY RANDOM SETS , 1982 .

[17]  L. Heinrich.  Asymptotic Behaviour of an Empirical Nearest‐Neighbour Distance Function for Stationary Poisson Cluster Processes , 1988 .

[18]  K. Krickeberg,et al.  Processus ponctuels en statistique , 1982 .

[19]  A. Kellerer.  The variance of a Poisson process of domains , 1986, Journal of Applied Probability.

[20]  Michel Schmitt.  Estimation of the density in a stationary Boolean model , 1991 .

[21]  G. Matheron.  Random Sets and Integral Geometry , 1976 .

[22]  W. Weil.  Stereology: A Survey for Geometers , 1983 .

[23]  Dietrich Stoyan,et al.  Fraktale, Formen, Punktfelder : Methoden der Geometrie-Statistik , 1992 .

[24]  D. Stoyan,et al.  Directional analysis of fibre processes related to Boolean models , 1994 .

[25]  Albrecht M. Kellerer,et al.  Counting figures in planar random configurations , 1985, Journal of Applied Probability.

[26]  N. Cressie,et al.  Random set theory and problems of modeling , 1987 .

[27]  Ilya Molchanov,et al.  Uniform Laws of Large Numbers for Empirical Associated Functionals of Random Closed Sets , 1988 .