Hostname: page-component-7b7567c8c-wxvwp Total loading time: 0 Render date: 2026-11-14T08:05:24.824Z Has data issue: false hasContentIssue false
  • English
  • Français

Estimation of the parameters of a branching process from migrating binomial observations

Part of:Parametric inferenceApplicationsMarkov processes

Published online by Cambridge University Press:  01 July 2016

C. Jacob and
J. Peccoud
C. Jacob*
Affiliation:
Institut National de la Recherche Agronomique
J. Peccoud*
Affiliation:
Université Joseph Fourier
*
∗ Postal address: INRA, Laboratoire de Biométrie, 78352 Jouy-en-Josas Cedex, France. Email address: Christine.Jacob@jouy.inra.fr
∗∗ Postal address: TIMC, IMAG, Faculté de médecine de Grenoble, Domaine de la Merci, 38706 La Tronche Cedex, France.
Get accessRights & Permissions [Opens in a new window]

Abstract

This paper considers a branching process generated by an offspring distribution F with mean m < ∞ and variance σ2 < ∞ and such that, at each generation n, there is an observed δ-migration, according to a binomial law B p v n *N nbef which depends on the total population size N n bef. The δ-migration is defined as an emigration, an immigration or a null migration, depending on the value of δ, which is assumed constant throughout the different generations. The process with δ-migration is a generation-dependent Galton-Watson process, whereas the observed process is not in general a martingale. Under the assumption that the process with δ-migration is supercritical, we generalize for the observed migrating process the results relative to the Galton-Watson supercritical case that concern the asymptotic behaviour of the process and the estimation of m and σ2, as n → ∞. Moreover, an asymptotic confidence interval of the initial population size is given.

Keywords

Branching processesmigrationgeneration-dependence

MSC classification

Primary:60J80: Branching processes (Galton-Watson, birth-and-death, etc.)62F12: Asymptotic properties of estimators
Secondary:62P10: Applications to biology and medical sciences

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 30 , Issue 4 , December 1998, pp. 948 - 967
Copyright
Copyright © Applied Probability Trust 1998 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Asmussen, S. and Hering, H. (1983). Branching Processes. Birkhaüser, Boston.CrossRefGoogle Scholar
Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, Berlin.Google Scholar
Basawa, I. V. and Scott, D. J. (1976). Efficient tests for branching processes. Biometrika 63, 531–536.Google Scholar
Bhat, B. R. and Adke, S. R. (1981). Maximum likelihood estimation for branching processes with immigration. Adv. Appl. Prob. 13, 498–509.Google Scholar
Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.Google Scholar
Brown, B. M. (1971). Martingale central limit theorems. Ann. Math. Statist. 42, 59–66.Google Scholar
Dacunha-Castelle, D. and Duflo, M. (1983). Probabilités et statistiques 2. Problèmes à temps mobile. Masson, Paris.Google Scholar
Dion, J. P. and Yanev, N. M. (1997). Limit theorems and estimation theory for branching processes with an increasing random number of ancestors. J. Appl. Prob. 34, 309–327.Google Scholar
D'Souza, J. C. and Biggins, J. D. (1992). The supercritical Galton–Watson process in varying environments. Stoch. Proc. Appl. 42, 39–47.Google Scholar
Egorov, V. A. (1987). On the strong law of large numbers and the law of the iterated logarithm for martingales and sums of independent random variables. Theory Prob. Appl. 35, 653–666.Google Scholar
Fearn, D. H. (1972). Galton–Watson processes with generation dependence. Proc. 6th Berkeley Symp. Math. Statist. Prob. 4, 159–172.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edition. Wiley, New York.Google Scholar
Foster, J. H. (1971). A limit theorem for a branching process with state-dependent immigration. Ann. Math. Statist. 42, 1773–1776.Google Scholar
Gupta, S. C., Srivastava, O. P. and Singh, Mahendra (1992). Branching process with emigration – a genetic model. Math. Biosci. 111, 159–168.Google Scholar
Guttorp, P. (1991). Statistical Inference for Branching Processes. Wiley, New York.Google Scholar
Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and its Application. Academic Press, New York.Google Scholar
Harris, T. E. (1963). The Theory of Branching Processes. Springer, Berlin.Google Scholar
Heyde, C. C. (1970). Extension of a result of Seneta for the super-critical Galton–Watson process. Ann. Math. Statist. 41, 739–742.Google Scholar
Heyde, C. C. (1971). Some central limit analogues for super-critical Galton–Watson processes. J. Appl. Prob. 8, 52–59.Google Scholar
Heyde, C. C. (1974). On estimating the variance of the offspring distribution in a simple branching process. Adv. Appl. Prob. 6, 421–433.Google Scholar
Heyde, C. C. and Leslie, J. R. (1971). Improved classical limit analogues for Galton–Watson processes with or without immigration. Bull. Austr. Math. Soc. 5, 145–155.Google Scholar
Heyde, C. C. and Seneta, E. (1972). Estimation theory for growth and immigration rates in a multiplicative process. J. Appl. Prob. 9, 235–256.Google Scholar
Heyde, C. C. and Seneta, E. (1974). Notes on ‘Estimation theory for growth and immigration rates in a multiplicative process’. J. Appl. Prob. 11, 572–577.Google Scholar
Hudson, I. L. (1982). Large sample inference for Markovian exponential families with application to branching processes with immigration. Austr. J. Statist. 24, 98–112.Google Scholar
Pakes, A. G. (1971). A branching process with a state-dependent immigration component. Adv. Appl. Prob. 3, 301–314.CrossRefGoogle Scholar
Quine, M. P. (1976). Asymptotic results for estimators in a subcritical branching process with immigration. Ann. Prob. 4, 319–325. Correction note (1977). 5, 318.Google Scholar
Rahimov, I. (1987). Statistical estimates for parameters of a subcritical Galton–Watson process with a reflecting screen. In Probabilistic Models and Mathematical Statistics, FAN, Tashkent, pp. 76–84.Google Scholar
Rahimov, I. (1995). Random Sums and Branching Stochastic Processes (Lecture Notes in Statist. 96). Springer, Berlin.Google Scholar
Sriram, T. N. (1991). On the uniform strong consistency of an estimator of the offspring mean in a branching process with immigration. Statist. Prob. Lett. 12, 151–155.Google Scholar
Sriram, T. N., Basawa, I. V. and Huggins, R. M. (1991). Sequential estimation for branching processes with immigration. Ann. Statist. 19, 2232–2243.Google Scholar
Venkataraman, K. N. (1982). A time series approach to the study of the simple subcritical Galton–Watson process with immigration. Adv. Appl. Prob. 14, 1–20.Google Scholar
Wei, C. Z. (1991). Convergence rates for the critical branching process with immigration. Statist. Sinica. 1, 175–184.Google Scholar
Wei, C. Z. and Winnicki, J. (1990). Estimation of the means in the branching process with immigration. Ann. Statist. 18, 1757–1773.Google Scholar
Winnicki, J. (1988). Estimation theory for the branching process with immigration. Contemp. Math. 80, 301–322.Google Scholar
Winnicki, J. (1991). Estimation of the variances in the branching process with immigration. Prob. Theory Rel. Fields 88, 77–106.Google Scholar
Yanev, G. P. and Yanev, N. M. (1995). Critical branching processes with random migration. In Branching Processes, ed. Heyde, C. C. (Lecture Notes in Statist. 99). Springer, Berlin, pp. 36–46.Google Scholar