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Nonparametric estimation for a two-dimensional renewal process. (English) Zbl 1337.62300

Summary: This paper introduces nonparametric estimation methods for spatial point processes satisfying a particular renewal property. Martingale methods yield a unified approach for renewal processes in both one and two dimensions, and can be used for both synchronous and asynchronous data. In each case, we obtain martingale estimators of the avoidance probabilities that characterize the renewal process. Asymptotic properties of the estimators are studied analytically and empirically.

MSC:

62M30 Inference from spatial processes
60K05 Renewal theory
62G05 Nonparametric estimation
60G48 Generalizations of martingales
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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References:

[1] Andersen, P. K., Borgan, O., Gill, R. D. and Keiding, N., Statistical Models Based on Counting Processes , Springer Series in Statistics, Springer-Verlag, New York, 1993. · Zbl 0769.62061
[2] Cairoli, R. and Walsh, J. B., Stochastic integrals in the plane., Acta Math. 134, 1975, 111-183. · Zbl 0334.60026 · doi:10.1007/BF02392100
[3] Carabarin-Aguirre, A. and Ivanoff, B. G., Multidimensional hazard estimation under generalized censoring, Electronic J. of Statistics , 3, 2009, 349-375. · Zbl 1326.62200 · doi:10.1214/08-EJS340
[4] Davies, K. F. and Braun, W. J., Modelling point patterns with the Ivanoff-Merzbach renewal process., Environmetrics 21, 2010, 85-101.
[5] Davies, K. F. and Braun, W. J., A two-dimensional Markov renewal process., Environmetrics 22, 2011, 98-108. · doi:10.1002/env.1049
[6] Gill, R. D., van der Laan, L. J. and Wellner, J. A., Inefficient estimators of the bivariate survival function for three models, Ann. Inst. Herni Poincaré 31, 1995, 545-597. · Zbl 0855.62024
[7] Ivanoff, B. G., Multiparameter renewal processes and simulations of forest fires, Environ. Ecol. Stat. 16, 2009, 153-171. · doi:10.1007/s10651-007-0082-4
[8] Ivanoff, B. G. and Merzbach, E., Random clouds and an application to censoring in survival analysis, Stochastic Process. Appl. 111, 2004, 259-279. · Zbl 1075.62088 · doi:10.1016/j.spa.2004.01.004
[9] Ivanoff, B. G. and Merzbach, E., What is a multiparameter renewal process?, Stochastics 78, 2006, 411-441. · Zbl 1109.60070 · doi:10.1080/17442500600965239
[10] Ivanoff, B. G., Merzbach, E. and Plante, M., A compensator characterization of point processes on topological lattices, Electronic J. of Probability 12, 2007, 47-74. · Zbl 1127.60085 · doi:10.1214/EJP.v12-390
[11] Karr, A. F., Point Processes and Their Statistical Inference, second ed. , Marcel Dekker, New York, 1991. · Zbl 0733.62088
[12] van der Vaart, A. W. and Wellner, J. A., Weak Convergence and Empirical Processes , Springer-Verlag, New York, 1996. · Zbl 0862.60002
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