Kernel estimation of the diameter distribution in boolean models with spherical grains

Abstract

We investigate asymptotic properties Including MSE of kernel estimators of the second-order product density of the point process of ‘exposed tangent points’ (for given direction ) associated with a stationary d-dimensional Boolean model with convex compact grains. Under minimal conditions on the typical grain we prove that the square root of the kernel estimators is asymptotically normally distributed with constant variance which only depends on the chosen kernel function. In the particular case of spherical grains, as first shown by Molchanov and Stoyan (1994), the diameter distribution function F(t) is just equal to the product of and some function of t>=0 which can be estimated by standard methods. Using this fact we are able to derive a multivariate CLT for a suitably defined empirical diameter distribution function. Owing to this result we suggest a χ2 ‐ goodness-of-fit-test for testing a hypothetical diameter distribution.