International Journal of Stochastic Analysis
Volume 2013 (2013), Article ID 963831, 10 pages
http://dx.doi.org/10.1155/2013/963831
Research Article

Modeling Neutral Evolution Using an Infinite-Allele Markov Branching Process

Xiaowei Wu1 and Marek Kimmel2,3

1Department of Statistics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA
2Departments of Statistics and Bioengineering, Rice University, Houston, TX 77005, USA
3Systems Engineering Group, Silesian University of Technology, 44-100 Gliwice, Poland

Received 6 November 2012; Revised 14 February 2013; Accepted 18 February 2013

Academic Editor: Peter Olofsson

Copyright © 2013 Xiaowei Wu and Marek Kimmel. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider an infinite-allele Markov branching process (IAMBP). Our main focus is the frequency spectrum of this process, that is, the proportion of alleles having a given number of copies at a specified time point. We derive the variance of the frequency spectrum, which is useful for interval estimation and hypothesis testing for process parameters. In addition, for a class of special IAMBP with birth and death offspring distribution, we show that the mean of its limiting frequency spectrum has an explicit form in terms of the hypergeometric function. We also derive an asymptotic expression for convergence rate to the limit. Simulations are used to illustrate the results for the birth and death process.