Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-23T17:28:09.906Z Has data issue: false hasContentIssue false
  • English
  • Français

Parametric inference in Markov branching processes with time-dependent random immigration rate

Published online by Cambridge University Press:  14 July 2016

Helmut Pruscha
Helmut Pruscha*
Affiliation:
University of Munich
*
∗Postal address: University of Munich, Institute of Mathematics, Theresienstr. 39, D-8000 München, West Germany.
Get access Rights & Permissions [Opens in a new window]

Abstract

The present paper deals with continuous-time Markov branching processes allowing immigration. The immigration rate is allowed to be random and time-dependent where randomness may stem from an external source or from state-dependence. Unlike the traditional approach, we base the analysis of these processes on the theory of multivariate point processes. Using the tools of this theory, asymptotic results on parametric inference are derived for the subcritical case. In particular, the limit distributions of some parametric estimators and of Pearson-type statistics for testing simple and composite hypotheses are established.

Keywords

MULTIVARIATE POINT PROCESSESASYMPTOTIC PARAMETRIC INFERENCELIKELIHOOD RATIO TEST STATISTICS
Type
Research Papers
Information
Journal of Applied Probability , Volume 22 , Issue 3 , September 1985 , pp. 503 - 517
Copyright
Copyright © Applied Probability Trust 1985 

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.)

References

Aalen, O. O. (1976) Statistical inference for a family of counting processes. Inst, of Math. Statist. University of Copenhagen.Google Scholar
Andersen, P. K., Borgan, Ø., Gill, R. D. and Keiding, N. (1982) Linear nonparametric tests for comparison of counting processes, with applications to censored survival data. Internat. Statist. Rev. 50, 219–258.CrossRefGoogle Scholar
Andersen, P. K. and Gill, R. D. (1982) Cox's regression model for counting processes: A large sample study. Ann. Statist. 10, 1100–1120.CrossRefGoogle Scholar
Bath, B. R. and Adke, S. R. (1981) Maximum likelihood estimation for branching processes with immigration. Adv. Appl. Prob. 13, 498–509.Google Scholar
Boel, R., Varaiya, P. and Wong, E. (1975) Martingales of jump processes I. SIAM J. Control 13, 999–1021.CrossRefGoogle Scholar
Bremaud, P. (1981) Point Processes and Queues. Springer-Verlag, New York.CrossRefGoogle Scholar
Foster, J. H. and Williamson, J. A. (1971) Limit theorems for the Galton–Watson process with time-dependent immigration. Z. Wahrscheinlichkeitsth. 20, 227–235.CrossRefGoogle Scholar
Franz, J. (1982) Sequential estimation and asymptotic properties in birth- and death-processes. Math. Operationsforschung. Statist., Ser. Statist. 13, 231–244.Google Scholar
Hudson, I. L. (1983) Asymptotic tests for the continuous time Markov branching process with immigration. Austral. J. Statist. 25, 47–57.CrossRefGoogle Scholar
Jacobsen, M. (1982) Statistical Analysis of Counting Processes. Lecture Notes in Statistics 12, Springer-Verlag, New York.CrossRefGoogle Scholar
Jacod, J. (1975) Multivariate point processes. Z. Wahrscheinlichkeitsth. 31, 235–253.CrossRefGoogle Scholar
Johansen, S. (1981) The statistical analysis of a Markov branching process. Inst, of Math. Statist., University of Copenhagen.Google Scholar
Kabanov, Ju. M, Liptser, R. S. and Shiryayev, A. N. (1980). Absolute continuity and singularity of locally absolutely continuous probability distributions II. Math. USSR Sb. 36, 31–58.CrossRefGoogle Scholar
Lepingle, D. (1978) Sur le comportement asymptotic des martingales locales. In Lecture Notes in Mathematics 649, Springer-Verlag, Berlin, 148–161.Google Scholar
Liptser, R. S. and Shiryayev, A. N. (1978) Statistics of Random Processes, Vol. II. Springer-Verlag, New York.CrossRefGoogle Scholar
Pakes, A. G. (1971) A branching process with a state dependent immigration component. Adv. Appl. Prob. 3, 301–314.CrossRefGoogle Scholar
Pakes, A. G. (1975) On Markov branching processes with immigration. Sankhya A 37, 129–138.Google Scholar
Rebolledo, R. (1980) Central limit theorems for local martingales. Z. Wahrscheinlichkeitsth. 51, 269–286.CrossRefGoogle Scholar
Sevast'yanov, B. A. (1957) Limit theorems for branching stochastic processes of special form. Theory Prob. Appl. 2, 321–331.CrossRefGoogle Scholar
Venrataraman, K. N. and Nanthi, K. (1982) A limit theorem on a subcritical Galton–Watson process with immigration. Ann. Prob. 10, 1069–1074.Google Scholar
Yang, Y. S. (1972) On branching processes allowing immigration. J. Appl. Prob. 9, 24–31.CrossRefGoogle Scholar