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Asymptotic Gaussianity of some estimators for reduced factorial moment measures and product densities of stationary Poisson cluster processes. (English) Zbl 0666.62032
The asymptotic normality of a class of estimators related to reduced factorial moment measures is established, using the central limit theorem in which the moment conditions are reduced to a minimum. The weak convergence theorem is also provided in the case of a stationary simple Poisson cluster process (pcp) and is applied to the investigation of the asymptotic behaviour of the empirical second-order moment function. Further, the asymptotic properties of kernel-type estimators for second- order product densities of pcp’s are discussed.
Reviewer: M.Akahira

62F12 Asymptotic properties of parametric estimators
60F05 Central limit and other weak theorems
60F17 Functional limit theorems; invariance principles
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
Full Text: DOI
[1] Fiksel Th, Elektron.Informationsverab.u.Kybernet 20 pp 270– (1984)
[2] Fiksel Th, Sehatzverfahren fur Weehselwirkungspotentiale und Dichtefunktionen markierter und nichtmarkierter Punktprozesse (1985)
[3] Fiksel Th, Edge-corrected density estimators for point processes (1988) · Zbl 0644.62044
[4] Glotzl E., Math.Nachr., 96 pp 195– (1980) · Zbl 0455.60084
[5] Glotzl E., On the Statistics of Gibbsian Point Processes (1981)
[6] Hanisch K.H., Math.Nachr. 106 pp 171– (1982) · Zbl 0505.60060
[7] Hanisch K.H., Math.Operationsforsch.u.Statist., ser.statist 14 pp 559– (1983)
[8] Isham V., Proc.R.Soc.Lond., 391 pp 39– (1984) · Zbl 0529.60048
[9] Kallenbero O., Random Measures (1983)
[10] Kirkwood J.G., J.Chem.phys. 20 pp 929– (1952)
[11] Krickeberg K., Processus poncturels en statistique. Ecote d’Ete de Probabilities de Saint Flour- (1982) · Zbl 0486.62089
[12] Matthes K., Math.Nachr 88 pp 117– (1979) · Zbl 0417.60062
[13] Ogata Y., Ann.Inst.Statist.Mathpart 33 pp 315– (1981) · Zbl 0478.62078
[14] Ogata Y., Approximation of Likelihood Funciton in Estimating the Interaciton Potentials from Spatial Point Patterns.The Institute of Statistical Mathematics Research Memorandum (1981)
[15] Ogata Y., Estimation of Interaciton Potentials of-Marked Spatial Point Patterns Through the Maximum Likelihood Procedure (1983)
[16] Penttinen A., Modelling Interaciton in spatial Point Patterns: Parameter Estimation by the Maximum Likelihood Method (1984)
[17] Preston, Random Fields. Lecture Notes in Mathematics (1976)
[18] Ripley B.D., J.Appl.Prob 13 pp 255– (1976) · Zbl 0364.60087
[19] Ripley B.D., J.Roy.Statist.Soc 2 pp 172– (1977)
[20] Ripley B.D., Spatial Statistics (1981) · Zbl 0583.62087
[21] Rowlinson J.S., Liquids and Liquid Mixtures.Butterworth (1959)
[22] Stoyan D., Stochastic Geometry Its Applications. (1987) · Zbl 0683.60014
[23] Stoyan D., Stochastische Geometric (1983)
[24] Takacs R., Estimator for the Pair-Poterntial of a Gibbsian Point Process.Juhanners Kepler Universitat Linz (1983)
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