Smoothing techniques and estimation methods for nonstationary Boolean models with applications to coverage processes

Kernel smoothing methods are applied to nonparametric estimation for nonstationary Boolean models. In many applications only exposed tangent points of the models are observable rather than full realisations. Several methods are developed for estimating the distribution of the underlying Boolean model from observation of the exposed tangent points. In particular, estimation methods for coverage processes are studied in detail and applied to neurobiological data.

[1]  D. Stoyan,et al.  Estimators of distance distributions for spatial patterns , 1998 .

[2]  S. Chiu A central limit theorem for linear Kolmogorov's birth-growth models , 1997 .

[3]  A. Shiryayev On The Statistical Theory of Metal Crystallization , 1992 .

[4]  B. Ripley Statistical inference for spatial processes , 1990 .

[5]  Ilya Molchanov,et al.  Asymptotic properties of estimators for parameters of the Boolean model , 1994, Advances in Applied Probability.

[6]  R. Gill,et al.  Kaplan-Meier estimators of distance distributions for spatial point processes , 1997 .

[7]  M. Bennett,et al.  Probabilistic secretion of quanta from nerve terminals at synaptic sites on muscle cells: non-uniformity, autoinhibition and the binomial hypothesis , 1990, Proceedings of the Royal Society of London. B. Biological Sciences.

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

[9]  U. R. Evans The laws of expanding circles and spheres in relation to the lateral growth of surface films and the grain-size of metals , 1945 .

[10]  J. Simonoff Smoothing Methods in Statistics , 1998 .

[11]  Heinrich Statistics of the Boolean Model for Practitioners and Mathematicians , 1998 .

[12]  Nonparametric and Parametric Estimation for a Linear Germination‐Growth Model , 2000, Biometrics.

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

[14]  M. Avrami,et al.  Kinetics of Phase Change 2 , 1940 .

[15]  W. A. Johnson Reaction Kinetics in Processes of Nucleation and Growth , 1939 .

[16]  Ilya S. Molchanov,et al.  Statistics of the Boolean model: from the estimation of means to the estimation of distributions , 1995, Advances in Applied Probability.

[17]  Lijian Yang,et al.  Multivariate bandwidth selection for local linear regression , 1999 .

[18]  E. Gilbert Random Subdivisions of Space into Crystals , 1962 .

[19]  J. Møller,et al.  Random Johnson-Mehl tessellations , 1992, Advances in Applied Probability.

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

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

[22]  A. Bowman,et al.  Applied smoothing techniques for data analysis : the kernel approach with S-plus illustrations , 1999 .

[23]  A. Faleiros,et al.  Kinetics of phase change , 2000 .

[24]  P. J. Green,et al.  Density Estimation for Statistics and Data Analysis , 1987 .

[25]  P. Diggle A Kernel Method for Smoothing Point Process Data , 1985 .

[26]  S. Chiu Limit theorems for the time of completion of Johnson-Mehl tessellations , 1995, Advances in Applied Probability.