Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-23T12:10:30.589Z Has data issue: false hasContentIssue false
  • English
  • Français

Maximum likelihood estimation for branching processes with immigration

Published online by Cambridge University Press:  01 July 2016

B. R. Bhat  and
S. R. Adke
B. R. Bhat*
Affiliation:
Karnatak University
S. R. Adke*
Affiliation:
University of Poona
*
∗Postal address: Department of Statistics, Karnatak University, Dharwad-580 003, India.
∗∗Postal address: Department of Statistics, University of Poona, Pune 411 007, India.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper establishes the strong consistency of the maximum likelihood estimators of the parameters of discrete- and continuous-time Markov branching processes with immigration. The asymptotic distributions of the maximum likelihood estimators of the parameters of a Galton–Watson branching process with immigration are also obtained.

Keywords

ASYMPTOTIC DISTRIBUTIONSMIXING FORM OF CENTRAL LIMIT THEOREMSTRONG CONSISTENCY
Type
Research Article
Information
Advances in Applied Probability , Volume 13 , Issue 3 , September 1981 , pp. 498 - 509
Copyright
Copyright © Applied Probability Trust 1981 

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

Aldous, D. J. and Eagleson, G. K. (1978) On mixing and stability of limit theorems. Ann. Prob. 6, 325–351.CrossRefGoogle Scholar
Athreya, K. B. and Keiding, N. (1977) Estimation theory for continuous-time branching processes. Sankhyā A 39, 101–123.Google Scholar
Basawa, I. V. and Scott, D. J. (1976) Efficient tests for branching processes. Biometrika 63, 531–536.CrossRefGoogle Scholar
Basawa, I. V. and Scott, D. J. (1977) Efficient tests for stochastic processes. Sankhyā A 39, 21–31.Google Scholar
Bhat, B. R. (1977) Statistical analysis for mixtures of discrete and continuous distributions. Proc. 41st Session Internat. Statist. Inst., New Delhi, 65–68.Google Scholar
Billingsley, P. (1961) Statistical Inference for Markov Processes. University of Chicago Press.Google Scholar
Chung, K. L. (1967) Markov Chains with Stationary Transition Probabilities, 2nd edn. Springer-Verlag, Berlin.Google Scholar
Cramér, H. (1946) Mathematical Methods of Statistics. Princeton University Press, Princeton, N.J. Google Scholar
Csörgo, M. and Fischler, R. (1973) Some examples and results in the theory of mixing and random sum central limit theorems. Period. Math. Hungar. 3, 41–67.CrossRefGoogle Scholar
Dion, J.-P. and Keiding, N. (1978) Statistical inference in branching processes. In Advances in Probability 5, ed. Ney, P.. Dekker, New York.Google Scholar
Heyde, C. C. (1975) Remarks on efficiency in estimation for branching processes. Biometrika 62, 49–55.CrossRefGoogle Scholar
Heyde, C. C. (1977) An optimal property of maximum likelihood with application to branching process estimation. Proc. 41st Session Internat. Statist. Inst., New Delhi.Google Scholar
Heyde, C. C. and Feigin, P. D. (1975) On efficiency and exponential families in stochastic processes. In Statistical Distributions in Scientific Work 1, ed. Patil, G. P. et al. Reidel, Dordrecht, 227–240.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.CrossRefGoogle 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.CrossRefGoogle Scholar
Keiding, N. (1974) Estimation in the birth process. Biometrika 61, 71–80 and 647.CrossRefGoogle Scholar
Keiding, N. (1975) Maximum likelihood estimation in birth and death process. Ann. Statist. 3, 363–372.CrossRefGoogle Scholar
Loève, M. (1968) Probability Theory, 3rd edn. Van Nostrand, New York.Google Scholar
Nagaev, A. N. (1967) On estimating the expected number of direct descendants of a particle in a branching process. Theory Prob. Appl. 12, 314–320.CrossRefGoogle Scholar
Pakes, A. G. (1971) Branching processes with immigration. J. Appl. Prob. 8, 32–42.CrossRefGoogle Scholar
Rao, C. R. (1973) Linear Statistical Inference and its Applications. Wiley, New York.CrossRefGoogle Scholar
Seneta, E. (1970a) A note on the super critical Galton–Watson process with immigration. Math. Biosci. 6, 305–311.CrossRefGoogle Scholar
Seneta, E. (1970b) An explicit limit theorem for the critical Galton–Watson process with immigration. J. R. Statist. Soc. B 32, 149–152.CrossRefGoogle Scholar
Zacks, S. (1971) The Theory of Statistical Inference. Wiley, New York.Google Scholar