Central limit theorem for Gibbsian U-statistics of facet processes

Abstract

Special case of a Gibbsian facet process on a fixed window with a discrete orientation distribution and with increasing intensity of the underlying Poisson process is studied. All asymptotic joint moments for interaction U-statistics are calculated and using the method of moments the central limit theorem is derived.

Planar Segment Processes with Reference Mark Distributions, Modeling and Estimation

“…and solve them for various test functions q. We take λ * from ( 14) where we insert approximation (18) for unknown g. First the density f X is estimated using a kernel estimator for directional data [11]. Then put β(a, r, C, ϕ) = exp((e a − 1)r 2 CJ(ϕ)), and we estimate C, a from the system of Takacs-Fiksel equations:…”

“…Here in the innovations equations we take score functions q(u, x) = N x (u) and q = 1, respectively, the integrals in the second term of ( 19) are evaluated by Monte Carlo method using J independent simulations of segments uniformly distributed in Y. Then we plug the estimators of C, a in a formula obtained by integrating (18):…”

“…The uniqueness issue is more complex, see [8] for Gibbs point process in R d . Here the uniqueness is not investigated, we deal with an existing stationary process in order to use the approximation (18).…”

“…Interactions enter the model by means of intersections, cf. [19], [18]. An approximation from [3] is used to avoid the problem with unknown moments.…”

Estimation of Parameters in a Planar Segment Process With a Biological Application

“…Particularly we are interested in parameter estimation in models given by a density with respect to the Poisson process (Møller and Waagepetersen, 2004). For systems of objects the disc process case was developed in Møller and Helisova (2008;, the facet process in Večeřa and Beneš (2016); Večeřa (2016), where facets are compact subsets of hyperplanes in R d . Because of problems with normalizing constant the Takacz-Fiksel method or maximum pseudolikelihood method are suitable choices, cf.…”