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Asymptotic properties of intensity estimators for Poisson shot-noise processes

Published online by Cambridge University Press:  14 July 2016

Friedrich Liese  and
Volker Schmidt
Friedrich Liese*
Affiliation:
University of Rostock
Volker Schmidt*
Affiliation:
Mining Academy of Freiberg
*
∗ Postal address: Universität Rostock, Fachbereich Mathematik, Universitätsplatz 1, Rostock, D-02500, Germany.
∗∗ Postal address: Bergakademie Freiberg, Fachbereich Mathematik, Bernhard-von-Cotta-Str. 2, Freiberg, D-09200, Germany.
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Abstract

Stochastic processes {X(t)} of the form X(t) = Σ n f(t – Tn) are considered, where {Tn} is a stationary Poisson point process with intensity λ and f: R → R is an unknown response function. Conditions are obtained for weak consistency and asymptotic normality of estimators of λ based on long-run observations of {X(t)}.

Keywords

MOMENT METHODEMPIRICAL COVARIANCE FUNCTIONCONSISTENCYCONVERGENCE RATECENTRAL LIMIT THEOREM
Type
Research Papers
Information
Journal of Applied Probability , Volume 28 , Issue 3 , September 1991 , pp. 568 - 583
Copyright
Copyright © Applied Probability Trust 1991 

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References

[1] Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
[2] Cox, D. R. and Isham, V. (1980) Point Processes. Chapman and Hall, London.Google Scholar
[3] Heinrich, L. (1985) Non-uniform estimates, moderate and large deviations in the central limit theorem for m-dependent random variables. Math. Nachr. 121, 107–121.Google Scholar
[4] Lane, J. A. (1984) The central limit theorem for the Poisson shot-noise process. J. Appl. Prob. 21, 287–301.Google Scholar
[5] Lehmann, E. L. (1983) Theory of Point Estimation. Wiley, New York.Google Scholar
[6] Ivanov, A. V. and Leonenko, N. N. (1989) Statistical Analysis of Random Fields. Kluwer Academic Publishers, Dordrecht.CrossRefGoogle Scholar
[7] Liese, F. and Vom Scheidt, J. (1983). A limit theorem for sequences of weakly dependent stochastic processes. Serdica 9, 18–30.Google Scholar
[8] Loève, M. (1977) Probability Theory I. Springer, New York.Google Scholar
[9] Matthes, K., Kerstan, J. and Mecke, J. (1978) Infinitely Divisible Point Processes. Wiley, Chichester.Google Scholar
[10] Rice, J. (1977) On generalized shot noise. Adv. Appl. Prob. 9, 553–565.Google Scholar
[11] Schmidt, V. (1987) Qualitative and asymptotic properties of stochastic integrals related to random marked point processes. Optimization 18, 737–759.CrossRefGoogle Scholar
[12] Utev, S. A. (1984) Inequalities for sums of weakly dependent random variables and bounds for the rate of convergence in the invariance principle. In Limit theorems for sums of random variables. Trudy Inst. Mat. 3, 50–76, Akad. Nauk SSSR, Sibir. Otdel (in Russian).Google Scholar