Estimation for the simple linear Boolean model
The model presented in the paper concerns the distribution in a (\(n\)-dimensional) space (\(n = 1, 2, 3\)) of segments of (iid) lengths, located at the points of a stationary Poisson process (independent of the segments length), equivalent (from the queuing theory point of view) to a Markov/General/infinity queue. The segments may overlap and so, the space may be multiply covered or not covered at all in some points. The aim of the paper is to estimate the intensity of the Poisson process and segment length distribution from a sample `clump' and spacing lengths. The authors develop a procedure to obtain explicit representations of the `clump' length distribution and pdf, representations taking the form of an integral equation, which can be numerically solved. Moreover, they show how this methodology can be used in the case of a sample of `clumps' and spacing. As application, they consider both the measurement of particle mass flow and the estimation of the frequency and duration of a recurrent viral infection. The methodology used in this paper includes the fast Fourier transform and the maximum likelihood technique. A comparison to a discrete approximation (Handley, 2004) is also given, highlighting the idea that this continuous Boolean model has the advantage of providing accurate approximation to partial derivatives, essential to fast optimization. To conclude, the authors obtain an analytical expression for the `clump' length density and numerical solution, which can be applied with arbitrary segment length distributions.. Further work will include two or three dimensional models and a possible gateway to queuing modeling.
scientific article; zbMATH DE number 5119603