Identification of multitype branching processes

Annals of Statistics

Volumn 33, Issue 6, 2005, Pages 2655-2694

  Maaouia, F ^a   Touati, A ^b  

^a FACULTÉ DES SCIENCES DE TUNIS   (Tunisia)

^b University of Carthage   (Tunisia)

Author keywords

Central limit theorem; Law of the iterated logarithm; Least squares estimator; Maximum likelihood estimator; Multidimensional Lotka Nagaev estimator; Multitype branching process; Quadratic strong law of large numbers

Indexed keywords

EID: 33644913936     PISSN: 00905364     EISSN: None     Source Type: Journal    
DOI: 10.1214/009053605000000561     Document Type: Article

References (29)

1. ADÈS, M., DION, J.-P., LABELLE, G. and NANTHI, K. (1982). Recurrence formula and the maximum likelihood estimation of the age in a simple branching process. J. Appl. Probability 19 776-784. MR0675141
2. ASMUSSEN, S. and KEIDING, N. (1978). Martingale central limit theorems and asymptotic estimation theory for multitype branching processes. Adv. in Appl. Probab. 10 109-129. MR0471027
3. ATHREYA, K. B. and NEY, P. E. (1972). Branching Processes. Springer, Berlin. MR0373040
4. BADALBAEV, I. S. and MUKHITDINOV, A. (1990). On the limit distribution of some functionals in multitype branching processes. Theory Probab. Appl. 35 625-638. MR1090491
5. CARVALHO, M. L. (1997). A joint estimator for the eigenvalues of the reproduction mean matrix of a multitype Gallon-Watson process. Linear Algebra Appl. 264 189-203. MR1465865
6. CHAÂBANE, F. (1996). Version forte du théorème de la limite centrale fonctionnel pour les martingales. C. R. Acad. Sci. Paris Sér. I Math. 323 195-198. MR1402542
7. CHAÂBANE, F. (2001). Invariance principles with logarithmic averaging for martingales. Studia Sci. Math. Hungar. 37 21-52. MR1834322
8. CHAÂBANE, F. and MAÂOUIA, F. (2000). Théorèmes limites avec poids pour les martingales vectorielles. ESAIM Probab. Statist. 4 137-189. MR1808925
9. CHAÂBANE, F., MAÂOUIA, F. and TOUATI, A. (1998). Généralisation du théorème de la limite centrale presque-sûr pour les martingales vectorielles. C. R. Acad. Sci. Paris Sér. I Math. 326 229-232. MR1646985
10. CHAÂBANE, F., MAÂOUIA, F. and TOUATI, A. (1998). Versions fortes des théorèmes limites en loi pour les martingales vectorielles. Preprint.
11. DACUNHA-CASTELLE, D. and DUFLO, M. (1983). Probabilités et Statistiques 2. Masson, Paris. MR0732786
12. DION, J.-P. (1974). Estimation of the mean and the initial probabilities of a branching process. J. Appl. Probability 11 687-694. MR0375488
13. DION, J.-P. (1975). Estimation of the variance of a branching processes. Ann. Statist. 3 1183-1187. MR0378308
14. DION, J.-P. (1993). Statistical inference for discrete time branching processes. In Lecture Notes of the 7th Summer School on Probability Theory and Mathematical Statistics (Varna, 1991) 60-121. Scientific, Cultural and Technical Publications, Singapore.
15. DUFLO, M. (1997). Random Iterative Models. Springer, Berlin. MR1485774
16. GUTTORP, P. (1991). Statistical Inference for Branching Processes. Wiley, New York. MR1254434
17. HALL, P. and HEYDE, C. C. (1980). Martingale Limit Theory and Its Application. Academic Press, San Diego. MR0624435
18. HEYDE, C. C. (1974). On estimating the variance of the offspring distribution in a simple branching process. Adv. in Appl. Probab. 6 421-433. MR0348855
19. HEYDE, C. C. (1997). Quasi-Likelihood and Its Application. A General Approach to Optimal Parameter Estimation. Springer, Berlin. MR1461808
20. HEYDE, C. C. and LESLIE, J. R. (1971). Improved classical limit analogues for Galton-Watson processes with or without immigration. Bull. Austral. Math. Soc. 5 145-155. MR0293731
21. HEYDE, C. C. and SENATA, E. (1971). Analogues of classical limit theorems for the supercritical Galton-Watson process with immigration. Math. Biosci. 11 249-259. MR0292185
22. KEIDING, N. and LAURITZEN, S. L. (1978). Marginal maximum likelihood estimates and estimation of the offspring mean in a branching process. Scand. J. Statist. 5 106-110. MR0497064
23. KESTEN, H. and STIGUM, B. P. (1966). A limit theorem for multidimensional Galton-Watson processes. Ann. Math. Statist. 37 1211-1223. MR0198552
24. KESTEN, H. and STIGUM, B. P. (1966). Additional limit theorems for indecomposable multidimensional Galton-Watson processes. Ann. Math. Statist. 37 1463-1481. MR0200979
25. MODE, C. J. (1971). Multitype Branching Processes. North-Holland, New York.
26. NANTHI, K. (1982). Estimation of the variance for the multitype Galton-Watson process. J. Appl. Probab. 19 408-414. MR0649978
27. TOUATI, A. (1993). Two theorems on convergence in distribution for stochastic integrals and statistical applications. Theory Probab. Appl. 38 95-117. MR1317786
28. TOUATI, A. (1996). Vitesse de convergence en loi de 1'estimateur des moindres carrés d'un modèle autorégressif (cas mixte). Ann. Inst. H. Poincaré Probab. Statist. 32 211-230. MR1386219
29. WEI, C. Z. (1985). Asymptotic properties of least-squares estimates in stochastic regression models. Ann. Statist. 13 1498-1508. MR0811506