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Non-parametric estimation of the death rate in branching processes. (English) Zbl 1035.62085

A branching process with immigration is considered where the particles move independently of each other on diffusion paths \(d\xi_t=b(\xi_t)dt+\sigma(\xi_t)dw_t\). A particle sitting in \(x\) at time \(t\) dies in \((t,t+h)\) with probability \(\kappa(x)h+o(h)\). The authors propose a kernel-type estimator for \(\kappa\). The asymptotic normality of the estimator is demonstrated. It is shown that it is asymptotically optimal for the minimax risk on Hölder classes of \(\kappa\). A local time for the particle process is introduced and its asymptotics are obtained via the Tanaka formula.

MSC:

62M05 Markov processes: estimation; hidden Markov models
62F12 Asymptotic properties of parametric estimators
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[1] Adke S. R., J. Soviet Math. 21 pp 1– (1983)
[2] 2K. B. Athreya, and P. E. Ney, 1972 .Branching processes. Springer-Verlag, Berlin, Heidelberg, New York. · Zbl 0259.60002
[3] 3V. Bally, and E. Locherbach (2001 ). On the invariant density of branching diffusions . Preprint 688, Laboratoire de Probabilites, Universite Paris VI.
[4] 4J. Crow, and M. Kimura (1970 ).An introduction to population genetics.Harper & Row, New York. · Zbl 0246.92003
[5] 5D. Daley, and E. Vere-Jones (1988 ).An introduction to the theory of point processes.Springer Verlag, New York. · Zbl 0657.60069
[6] Etheridge A. M., Proc. Amer. Math. Soc. 118 pp 1251– (1993)
[7] 7W. Ewens (1969 ).Population genetics. Methuen, New York.
[8] Fleming W., Theoret. Population Biol. 5 pp 431– (1974)
[9] 9P. Guttorp (1991 ).Statistical inference for branching processes.Wiley Series in Probability and Mathematical Statistics. Wiley, New York. · Zbl 0778.62077
[10] 10W. Hardle, G. Kerkyacharian, D. Picard, and A. Tsybakov (1998 ).Wavelets, approximation, and statistical applications. Lecture Notes in Statistics,129, Springer Verlag, New York.
[11] Holmes E. E., Ecology 75 pp 17– (1994)
[12] Hoffmann M., Bernoulli 5 pp 447– (1999)
[13] DOI: 10.1023/A:1009944302344 · Zbl 1061.62562
[14] 14R. Hopfner, and E. Locherbach (1999 ). On invariant measure for branching diffusions . Unpublished note, see http://www.mathematik.uni-mainz.de/hoepfner.
[15] 15R. Hopfner, and E. Locherbach (2000 ). Limit theorems for null recurrent Markov processes . To appear inMemoirs AMS. Preprint 22-2000 at Mainz, Fachbereich Mathematik, University of Mainz.
[16] 16I. A. Ibragimov, and R. Z. Khas’minskii (1981 ).Statistical estimation-asymptotic theory. Springer Verlag, Berlin.
[17] Jacod J., Scand. J. Statist. 27 pp 83– (2000)
[18] 18J. Jacod, and A.N. Shiryaev (1987 ).Limit theorems for stochastic processes.Springer, Berlin. · Zbl 0635.60021
[19] 19J. Karatzas, and S. Shreve (1991 ).Brownian motion and stochastic calculus,2nd edn. Springer Verlag, New York. · Zbl 0734.60060
[20] 20R. Khasminskii (1980 ).Stochastic stability of differential equations.Sijthoff & Noordhoff, Alphen.
[21] Kimura M., Genetics 49 pp 561– (1964) · Zbl 0134.38103
[22] 22A. P. Korostelev, and A. D. Tsybakov (1993 ).Minimax theory of image reconstruction.Lecture Notes in Statistics,82, Springer Verlag, New York. · Zbl 0833.62039
[23] Kulperger R., J. Multivariate Anal. 18 pp 225– (1986)
[24] 24Y. Kutoyants (1998 ).Statistical inference for spatial Poisson processes.Lecture Notes in Statistics,134, Springer Verlag, New York. · Zbl 0904.62108
[25] Lepski O. V., Theory Probab. Appl. 36 pp 682– (1991)
[26] 26E. Locherbach (1999 ). Statistical models and likelihood ratio processes for interacting particle systems with branching and immigration . Thesis, Universitat Paderborn.
[27] 27E. Locherbach (1999 ). Likelihood ratio processes for Markovian particle systems with killing and jumps . Preprint 551, Laboratoire de Probabilites, Universite Paris VI, to appear inStat. Inference Stoch. Process.
[28] DOI: 10.1016/S0246-0203(01)01077-9 · Zbl 1004.60078
[29] Malecot G., Proc. 5th Berkeley Symp. Math. Statist. Probab. 4 pp 317– (1967)
[30] Nagylaki T., J. Math. Biol. 6 pp 375– (1978)
[31] 31J. Neyman, and E. Scott (1964 ). A stochastic model of epidemics . InStochastic Models Med. Biol., Proc. Sympos. Univ. Wisconsin 1963(ed. J. Gurland), pp. 45-83, University of Wisconsin Press, Madison.
[32] Pakes A., Sankhya 37 pp 129– (1975)
[33] 33D. Revuz, and M. Yor (1991 ).Continuous martingales and Brownian motion.Springer Verlag, Berlin. · Zbl 0731.60002
[34] Sawyer S., Adv. Appl. Probab. 8 pp 659– (1976)
[35] 35B. A. Sewastjanow (1975 ).Verzweigungsprozesse. Oldenburg Verlag, Munchen Wien.
[36] Wakolbinger A., Bernoulli 1 pp 171– (1995)
[37] Zubkov A., Theory Probab. Appl. 17 pp 174– (1972)
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