Colloquia — Summer 2005
Friday, May 27, 2005
Statistical inference for branching processes
Professor Nikolay Yanev
Department of Probability and Statistics
Institute of Mathematics and Informatics
Bulgarian Academy of Sciences
It is well known that branching processes have a lot of applications, in particular in biology and medicine. We will discuss the asymptotic behavior of branching populations having an increasing random number of ancestors. We will present an estimation theory for the mean, variance, and offspring distributions of the processes \(Z_t(n)\) with a random number of ancestors \(Z_0(n)\), as both \(n\) (and thus \(Z_0(n)\) in some sense) and \(t\to\infty\). Non-parametric, consistent, and asymptotically normal estimators will be proposed. Some censored estimators will also be considered. We will show that all results can be transferred to branching processes with immigration, under an appropriate sampling scheme. A software system for simulation and estimation of branching processes will be demonstrated.
Friday, May 20, 2005
Variational comparison method and qualitative analysis of time series
Professor G. S. Ladde
Department of Mathematics
The University of Texas at Arlington
In this work, the study of convergence and stability analysis of stochastic iterative processes under both structural perturbations is outlined. The random structural perturbations are described by a Markov chain with a finite number of states. Under algebraic conditions on rate functions and an intensity matrix associated with the Markov chain, convergence and stability results are obtained. This is achieved through the development and the introduction of variational comparison theorems in the context of a Lyapunov-like function. Furthermore,
hereditary effects and the effects of random structural perturbations are analyzed. The mathematical conditions are algebraically simple, easy to verify, and robust to the parametric changes in the system. Several examples are given to illustrate the usefulness of the technique.