×

zbMATH — the first resource for mathematics

Contrast estimation for parametric stationary determinantal point processes. (English) Zbl 1361.60034
Summary: We study minimum contrast estimation for parametric stationary determinantal point processes. These processes form a useful class of models for repulsive (or regular, or inhibitive) point patterns and are already applied in numerous statistical applications. Our main focus is on minimum contrast methods based on the Ripley’s \(K\)-function or on the pair correlation function. Strong consistency and asymptotic normality of theses procedures are proved under general conditions that only concern the existence of the process and its regularity with respect to the parameters. A key ingredient of the proofs is the recently established Brillinger mixing property of stationary determinantal point processes. This work may be viewed as a complement to the study of Y. Guan and M. Sherman who establish the same kind of asymptotic properties for a large class of Cox processes, which in turn are models for clustering (or aggregation).

MSC:
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60F05 Central limit and other weak theorems
62F10 Point estimation
62F12 Asymptotic properties of parametric estimators
62E20 Asymptotic distribution theory in statistics
Software:
R; spatstat
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Baddeley, Spatial point patterns: Methodology and applications (2015) · Zbl 1296.60125
[2] Baddeley, spatstat: An R package for analyzing spatial point patterns, J. Stat. Software 12 pp 1– (2005) · doi:10.18637/jss.v012.i06
[3] Billingsley, Probability and measure (1979) · Zbl 0822.60002
[4] Biscio, Brillinger mixing of determinantal point processes and statistical applications, Electron. J. Stat. 10 pp 582– (2016a) · Zbl 1403.60039 · doi:10.1214/16-EJS1116
[5] Biscio, Quantifying repulsiveness of determinantal point processes, Bernoulli 22 pp 2001– (2016b) · Zbl 1343.60058 · doi:10.3150/15-BEJ718
[6] Daley, An Introduction to the theory of point processes, elementary theory and methods I (2003) · Zbl 1026.60061
[7] Daley, An introduction to the theory of point processes, general theory and structure II (2008) · Zbl 1159.60003 · doi:10.1007/978-0-387-49835-5
[8] Deng, The Ginibre point process as a model for wireless networks with repulsion, IEEE Trans. Wireless Commun. 14 pp 107– (2015) · doi:10.1109/TWC.2014.2332335
[9] Diggle, Statistical analysis of spatial point patterns (2003) · Zbl 1021.62076
[10] Guan, On least squares fitting for stationary spatial point processes, J. R. Stat. Soc. Ser. B. Stat Method. 69 pp 31– (2007)
[11] Heinrich , L. 1992 Minimum contrast estimates for parameters of spatial ergodic point processes Transactions of the 11th Prague Conference on Information Theory, Statistical Decision Functions, Random processes 479 492 Prague · Zbl 0770.62082
[12] Heinrich, Lect. Notes Math., in: Stochastic Geometry, Spatial Statistics and Random Fields pp 115– (2013) · Zbl 1296.62163 · doi:10.1007/978-3-642-33305-7_4
[13] Hough, University Lecture Series 51, in: Zeros of Gaussian analytic functions and determinantal point processes (2009) · doi:10.1090/ulect/051
[14] Jolivet, Colloq. Math. Soc. jános Bolyai, in: Point processes and queuing problems (Colloqium, Keszthely, 1978) pp 117– (1981)
[15] Krickeberg, Lecture Notes in Math., in: Tenth Saint Flour Probability Summer School-1980 (Saint Flour, 1980) pp 205– (1982) · doi:10.1007/BFb0095620
[16] Kulesza, Determinantal point processes for machine learning, Foundations and Trends in Machine Learning 5 pp 123– (2012) · Zbl 1278.68240 · doi:10.1561/2200000044
[17] Lavancier, Modelling aggregation on the large scale and regularity on the small scale in spatial point pattern datasets, Scand. J. Stat. 43 pp 587– (2016) · Zbl 1419.62268 · doi:10.1111/sjos.12193
[18] Lavancier, Determinantal point process models and statistical inference : Extended version, arXiv:1205.4818v5 (2014)
[19] Lavancier, Determinantal point process models and statistical inference, J. R. Stat. Soc. Ser. B. Stat Method. 77 pp 853– (2015) · doi:10.1111/rssb.12096
[20] Lavancier, A general procedure to combine estimators, Comput. Stat. Data Anal. 94 pp 175– (2016) · Zbl 06918660 · doi:10.1016/j.csda.2015.08.001
[21] Macchi, The coincidence approach to stochastic point processes, Adv. Appl. Probab. 7 pp 83– (1975) · Zbl 0366.60081 · doi:10.1017/S0001867800040313
[22] Miyoshi , N. Shirai , T. 2013 A cellular network model with Ginibre configurated base stations · Zbl 1344.60050
[23] Møller, Monographs on Statistics and Applied Probability 100, in: Statistical inference and simulation for spatial point processes (2004)
[24] Nguyen, Ergodic theorems for spatial processes, Z. Wahrsch. Verw. Gebiete 48 pp 133– (1979) · Zbl 0397.60080 · doi:10.1007/BF01886869
[25] Pfanzagl, On measurability and consistency of minimum contrast estimates, Metrika 14 pp 249– (1969) · Zbl 0181.45501 · doi:10.1007/BF02613654
[26] R Core Team, R: A language and environment for statistical computing (2016)
[27] Sasvári, Multivariate characteristic and correlation functions 50 (2013) · Zbl 1276.62034 · doi:10.1515/9783110223996
[28] Shirai, Random point fields associated with certain Fredholm determinants. I. Fermion, Poisson and boson point processes, J. Funct. Anal. 205 pp 414– (2003) · Zbl 1051.60052 · doi:10.1016/S0022-1236(03)00171-X
[29] Soshnikov, Determinantal random point fields, Russ. Math. Surv. 55 pp 923– (2000) · Zbl 0991.60038 · doi:10.1070/RM2000v055n05ABEH000321
[30] Soshnikov, Gaussian limit for determinantal random point fields, Ann. Probab. 30 pp 171– (2002) · Zbl 1033.60063 · doi:10.1214/aop/1020107764
[31] Stein, Introduction to Fourier analysis on Euclidean spaces (PMS-32) 1 (1971)
[32] Vaart, Cambridge Series in Statistical and Probabilistic Mathematics 3, in: Asymptotic statistics (1998) · Zbl 0910.62001 · doi:10.1017/CBO9780511802256
[33] Waagepetersen, Two-step estimation for inhomogeneous spatial point processes, J. R. Stat. Soc. Ser. B. Stat Method. 71 pp 685– (2009) · Zbl 1250.62047 · doi:10.1111/j.1467-9868.2008.00702.x
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.