Multitype branching process: extensions, inferences and applications

Abstract

The idea of a simple branching process is that a parent produces offsprings. They then further take the role of parents and produce o springs, and the processes continue either until infinity or until extinction. We relax the assumptions of simple branching process and introduce age structure in the population, where an individual is allowed to survive for many time periods and can also give birth more than once. In this thesis, the following three issues are examined - 1. We model the process using multitype branching process and derive conditions on the mean matrix that determines the long-run behaviour of the process. Next, we analyze the distribution of the number of forefathers in a given generation. Here, number of forefathers of an individual is defined as all the individuals since zeroth generation who have contributed to the birth of the individual under consideration. We derive an exact expression for expected number of individuals in a given generation having a specified number of forefathers and do further analysis of a special case. 2. Inference Problem - In the same setup as given above, for the special case, we study the problem of estimation of parameters using di erent methods and give the comparative performance of estimates under some assumptions. 3. Coalescence Problem in branching process is as follows: Pick two individuals from nth generation at random and trace their lines of descent back till they meet. Call that random generation as X(n) and study the properties of X(n). We study this problem for some deterministic and random cases considering age structure in the population. Explicit expressions about some mathematical properties of X(n) have been derived for some cases of deterministic trees. For random trees, we provide explicit expression for some special cases. We also derive properties of X(n) as n goes to infinity.