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Statistical inference for partially observed branching processes with immigration

Part of: Parametric inference Stochastic processes Markov processes

Published online by Cambridge University Press:  04 April 2017

Ibrahim Rahimov
Ibrahim Rahimov*
Affiliation:
Zayed University
*
* Postal address: Department of Mathematics and Statistics, Zayed University, Box 19282, Dubai, UAE. Email address: ibrahim.rahimov@zu.ac.ae
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Abstract

In the paper we consider the following modification of a discrete-time branching process with stationary immigration. In each generation a binomially distributed subset of the population will be observed. The number of observed individuals constitute a partially observed branching process. After inspection both observed and unobserved individuals may change their offspring distributions. In the subcritical case we investigate the possibility of using the known estimators for the offspring mean and for the mean of the stationary-limiting distribution of the process when the observation of the population sizes is restricted. We prove that, if both the population and the number of immigrants are partially observed, the estimators are still strongly consistent. We also prove that the `skipped' version of the estimator for the offspring mean is asymptotically normal and the estimator of the stationary distribution's mean is asymptotically normal under additional assumptions.

Keywords

Branching processsubcriticalimmigrationrestricted observationoffspring meanstationary distributionrandom sumlimit theorem

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 62F12: Asymptotic properties of estimators 60G99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 1 , March 2017 , pp. 82 - 95
Copyright
Copyright © Applied Probability Trust 2017 

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