Abstract
We consider a flexible class of space-time point process models—inhomogeneous shot-noise Cox point processes. They are suitable for modelling clustering phenomena, e.g. in epidemiology, seismology, etc. The particular structure of the model enables the use of projections to the spatial and temporal domain. They are used to formulate a stepwise estimation method to estimate different parts of the model separately. In the first step, the Poisson likelihood approach is used to estimate the inhomogeneity parameters. In the second and third steps, the minimum contrast estimation based on K-functions of the projected processes is used to estimate the interaction parameters. We study the asymptotic properties of the resulting estimators and formulate a set of conditions sufficient for establishing consistency and asymptotic normality of the estimators under the increasing domain asymptotics.
Similar content being viewed by others
References
A. J. Baddeley, J. Møller, R. Waagepetersen: Non- and semi-parametric estimation of interaction in inhomogeneous point patterns. Stat. Neerl. 54 (2000), 329–350.
D. J. Daley, D. Vere-Jones: An Introduction to the Theory of Point Processes. Vol. II: General Theory and Structure. Probability and Its Applications, Springer, New York, 2008.
P. J. Diggle: Spatio-temporal point processes: methods and applications. Statistical Methods for Spatio-temporal Systems (B. Finkenstädt, et al., eds.). Selected Invited Papers Based on the Presentations at the 6th Séminaire Européen de Statistique Sem-Stat Held as a Summer School of the European Mathematical Society, Bernried, 2004, Chapman and Hall/CRC, Boca Raton, 2007, Monographs on Statistics and Applied Probability 107 (2007), 1–45.
P. Doukhan: Mixing: Properties and Examples. Lecture Notes in Statistics 85, Springer, New York, 1994.
J. Dvořák, M. Prokešová: Parameter estimation for inhomogeneous space-time shot-noise Cox point processes. To appear in Scand. J. Stat.
E. Gabriel: Estimating second-order characteristics of inhomogeneous spatio-temporal point processes. Methodol. Comput. Appl. Probab. 16 (2014), 411–431.
E. Gabriel, P. J. Diggle: Second-order analysis of inhomogeneous spatio-temporal point process data. Stat. Neerl. 63 (2009), 43–51.
X. Guyon: Random Fields on a Network. Modeling, Statistics, and Applications. Probability and Its Applications, Springer, New York, 1995.
W. W. Hager: Minimizing a quadratic over a sphere. SIAM J. Optim. 12 (2001), 188–208.
G. Hellmund, M. Prokešová, E. B. V. Jensen: Lévy-based Cox point processes. Adv. Appl. Probab. 40 (2008), 603–629.
Z. Karácsony: A central limit theorem for mixing random fields. Miskolc Math. Notes 7 (2006), 147–160.
Th. Motzkin: From among n conjugate algebraic integers, n - 1 can be approximately given. Bull. Am. Math. Soc. 53 (1947), 156–162.
J. Møller: Shot noise Cox processes. Adv. Appl. Probab. 35 (2003), 614–640.
J. Møller, M. Ghorbani: Aspects of second-order analysis of structured inhomogeneous spatio-temporal point processes. Stat. Neerl. 66 (2012), 472–491.
J. Møller, R. P. Waagepetersen: Statistical Inference and Simulation for Spatial Point Processes. Monographs on Statistics and Applied Probability 100, Chapman & Hall/CRC, Boca Raton, 2004.
M. Prokešová, J. Dvořák: Statistics for inhomogeneous space-time shot-noise Cox processes. Methodol. Comput. Appl. Probab. 16 (2014), 433–449.
M. Prokešová, J. Dvořák, E. B. V. Jensen: Two-step estimation procedures for inhomogeneous shot-noise Cox processes. To appear in Ann. Inst. Stat. Math.
B. D. Ripley: Statistical Inference for Spatial Processes. Cambridge University Press, Cambridge, 1988.
D. Stoyan, W. S. Kendall, J. Mecke: Stochastic Geometry and Its Applications. Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Chichester, 1995.
A. W. van der Vaart: Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics 3, Cambridge University Press, Cambridge, 1998.
R. Waagepetersen, Y. Guan: Two-step estimation for inhomogeneous spatial point processes. J. R. Stat. Soc., Ser. B, Stat. Methodol. 71 (2009), 685–702.
Author information
Authors and Affiliations
Corresponding author
Additional information
The research has been supported by the Czech Science Foundation, project no. 16-03708S.
Rights and permissions
About this article
Cite this article
Dvořák, J., Prokešová, M. Asymptotic properties of the minimum contrast estimators for projections of inhomogeneous space-time shot-noise Cox processes. Appl Math 61, 387–411 (2016). https://doi.org/10.1007/s10492-016-0138-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10492-016-0138-6
Keywords
- space-time point process
- shot-noise Cox process
- minimum contrast estimation
- projection process
- increasing domain asymptotics