Estimation theory for multitype branching processes

  • Søren Asmussen


We consider a p-type Galton-Watson process \( {\left\{ {{Z_n}} \right\}_{n \in {\Bbb N}}}\) i.e. Zn= (Zn(1)...Zn((p)),
$$ {Z_n} = \left( {{Z_n}\left( 1 \right) \ldots {Z_n}\left( p \right)} \right),\;{Z_{n + 1}} = \mathop \Sigma \limits_{i = l}^p \mathop \Sigma \limits_{k = l}^{{Z_n}\left( i \right)} Z_{n,k}^{\left( i \right)}$$
where the \( {Z_{n + 1}} = \mathop \Sigma \limits_{i = 1}^p \mathop \Sigma \limits_{k = 1}^{{Z_n}\left( i \right)} Z_{n,k}^{\left( i \right)}\) are independent for all n, i, k and with the same distribution for any fixed i.




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  1. Asmussen, S. (1977). Almost sure behaviour of linear functionals of supercritical branching processes. Trans.Amer.Math.Soc. 231, 233–248.MathSciNetMATHCrossRefGoogle Scholar
  2. Asmussen, S. (1982). On the role of a certain eigenvalue in estimating the growth rate of a branching process. Austr.J.Statist. 24, 151–159.MathSciNetMATHCrossRefGoogle Scholar
  3. Asmussen, S. and H. Hering (1983). Branching processes. Birkhäuser, Basel Boston Stuttgart.MATHCrossRefGoogle Scholar
  4. Asmussen, S. and N. Keiding (1978). Martingale central limit theorems and asymptotic estimation theory for multitype branching processes. Adv.Appl.Probab. 10, 109–129.MathSciNetMATHCrossRefGoogle Scholar
  5. Athreya, K.B. (1969). Limit theorems for multitype continuous time Markov branching processes. Z. Wahrscheinlichkeitsth.verw.Geb. 12, 320–332; ibid. 13, 204–214.MathSciNetMATHCrossRefGoogle Scholar
  6. Athreya, K.B. (1971). Some refinements in the limit theory of supercritical multitype Markov branching processes. Z.Wahrscheinlichkeitsth.verw.Geb. 20, 47–57.MathSciNetMATHCrossRefGoogle Scholar
  7. Becker, N. (1977). Estimation for discrete time branching processes with applications to epidemics. Biometrics 33, 515–522.MathSciNetMATHCrossRefGoogle Scholar
  8. Carvalho, M.L.S. (1983). Etude asymptotique de quelques estimateurs du processus de ramification multitype. Doctoral dissertation, l’Université Pierre et Marie Curie, Paris.Google Scholar
  9. Dion, J.-P. and N. Keiding (1978). Statistical inference in branching processes. In: Advances in Probability 5 (Joffe-Ney ed.). Marcel Dekker, New York.Google Scholar
  10. Dion, J.-P. and K. Nanthi (1982). Estimation in multitype branching processes. Technical report, Université de Montréal.Google Scholar
  11. Heyde, C.C. (1981). On Fibonacci (or lagged Bienaymé-Galton-Watson) processes. J.Appl.Probab. 17, 1079–1082.MathSciNetCrossRefGoogle Scholar
  12. Keiding, N. and S.L. Lauritzen (1978). Marginal maximum likelihood estimates and estimation of the offspring mean in a branching process. Scand.J. Statist. 5, 106–110.MathSciNetMATHGoogle Scholar
  13. Kesten, H. and B.P. Stigum (1966). Additional limit theorems for indecomposable multidimensional Galton-Watson processes. Ann. Math.Statist. 37, 1211–1223.MathSciNetMATHCrossRefGoogle Scholar
  14. Khan, I.A. and S. Rehman (1980). On estimation of the matrix of the first moments of a two-dimensional branching process. Pure Appl.Math.Sci. 12, 41–48.MathSciNetMATHGoogle Scholar
  15. Lockhart, R. (1982). On the non-existence of consistent estimates in Galton-Watson processes. J.Appl.Probab. 19, 842–846.MathSciNetMATHCrossRefGoogle Scholar
  16. Nanthi, K. (1982). Estimation of the variance for the multitype Galton-Watson process. J.Appl.Probab. 19, 421–426.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1986

Authors and Affiliations

  • Søren Asmussen
    • 1
  1. 1.Institute of Mathematical StatisticsUniversity of CopenhagenDenmark

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