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Independence of thinned processes characterizes the Poisson process: an elementary proof and a statistical application. (English) Zbl 1119.62089
Summary: Let \(N, N_{1}\) and \(N_{2}\) be point processes such that \(N_{1}\) is obtained from \(N\) by homogeneous independent thinning and \(N_{2}=N - N _{1}\). We give a new elementary proof that \(N_{1}\) and \(N_{2}\) are independent if and only if \(N\) is a Poisson point process. We also present an application of this result to test if a homogeneous point process is a Poisson point process.

MSC:
62M30 Inference from spatial processes
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
62M07 Non-Markovian processes: hypothesis testing
62P10 Applications of statistics to biology and medical sciences; meta analysis
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References:
[1] Besag J, Diggle PJ (1977) Simple Monte Carlo tests for spatial patterns. Appl Stat 26:327–333
[2] Chetwynd AG, Diggle PJ (1998) On estimating the reduced second moment measure of a stationary point process. Aust New Zeal J Stat 40:11–15 · Zbl 0924.62098
[3] Cressie N (1993) Statistics for spatial data, revised edn. Wiley, New York
[4] Daley DJ, Vere-Jones D (1988) An introduction to the theory of point processes, 2nd edn. Springer, New York · Zbl 0657.60069
[5] Diggle PJ (1990) A point process modelling approach to raised incidence of a rare phenomenon in the vicinity of a pre-specified point. J Roy Stat Soc Ser A 153:349–362 · Zbl 04579516
[6] Diggle PJ (1993) Point process modelling in environmental epidemiology. In: Barnett V, Turkman K (eds) Statistics for the environment. Wiley, Chichester, pp 89–110
[7] Diggle PJ (2003) Statistical analysis of spatial point patterns. Arnold, London · Zbl 1021.62076
[8] Diggle PJ, Rowlingson BS (1994) A conditional approach to point process modelling of raised incidence. J Roy Stat Soc Ser A 157:433–440 · Zbl 04522680
[9] Fichtner VK-H (1975) Charakterisierung Poissonscher zufälliger Punkfolgen und infinitesemale Verdünnungsschemata. Math Nachr 193:93–104 · Zbl 0328.60033
[10] Hanisch KH, Stoyan D (1979) Formulas for second-order analysis of marked point processes. Math Oper Stat Ser Stat 10:555–560 · Zbl 0451.62066
[11] Kelsall J, Diggle PJ (1995) Kernel estimation of relative risk. Bernoulli 1:3–16 · Zbl 0830.62039
[12] Lotwick HW (1984) Some models for multitype spatial point processes, with remarks on analyzing multitype patterns. J Appl Probab 21:575–582 · Zbl 0553.60049
[13] Lotwick HW, Silverman BW (1982) Methods for analyzing spatial processes of several types of points. J Roy Stat Soc Ser B 44:406–413
[14] Mateu J (2001) Parametric procedures in the analysis of replicated pairwise interaction point patterns. Biom J 43:375–394 · Zbl 1002.62073
[15] Moran PAP (1952) A characterization of the Poisson distribution. Proc Camb Philos Soc 48:206–207 · Zbl 0047.37301
[16] Rényi A (1967) Remarks on the Poisson process. In: Symposium on probability methods in analysis, Loutraki, 1966. Lecture notes in mathematics, vol 31. Springer, Berlin
[17] Ripley BD (1977) Modelling spatial patterns (with discussion). J Roy Stat Soc Ser B 39:172–212
[18] Silverman BW (1978) Distances on circles, toruses and spheres. J Appl Probab 15:136–143 · Zbl 0388.62019
[19] Srivastava RC (1971) On a characterization of the Poisson process. J Appl Probab 8:615–616 · Zbl 0236.60074
[20] Stoyan D, Stoyan H (2000) Improving ratio estimators of second order point process characteristics. Scand J Stat 27:641–656 · Zbl 0963.62089
[21] Wiegand JFK, Ward D (2000) Do spatial effects play a role in the spatial distribution of desert-dwelling acacia raddiana? J Veg Sci 11:473–484
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