×

Asymptotic goodness-of-fit tests for point processes based on scaled empirical \(K\)-functions. (English) Zbl 1404.62022

In this paper methods of geometrical statistics are considered. Sequences of scaled edge-corrected empirical \(K\)-functions (modifying Ripley’s \(K\)-function) are studied. The Ripley’s \(K\)-function is a spatial analysis method. It is used to describe how point patterns occur over a given area of interest. The aim of paper is to establish asymptotic goodness-of-fit tests for checking random point process hypotheses provided the hypothesized \(d\)-dimensional simple point process with distribution \(P\) is fourth-order stationary with known intensity and a known generalized \(K\)-function. There are many interesting results in this article worth reading.

MSC:

62F05 Asymptotic properties of parametric tests
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60F17 Functional limit theorems; invariance principles
60D05 Geometric probability and stochastic geometry
62M30 Inference from spatial processes

Software:

GET
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Daley, DJ; Vere-Jones, D., An introduction to the theory of point processes, (1988), Springer, New York · Zbl 0657.60069
[2] Ripley, BD., The second-order analysis of stationary point processes, J Appl Probab, 13, 2, 255-266, (1976) · Zbl 0364.60087 · doi:10.2307/3212829
[3] Ohser, J; Stoyan, D., On the second-order and orientation analysis of planar stationary point processes, Biom J, 23, 6, 523-533, (1981) · Zbl 0494.60048 · doi:10.1002/bimj.4710230602
[4] Stein, ML., Asymptotically optimal estimation for the reduced second moment measure of point processes, Biometrika, 80, 2, 443-449, (1993) · Zbl 0779.62092 · doi:10.1093/biomet/80.2.443
[5] Diggle, PJ., Statistical analysis of spatial point patterns, (2003), Arnold, London · Zbl 1021.62076
[6] Chiu, SN; Stoyan, D; Kendall, WS, Stochastic geometry and its applications, (2013), Wiley, Chichester · Zbl 1291.60005
[7] Illian, J; Penttinen, A; Stoyan, H, Statistical analysis and modelling of spatial point patterns, (2008), Wiley, Chichester · Zbl 1197.62135
[8] Guan, Y; Sherman, M; Calman, JA., Assessing isotropy for spatial point processes, Biometrics, 62, 1, 119-125, (2006) · Zbl 1091.62096 · doi:10.1111/j.1541-0420.2005.00436.x
[9] Guan, Y; Sherman, M., On least squares Fitting for stationary spatial point processes, J R Statist Soc Ser B, 69, 1, 31-49, (2007) · Zbl 07555348 · doi:10.1111/j.1467-9868.2007.00575.x
[10] Baddeley, AJ; Møller, J; Waagepetersen, R., Non- and semi-parametric estimation of interaction in inhomogeneous point patterns, Statist Neerl, 54, 2, 329-350, (2000) · Zbl 1018.62027 · doi:10.1111/1467-9574.00144
[11] Gaetan, C; Guyon, X., Spatial statistics and modeling, (2010), Springer, New York · Zbl 1271.62214
[12] Adelfio, G; Schoenberg, FP., Point process diagnostics based on weighted second-order statistics and their asymptotic properties, Ann Inst Statist Math, 61, 4, 929-948, (2009) · Zbl 1332.60070 · doi:10.1007/s10463-008-0177-1
[13] Zhao, J; Wang, J., Asymptotic properties of an empirical K-function for inhomogeneous spatial point processes, Statistics, 44, 3, 261-267, (2010) · Zbl 1282.62052 · doi:10.1080/02331880903024132
[14] Billingsley, P., Convergence of probability measures, (1968), Wiley, New York · Zbl 0172.21201
[15] Heinrich, L., Goodness-of-fit tests for the second moment function of a stationary multidimensional Poisson process, Statistics, 22, 2, 245-268, (1991) · Zbl 0809.62075 · doi:10.1080/02331889108802308
[16] Asymptotic methods in statistics of random point processes. In: Spodarev E, editor. Stochastic geometry, spatial statistics and random fields. New York: Springer; 2013. p. 115-150 (Lecture Notes in Mathematics; 2068) · Zbl 1296.62163
[17] Heinrich, L., Gaussian limits of empirical multiparameter K-functions of homogeneous Poisson processes and tests for complete spatial randomness, Lithuan Math J, 55, 1, 72-90, (2015) · Zbl 1319.60068 · doi:10.1007/s10986-015-9266-z
[18] Schoenberg, FP., Transforming spatial point processes into Poisson processes, Stoch Process Appl, 81, 2, 155-164, (1999) · Zbl 0962.60029 · doi:10.1016/S0304-4149(98)00098-2
[19] Ho, LP; Chiu, SN., Testing the complete spatial randomness by Diggle’s test without an arbitrary upper limit, J Statist Comput Simul, 76, 7, 585-591, (2006) · Zbl 1089.62111 · doi:10.1080/00949650412331321043
[20] Marcon, E; Traissac, S; Lang, G., A statistical test for Ripley’s K-function rejection of Poisson hypothesis, ISRN Ecol, 2013, 1-9, (2013) · doi:10.1155/2013/753475
[21] Wiegand, T; Grabarnik, P; Stoyan, D., Envelope tests for spatial point patterns with and without simulation, Ecosphere, 7, 6, 641-656, (2016) · doi:10.1002/ecs2.1365
[22] Central limit theorem and convergence of empirical processes for stationary point processes. In: Bartfai P, Tomko J, editors. Point processes and queueing problems, 24th colloquia mathematica societatis J Bolyai; 1978; Keszthely, Hungary. Amsterdam: North-Holland; 1981. p. 117-161 · Zbl 0474.60040
[23] Karr, AF., Estimation of palm measures of stationary point processes, Probab Theory Relat Fields, 74, 1, 55-69, (1987) · Zbl 0586.60043 · doi:10.1007/BF01845639
[24] Kiêu, K; Mora, M., Estimating the reduced moments of a random measure, Adv Appl Probab, 31, 1, 48-62, (1999) · Zbl 0926.62093 · doi:10.1239/aap/1029954265
[25] Asymptotic goodness-of-fit tests for stationary point processes based on scaled empirical K-functions. arXiv:1706.01074v1 [math.ST], 33 pages, submitted 4 June 2017
[26] Heinrich, L., On the strong brillinger-mixing property of α-determinantal point processes with some applications, Appl Math, 61, 4, 443-461, (2016) · Zbl 1488.60126 · doi:10.1007/s10492-016-0141-y
[27] Biscio, CAN; Lavancier, F., Brillinger mixing of determinantal point processes and statistical applications, Electron J Statist, 10, 1, 582-607, (2016) · Zbl 1403.60039 · doi:10.1214/16-EJS1116
[28] Biscio, CAN; Lavancier, F., Contrast estimation for parametric stationary determinantal point processes, Scand J Statist, 44, 1, 204-229, (2017) · Zbl 1361.60034 · doi:10.1111/sjos.12249
[29] Baddeley, A; Silverman, BW., A cautionary example for the use of second-order methods for analyzing point patterns, Biometrics, 40, 4, 1089-1094, (1984) · doi:10.2307/2531159
[30] Heinrich, L., Normal approximation for some mean-value estimates of absolutely regular tessellations, Math Methods Statist, 3, 1, 1-24, (1994) · Zbl 0824.60011
[31] Heinrich, L; Prokešová, M., On estimating the asymptotic variance of stationary point processes, Methodol Comput Appl Probab, 12, 3, 451-471, (2010) · Zbl 1197.62122 · doi:10.1007/s11009-008-9113-3
[32] Heinrich, L., Asymptotic gaussianity of some estimators for reduced factorial moment measures and product densities of stationary Poisson cluster processes, Statistics, 19, 1, 87-106, (1988) · Zbl 0666.62032 · doi:10.1080/02331888808802075
[33] Heinrich, L; Schmidt, V., Normal convergence of multidimensional shot noise and rates of this convergence, Adv Appl Probab, 17, 4, 709-730, (1985) · Zbl 0609.60036 · doi:10.2307/1427084
[34] Myllymäki, M; Mrkvička, T; Grabarnik, P, Global envelope tests for spatial processes, J R Statist Soc Ser B, 79, 2, 381-404, (2017) · Zbl 1414.62404 · doi:10.1111/rssb.12172
[35] Doss, H., On estimating the dependence between two point processes, Ann Statist, 17, 2, 749-763, (1989) · Zbl 0672.62088 · doi:10.1214/aos/1176347140
[36] Hadwiger, H., Vorlesungen über Inhalt, Oberfläche und Isoperimetrie, (1957), Springer, Berlin · Zbl 0078.35703
[37] Heinrich, L; Pawlas, Z., Weak and strong convergence of empirical distribution functions from germ-grain processes, Statistics, 42, 1, 49-65, (2008) · Zbl 1151.62039 · doi:10.1080/02331880701538531
[38] Wills, JM., Zum verhältnis volumen zu oberfläche bei konvexen Körpern, Arch Math, 21, 5, 557-560, (1970) · Zbl 0204.55204 · doi:10.1007/BF01220963
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.