Inés M. del Puerto

Nonparametric estimation of the offspring distribution and the mean for a controlled branching process

In this paper, under a nonparametric context, some estimators for the mean and the offspring distribution of a controlled branching process are provided. Their conditional and unconditional moments and their asymptotic properties (consistency, normality) are investigated. Finally, as an illustration, a simulated example is presented.

Workshop on Branching Processes and Their Applications

Part I Population Growth Models in Random and Varying Environments.- Part II Special Branching Processes.- Part III Limit Theorems and Statistics.- Part IV Applications in Cell Kinetics and Genetics.- Part V Applications in Epidemiology.- Part VI Two-sex Branching Models.

Placebo effect characteristics observed in a single, international, longitudinal study in Huntington's disease.

Classically, clinical trials are based on the placebo‐control design. Our aim was to analyze the placebo effect in Huntingtons disease.

Minimum disparity estimation in controlled branching processes

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.

Branching Processes and Their Applications

We define a dynamical simple symmetric random walk in one dimension, and show that there almost surely exist exceptional times at which the walk tends to infinity. In fact the set of such times has Hausdorff dimension 1/2 almost surely. This is in contrast to the usual dynamical simple symmetric random walk in one dimension, for which such exceptional times are known not to exist. This is joint work with Martin Prigent.

Bayesian Analysis for Controlled Branching Processes

The branching model considered in the present work is the controlled branching process. This model is a generalization of the standard Bienayme-Galton-Watson (BGW) branching process, and, in the terminology of population dynamics, is used to describe the evolution of populations in which a control of the population size at each generation is needed. This control consists of determining mathematically the number of individuals with reproductive capacity at each generation through a random process. In practice, this branching model can describe reasonably well the probabilistic evolution of populations in which, for various reasons of an environmental, social, or other nature, there is a mechanism that establishes the number of progenitors which take part in each generation. For example, in an ecological context, one can think of an invasive animal species that is widely recognized as a threat to native ecosystems, but there is disagreement about plans to eradicate it, i.e., while the presence of the species is appreciated by a part of the society, if its numbers are left uncontrolled it is known to be very harmful to native ecosystems. In such a case, it is better to control the population to keep it within admissible limits even though this might mean periods when animals have to be culled. Another practical situation that can be modelled by this kind of process is the evolution of an animal population that is threatened by the existence of predators. In each generation, the survival of each animal (and therefore the possibility of giving new births) will be strongly affected by this factor, making the introduction of a random mechanism necessary to model the evolution of this kind of population.

Weighted conditional least squares estimation in controlled multitype branching processes

The multitype controlled branching process provides a useful way to model generation sizes in population dynamics studies, where several types of individuals coexist and a control on the growth of population size is necessary at each generation. From a probabilistic viewpoint this model has been studied in Gonz’alez et al. (Bernoulli 11(3):559–570, 2005; J. Appl. Probab. 42:1015–1030, 2005; Pliska Stud. Math. Bulgar. 17: 85–96, 2005; J. Appl. Probab. 43: 159–174, 2006; Pliska Stud. Math. Bulgar. 18: 103–110, 2007; Stoch. Models 24: 401–424, 2008). In this paper we are interested in developing its inferential theory, not considered until now. We propose a weighted conditional least squares estimator of the offspring mean matrix. For the supercritical case, we establish the strong consistency of the proposed estimator.