What is a multi-parameter renewal process?

Ivanoff, B. Gail; Merzbach, Ely

doi:10.1080/17442500600965239

Cited by 13 publications

(22 citation statements)

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“…Complications arise when the dimension increases. Ivanoff and Merzbach (2006) have overcome these complications by providing a new way of looking at the renewal process which has allowed them a natural generalization into higher dimensions. The present paper will explore some of the statistical properties of this Ivanoff-Merzbach process in the two-dimensional case.…”

Section: The Ivanoff-merzbach Renewal Processmentioning

confidence: 98%

“…Generalizing higher dimensional Poisson processes to renewal processes has been a trickier problem. A renewal process in higher dimensions which specializes to a Poisson process has been introduced by Ivanoff and Merzbach (2006), and a useful approach to simulating this process has been described in Ivanoff (2008). The theory and construction of this process have been thoroughly discussed in aforementioned papers.…”

Section: Introductionmentioning

confidence: 99%

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confidence: 99%

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Modelling point patterns with the Ivanoff–Merzbach renewal process

Davies

Braun

2009

Environmetrics

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SUMMARYThis paper demonstrates how to apply the Ivanoff-Merzbach renewal process (Ivanoff and Merzbach, 2006;Ivanoff, 2008) to the modelling of environmental and other forms of point process data. The model is explored via simulation, focusing on two special cases. Maximum likelihood estimation of model parameters is described and demonstrated for these cases. The methods are applied to a point pattern related to human settlement patterns and to a point pattern describing the locations of a species of pine.

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Section: The Ivanoff-merzbach Renewal Processmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

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Modelling point patterns with the Ivanoff–Merzbach renewal process

Davies

Braun

2009

Environmetrics

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“…Definition 4.2): we have ξ 1 (M ) = ξ 1 (M 1 ) = ξ 1 (N ). As shown in Example 7.4 of [9], it is easy to see that the (…”

Section: Definition 62 Given An Arbitrary σ-Field Fmentioning

confidence: 99%

“…• It is shown in [9] that the sets ξ k (N ) and ξ + k (N ) are both F -stopping sets. As well, both are F (ξ k (N ))-measurable for every k (i.e.…”

Section: Stopping Sets and Their Distributionsmentioning

confidence: 99%

A Compensator Characterization of Planar Point Processes

Ivanoff

2015

Asymptotic Laws and Methods in Stochastics

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Martingale techniques play a fundamental role in the analysis of point processes on [0, ∞). In particular, the compensator of a point process uniquely determines and is determined by its distribution, and an explicit formula involving conditional interarrival distributions is well-known. In two dimensions there are many possible definitions of a point process compensator and we focus here on the one that has been the most useful in practice: the so-called *-compensator. Although existence of the *-compensator is well understood, in general it does not determine the law of the point process and it must be calculated on a case-by-case basis. However, it will be proven that when the point process satisfies a certain property of conditional independence (usually denoted by (F4)), the *-compensator determines the law of the point process and an explicit regenerative formula can be given. The basic building block of the planar model is the single line process (a point process with incomparable jump points). Its law can be characterized by a class of avoidance probabilities that are the two-dimensional counterpart of the survival function on [0, ∞). Conditional avoidance probabilities then play the same role in the construction of the *-compensator as conditional survival probabilities do for compensators in one dimension.

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A two-dimensional Markov renewal process

Davies

Braun

2011

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This paper presents an extension to the the Ivanoff–Merzbach renewal process. The Ivanoff–Merzbach process is a natural analogue to the one‐dimensional renewal process. Restricting our attention to two dimensions, the focus of the current paper will be formation of this extension, which we will refer to as a Markov renewal process, as well as simulation, estimation and applications to data sets in ecology and forestry. Copyright © 2010 John Wiley & Sons, Ltd.

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What is a multi-parameter renewal process?

Cited by 13 publications

References 20 publications

Modelling point patterns with the Ivanoff–Merzbach renewal process

Modelling point patterns with the Ivanoff–Merzbach renewal process

A Compensator Characterization of Planar Point Processes

A two-dimensional Markov renewal process

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