A fixed-point algorithm to estimate the Yule–Simon distribution parameter

The Yule–Simon distribution is a discrete probability distribution related to preferential attachment processes such as the growth in the number of species per genus in some higher taxon of biotic organisms, the distribution of the sizes of cities, the wealth distribution among individuals, the number of links to pages in the World Wide Web, among others. In this paper we present an algorithm to, given a set of observations stemmed from a Yule process, obtain the parameter of the Yule–Simon distribution with maximum likelihood. In order to test our algorithm, we use a modified Polya urn process simulation to generate some data that was used as input to our algorithm. We make a comparison of our algorithm with other methods and also we show an application to some empirical data.