Controlled branching processes are stochastic growth population models in which the number of individuals with reproductive capacity in each generation is controlled through a random control function. The purpose of this work is to examine the sequential Monte Carlo Approximate Bayesian Computation method and to propose appropriate summary statistics in the context of these processes. We show that the success of this methodology lies on this latter issue. The accuracy of the proposed method is illustrated and compared with a "likelihood free" Markov chain Monte Carlo technique by means of a simulated example. Moreover we illustrate how to extend this methodology to a controlled multitype branching
The controlled branching process is a generalization of the classical Bienaymé-Galton-Watson branching process. It is a useful model for describing the evolution of populations in which the population size at each generation needs to be controlled. The maximum likelihood estimation of the parameters of interest for this process is addressed under various sample schemes. Firstly, assuming that the entire family tree can be observed, the corresponding estimators are obtained and their asymptotic properties investigated. Secondly, since in practice it is not usual to observe such a sample, the maximum likelihood estimation is initially considered using the sample given by the total number of individuals and progenitors of each generation, and then using the sample given by only the generation sizes. Expectation-maximization algorithms are developed to address these problems as incomplete data estimation problems. The accuracy of the procedures is illustrated by means of a simulated example.
Minimum disparity estimation in controlled branching processes is dealt with by assuming that the offspring law belongs to a general parametric family.Under some regularity conditions it is proved that the minimum disparity estimators proposed -based on the nonparametric maximum likelihood estimator of the offspring law when the entire family tree is observed-are consistent and asymptotic normally distributed. Moreover, it is discussed the robustness of the estimators proposed. Through a simulated example, focussing on the minimum Hellinger and negative exponential disparity estimators, it is shown that both are robust against outliers, being the negative exponential one also robust against inliers.
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