Estimation for the simple linear Boolean model
type: Item
instance of: scholarly article
title: Estimation for the simple linear Boolean model
zbMATH Open document ID: 1110.62112
author: Catherine M. Crespi
author: Kenneth Lange
published in: Methodology and Computing in Applied Probability
publication date: 2007-01-29T00:00:00Z
full work available at URL: 8qp6v3hh
review text: 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.
reviewed by: Florin Gorunescu
Mathematics Subject Classification ID: 62M09
Mathematics Subject Classification ID: 60K25
Mathematics Subject Classification ID: 65C60
Mathematics Subject Classification ID: 65C50
Mathematics Subject Classification ID: 65R20
zbMATH DE Number: 5119603
zbMATH Keywords: Boolean model
zbMATH Keywords: Coverage process
zbMATH Keywords: Markov/General/\(\infty\) queue
zbMATH Keywords: Type II counter
MaRDI profile type: Publication
OpenAlex ID: W2134233176
cites work: Non-parametric estimation for the \(M/G/\infty\) queue
cites work: A Queueing Model for Chronic Recurrent Conditions under Panel Observation
cites work: The busy period of the M/GI/\(\infty\) queue
cites work: Q4040734
cites work: Computationally efficient approximate likelihood procedures for the Boolean model
cites work: Q4101255
cites work: Numerical Analysis for Statisticians
cites work: Statistics of the Boolean model for practitioners and mathematicians
cites work: Q4344703
cites work: Mathematics of genome analysis
cites work: Q3997917
cites work: Q5526189
cites work: The busy period of the queueing system M/G/∞
cites work: A Combinatorial Method in the Theory of Queues
cites work: Q3263029
DOI: 10.1007/S11009-006-0429-6
description: scientific article; zbMATH DE number 5119603