Mixtures of Erlang distributions and renewal processes based on them

Erlang Mixtures are useful in many fields including data modeling, statistical genetics and network communications. They are formed by mixing different gamma distributions with positive integer shape parameter j (also called Erlang) and common scale parameter lambda with a discrete mixing distribution s = (s1, s2,...). Definitions, notations, properties, and examples are given and results about moments, moment generating function, and closeness of Erlang mixture pdfs and cdfs are developed. Computational and graphical tools necessary for the use of these distributions are also provided. Four methods for estimating Erlang mixture parameters are discussed: the method of moments, density and moment least squares, and maximum likelihood estimation methods. Simulations are used to compare and evaluate these methods. Rice recombination data and the Old Faithful data are modeled by Erlang Mixtures. Renewal processes with Erlang mixtures interevent distances are discussed and computational expressions for their multilocus probabilities are presented.