Manuel Mota

On the limit behavior of a supercritical bisexual Galton–Watson branching process with immigration of mating units

In this paper a bisexual Galton–Watson branching process allowing the immigration of mating units is introduced and its limit behaviour is investigated. For the supercritical case, i.e., asymptotic growth rate greater than one, necessary and sufficient conditions for the almost sure and L 1 convergence of the suitably normed underlying sequences are given. *Research supported by the Consejerı´a de Educación y Juventud de la Junta de Extremadura and the Fondo Social Europeo, grant IPR98A023.

Bayesian inference for bisexual galton-watson processes

In this work, an approach to the Bayesian estimation in a bisexual Galton-Watson process is considered. First we study an important parametric case assuming offspring distribution belonging to the bivariate series power family of distributions and then, we continue to investigate the nonparametric case. In both situations, Bayes estimators under weighted squared error loss function, for means, variances and covariance of the off spring distribution are obtained. For the superadditive case, the Bayes estimation of the asymptotic growth rate is also considered. Illustrative examples are given.

Limit behaviour for a subcritical bisexual Galton{Watson branching process with immigration (

A bisexual Galton-Watson branching process allowing immigration of females and males is considered and for the subcritical case, i.e. growth rate less than one, the limit behaviour of related sequences is investigated.

Bisexual Galton‐Watson Branching Process in Varying Environments

Abstract In this paper we introduce a bisexual Galton‐Watson branching process (BGWP) in which the offspring probability distribution is different in each generation. We obtain some relations among the probability generating functions (pgf) involved in the model and, making use of mean growth rates and fractional linear functions (flf), we provide sufficient and necessary conditions for its almost sure extinction.

Non-parametric Bayesian estimation for multitype branching processes through simulation-based methods

The problem of statistical inference from a Bayesian outlook is studied for the multitype Galton-Watson branching process, considering a non-parametric framework. The only data assumed to be available are each generations population size vectors. The Gibbs sampler is used in estimating the posterior distributions of the main parameters of the model, and the predictive distributions for as yet unobserved generations. The algorithm provided is independent of whether the process becomes extinct or not. The method is illustrated with simulated examples.

ESTIMATION OF THE OFFSPRING DISTRIBUTION AND THE MEAN VECTOR FOR A BISEXUAL GALTON-WATSON PROCESS

In this paper, we obtain estimators for the offspring distribution and the mean vector of a bisexual Galton-Watson process and derive, for these estimators, some conditional moments, given non extinction, and unconditional moments, asymptotic properties and confidence intervals. As an illustration, we consider an simulated example. ** Research supported by the Consejeria de Educación y Juventud de la Junta de Extremadura and the Fondo Social Europeo, grant IPR98A023

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.

Bisexual branching processes to model extinction conditions for Y-linked genes.

In a two-sex monogamic population, the evolution of the number of carriers of the two alleles of a Y-linked gene is considered. To this end, a multitype bisexual branching model is presented in which it is assumed that the gene has no influence on the mating process. It is deduced from this model that the average numbers of female and male descendants per mating unit constitute the key to determining the extinction or survival of each allele. Moreover, the destiny of each allele in the population is found not to depend on the behavior of the other.

Bisexual branching processes in a genetic context: rates of growth for Y-linked genes.

A multitype bisexual branching process is considered to model the behaviour of a Y-linked gene with two genotypes in a two-sex population. It is assumed perfect fidelity mating with preference of females for the males carrying certain allele of the gene. Under these assumptions, we study the rate of growth of each genotype on the event of non-extinction. The rate of growth of a genotype may depend on whether the other survives or becomes extinct and, in general, both genotype frequencies grow at different rates. We also investigate conditions for the simultaneous explosion of both genotypes to have positive or zero probability.

Limiting behaviour for superadditive bisexual Galton-Watson processes in varying environments

In this paper, the class of superadditive bisexual Galton-Watson processes in varying environments is considered and the limiting behaviour of the number of mating units per generation, suitably normalized, is investigated. Two sequences of normalizing constants are considered and, for each one of them, limit theorems are established. In particular, conditions for the almost sure,L1 andL2 convergence to a non degenerate at 0 random variable are determined.