A note on weighted third-order statistics for spatial point processes

  • Giada Adelfio Dipartimento di scienze economiche aziendali e statistiche, università di Palermo
Keywords: Third-order statistics, intensity function, point processes


In this paper a weighted version of the directional K-statistics, that is the T function, is introduced. The weighted directional statistics is obtained by weighting points by the inverse of the conditional intensity function of the generating point process. Some theoretical results are also provided as a generalization of theorems introduced in Adelfio and Schoenberg (2009).


Adelfio, G., Chiodi, M., 2009. Second-order diagnostics for space-time point processes with application to seismic events. Environmetrics 20, 895 - 911
Adelfio, G., Schoenberg, F. P., 2009. Point process diagnostics based on weighted second-order statistics and their asymptotic properties. Annals of the Institute of Statistical Mathematics 61 (4), 929–948.
Adelfio, G., Siino, M., Mateu, J. et al. 2020. Some properties of local weighted second-order statistics for spatio-temporal point processes. Stoch Environ Res Risk Assess 34, 149 - 168.
Baddeley, A. J., Silverman, B. W., 1984. A cautionary example on the use of second-order methods for analyzing point patterns. Biometrics 40, 1089–1093.
Cressie, N. , Collins, L. B., 2001. Analysis of spatial point patterns using bundles of product density lisa functions. Journal of Agricultural, Biological, and Environmental Statistics 6, 118–135.
Daley, D. J., Vere-Jones, D., 2003. An introduction to the theory of point processes, 2nd Edition. New York: Springer-Verlag.
Meyer, P., 1971. Dèmonstration simplifée d’un théorème de knight Lecture Notes in Mathematics 191, 191–195.
Ogata, Y., 1988. Statistical models for earthquake occurrences and residual analysis for point processes, Journal of the American Statistical Association 83, 9–27.
Ripley, B. D., 1976. The second-order analysis of stationary point processes. Journal of Applied Probability 13 (2), 255–266.
Ripley, B., 1977. Modelling spatial patterns (with discussion) Journal of the Royal Statistical Society, Series B 39, 172–212.
Schladitz, K., Baddeley, A. J., 2000. A third order point process characteristic. Scandinavian Journal of Statistics 27, 657–671.
Schoenberg, F. P., Stochastic Processes and their Applications 81 (1999) 155–164.
Schoenberg, F. P. , 2003. Multi-dimensional residual analysis of point process models for 420 earthquake occurrences Journal American Statistical Association 98, 789–795.
Stoyan, D., Kendall, W. S., Mecke, J., 1995. Stochastic geometry and its applications. Wiley, Chichester.
Stoyan, D., Stoyan, H., 1994. Fractals, random shapes and point fields. Wiley, Chichester.
Veen, A.,2006. Some Methods of Assessing and Estimating Point Processes Models for Earthquake Occurrences, Ph.D. thesis, UCLA.
Veen, A., F. P. Schoenberg, 2015. Assessing spatial point process models using weighted k-functions: Analysis of california earthquakes UC’s eScholarship Repository.
Velázquez, E., Martı́nez, I., Getzin, S., Moloney, K. A., & Wiegand, T. 2016. An evaluation of the state of spatial point pattern analysis in ecology. In Ecography. https://doi.org/10.1111/ecog.01579
Zhuang, J. , 2006. Second-order residual anaysis of spatio-timporal point processes and applications in model evaluation Journal of the Royal Statistical Society, Series B 68, 635–653.