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Estimation for multitype branching processes

Published online by Cambridge University Press:  14 July 2016

M. P. Quine  and
P. Durham
M. P. Quine
Affiliation:
University of Sydney
P. Durham
Affiliation:
University of Sydney
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Abstract

It is shown that certain estimators of the matrix of offspring means and the vector of stationary means in a subcritical multitype branching process with immigration are strongly consistent and obey the central limit theorem under fairly natural conditions. The results give some evidence of the robustness of the analogous autoregressive time series model in the case of large samples.

Keywords

MULTITYPE BRANCHING PROCESSIMMIGRATIONESTIMATIONCONSISTENCYCENTRAL LIMIT THEOREMAUTOREGRESSIVE TIME SERIES
Type
Short Communications
Information
Journal of Applied Probability , Volume 14 , Issue 4 , December 1977 , pp. 829 - 835
Copyright
Copyright © Applied Probability Trust 1977 

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References

Billingsley, P. (1961) Statistical Inference for Markov Processes. University of Chicago Press, Chicago.Google Scholar
Chung, K. L. (1967) Markov Chains with Stationary Transition Probabilities, 2nd. edn. Springer-Verlag, New York.Google Scholar
Hannan, E. J. (1970) Multiple Time Series. Wiley, New York.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
Quine, M. P. (1970a) A note on the moment structure of the multitype Galton-Watson process. Biometrika 57, 219–222.Google Scholar
Quine, M. P. (1970b) The multitype Galton-Watson process with immigration. J. Appl. Prob. 7, 411–422.Google Scholar
Quine, M. P. (1976) On the estimation of means in a subcritical branching process with immigration. Ann. Prob. 4, 319–325.Google Scholar