On Parameter Estimation for Pairwise Interaction Point Processes

1994 ◽  
Vol 62 (1) ◽  
pp. 99 ◽  
Author(s):  
Peter J. Diggle ◽  
Thomas Fiksel ◽  
Pavel Grabarnik ◽  
Yosihiko Ogata ◽  
Dietrich Stoyan ◽  
...  
Keyword(s):  
Parameter Estimation ◽  
Point Processes ◽  
Pairwise Interaction ◽  
Interaction Point

Poisson limits and nonparametric estimation for pairwise interaction point processes

2000 ◽  
Vol 37 (1) ◽  
pp. 252-260 ◽  
Author(s):  
Wei-Bin Chang ◽  
John A. Gubner
Keyword(s):  
Poisson Process ◽  
Point Processes ◽  
Pairwise Interaction ◽  
Nonparametric Estimator ◽  
Interpoint Distance ◽  
Interaction Point ◽  
Interaction Function ◽  
The Mean ◽  
Piecewise Continuous ◽  
Independent Identically Distributed

The distribution of the interpoint distance process of a sequence of pairwise interaction point processes is considered. It is shown that, if the interaction function is piecewise-continuous, then the sequence of interpoint distance processes converges weakly to an inhomogeneous Poisson process under certain sparseness conditions. Convergence of the expectation of the interpoint distance process to the mean of the limiting Poisson process is also established. This suggests a new nonparametric estimator for the interaction function if independent identically distributed samples of the point process are available.

Poisson limits and nonparametric estimation for pairwise interaction point processes

2000 ◽  
Vol 37 (01) ◽  
pp. 252-260 ◽  
Author(s):  
Wei-Bin Chang ◽  
John A. Gubner
Keyword(s):  
Poisson Process ◽  
Point Processes ◽  
Pairwise Interaction ◽  
Nonparametric Estimator ◽  
Interpoint Distance ◽  
Interaction Point ◽  
Interaction Function ◽  
The Mean ◽  
Piecewise Continuous ◽  
Independent Identically Distributed

The distribution of the interpoint distance process of a sequence of pairwise interaction point processes is considered. It is shown that, if the interaction function is piecewise-continuous, then the sequence of interpoint distance processes converges weakly to an inhomogeneous Poisson process under certain sparseness conditions. Convergence of the expectation of the interpoint distance process to the mean of the limiting Poisson process is also established. This suggests a new nonparametric estimator for the interaction function if independent identically distributed samples of the point process are available.

A nonparametric estimator for pairwise-interaction point processes

Biometrika ◽  
1987 ◽  
Vol 74 (4) ◽  
pp. 763-770 ◽  
Author(s):  
PETER J. DIGGLE ◽  
DAVID J. GATES ◽  
ALYSON STIBBARD
Keyword(s):  
Point Processes ◽  
Pairwise Interaction ◽  
Nonparametric Estimator ◽  
Interaction Point

scholarly journals Existence of Gibbs point processes with stable infinite range interaction

2020 ◽  
Vol 57 (3) ◽  
pp. 775-791
Author(s):  
David Dereudre ◽  
Thibaut Vasseur
Keyword(s):  
Point Processes ◽  
Existence Results ◽  
Pairwise Interaction ◽  
Range Interaction ◽  
Pairwise Interactions ◽  
Gibbs Point Processes ◽  
Infinite Range Interactions ◽  
Finite Configuration ◽  
Multi Body ◽  
Infinite Range

AbstractWe provide a new proof of the existence of Gibbs point processes with infinite range interactions, based on the compactness of entropy levels. Our main existence theorem holds under two assumptions. The first one is the standard stability assumption, which means that the energy of any finite configuration is superlinear with respect to the number of points. The second assumption is the so-called intensity regularity, which controls the long range of the interaction via the intensity of the process. This assumption is new and introduced here since it is well adapted to the entropy approach. As a corollary of our main result we improve the existence results by Ruelle (1970) for pairwise interactions by relaxing the superstabilty assumption. Note that our setting is not reduced to pairwise interaction and can contain infinite-range multi-body counterparts.

First-order autoregressive gamma sequences and point processes

1980 ◽  
Vol 12 (3) ◽  
pp. 727-745 ◽  
Author(s):  
D. P. Gaver ◽  
P. A. W. Lewis
Keyword(s):  
Parameter Estimation ◽  
Point Processes ◽  
Random Variables ◽  
Innovation Process ◽  
Sums Of Random Variables ◽  
First Order ◽  
Wide Range ◽  
Serial Correlations ◽  
Generating Sequences ◽  
Stieltjes Transforms

It is shown that there is an innovation process {∊n} such that the sequence of random variables {Xn} generated by the linear, additive first-order autoregressive scheme Xn = pXn-1 + ∊n are marginally distributed as gamma (λ, k) variables if 0 ≦p ≦ 1. This first-order autoregressive gamma sequence is useful for modelling a wide range of observed phenomena. Properties of sums of random variables from this process are studied, as well as Laplace-Stieltjes transforms of adjacent variables and joint moments of variables with different separations. The process is not time-reversible and has a zero-defect which makes parameter estimation straightforward. Other positive-valued variables generated by the first-order autoregressive scheme are studied, as well as extensions of the scheme for generating sequences with given marginal distributions and negative serial correlations.

First-order autoregressive gamma sequences and point processes

1980 ◽  
Vol 12 (03) ◽  
pp. 727-745 ◽  
Author(s):  
D. P. Gaver ◽  
P. A. W. Lewis
Keyword(s):  
Parameter Estimation ◽  
Point Processes ◽  
Random Variables ◽  
Innovation Process ◽  
Sums Of Random Variables ◽  
First Order ◽  
Wide Range ◽  
Serial Correlations ◽  
Generating Sequences ◽  
Stieltjes Transforms

It is shown that there is an innovation process {∊ n } such that the sequence of random variables {X n } generated by the linear, additive first-order autoregressive scheme X n = pXn-1 + ∊ n are marginally distributed as gamma (λ, k) variables if 0 ≦p ≦ 1. This first-order autoregressive gamma sequence is useful for modelling a wide range of observed phenomena. Properties of sums of random variables from this process are studied, as well as Laplace-Stieltjes transforms of adjacent variables and joint moments of variables with different separations. The process is not time-reversible and has a zero-defect which makes parameter estimation straightforward. Other positive-valued variables generated by the first-order autoregressive scheme are studied, as well as extensions of the scheme for generating sequences with given marginal distributions and negative serial correlations.

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