Log-Gaussian Cox processes in infinite-dimensional spaces
Authors:
A. Torres, M. P. Frías and M. D. Ruiz-Medina
Journal:
Theor. Probability and Math. Statist. 95 (2017), 173-193
MSC (2010):
Primary 60G55, 60J60, 60J05, 60J70
DOI:
https://doi.org/10.1090/tpms/1028
Published electronically:
February 28, 2018
MathSciNet review:
3631650
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Additional Information
Abstract: This paper introduces new results on doubly stochastic Poisson processes, with log-Gaussian Hilbert-valued random intensity (LGHRI), defined from the Ornstein–Uhlenbeck process (O-U process) in Hilbert spaces. Sufficient conditions are derived for the existence of a counting measure on $\ell ^{2}$ for this type of doubly stochastic Poisson processes. Functional parameter estimation and prediction is achieved from the discrete-time approximation of the Hilbert-valued O-U process by an autoregressive Hilbertian process of order one (ARH(1) process). The results derived are applied to functional prediction of spatiotemporal log-Gaussian Cox processes, and an application to functional disease mapping is developed. The numerical results given, from the conditional simulation study undertaken, are compared to those obtained when the random intensity is assumed to be a spatiotemporal long-range dependence (LRD) log-Gaussian process (see [19]).
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- P. Brémaud, Point Processes and Queues: Martingale Dynamics, Springer-Verlag, New York, 1972. MR 636252
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- J. Cox, J. Ingersoll, and S. Ross, A theory of the term structure of interest rates, Econometrica 53 (1985), 385–408. MR 785475
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- D. Duffie and R. Kan, A yield-factor model of interest rates, Mathematical Finance 6 (1996), 379–406.
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- J. Grandell, Doubly Stochastic Process, Springer-Verlag, New York, 1976. MR 0433591
- C. Heil and F. Walnut, Fundamental Papers in Wavelet Theory, Princeton University Press, Oxford, 2006. MR 2229251
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- N. Leonenko and E. Merzbach, Fractional Poisson fields, Methodology and Computing in Applied Probability 17 (2015), 155–168. MR 3306677
- N. Leonenko, E. Scalas, and M. Trinh, The fractional non-homogeneous Poisson process, Statist. Probab. Lett. 120 (2017), 147–156. MR 3567934
- D. Lando, On Cox processes and credit risky securities, Review of Derivatives Research 2 (1998), 99–120.
- M. M. Meerschaert, E. Nane, and P. Vellaisamy, The fractional Poisson process and the inverse stable subordinator, Electron. J. Probab. 16 (2011), 1600–1620. MR 2835248
- J. Moller, A. R. Syversveen, and R. Waagepetersen, Log Gaussian Cox processes, Scandinavian Journal of Statistics 25 (1998), 451–482. MR 1650019
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- G. Peccati and M. S. Taqqu, Wiener Chaos: Moments, Cumulants and Diagrams, Springer, New York, 2011. MR 2791919
- S. L. Rathbun and N. Cressie, A space-time survival point process for a longleaf pine forest in Southern Georgia, Journal of the American Statistical Association 89 (1994), 1164–1174.
- M. Riedle, Cylindrical Wiener processes, Research Report No. 7, Probability and Statistics Group School of Mathematics, The University of Manchester, Manchester, 2008.
- M. D. Ruiz-Medina, Functional analysis of variance for Hilbert-valued multivariate fixed effect models, Statistics 50 (2016), 689–715. MR 3506664
- M. D. Ruiz-Medina, E. Romano, and R. Fernández-Pascual, Plug-in interval prediction for a special class of standard ARH(1) processes, Journal of Multivariate Analysis 146 (2016), 138–150. MR 3477655
- G. Wei, P. Clifford, and J. Feng, Population death sequences and Cox processes driven by interacting Feller diffusions, Journal of Physics A: Mathematical and General 35 (2002), 9–31. MR 1956765
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Additional Information
A. Torres
Address at time of publication:
Department of Statistics and O.R., University of Granada, Granada, Spain
Email:
atisignes@gmail.com
M. P. Frías
Address at time of publication:
Department of Statistics and O.R., University of Jaén, Jaén, Spain
Email:
mpfrias@ujaen.es
M. D. Ruiz-Medina
Address at time of publication:
Department of Statistics and O.R., University of Granada, Granada, Spain
Email:
mruiz@ugr.es
Keywords:
ARH(1) process,
Hilbert-valued O-U process,
infinite dimension,
parameter estimation and prediction,
spatiotemporal log-Gaussian Cox process
Received by editor(s):
November 7, 2016
Published electronically:
February 28, 2018
Article copyright:
© Copyright 2018
American Mathematical Society