Approximation of Gamma Distribution by Two-Branch Mixture of Erlang Distributions

The Gamma and Erlang distributions are commonly utilized. The Gamma distribution is easier to use for parameter estimation and provides a better fit for the data, but it has poorer analytical computational properties. On the other hand, the Erlang distribution has better analytical computational properties, but it provides a weaker fit for the data than the Gamma distribution and is more difficult to use for parameter estimation. The paper aims to address the problem of harnessing the benefits of both the Gamma distribution for its ease of parameter estimation and the Erlang distribution for its good analytical properties. Our study focuses on utilizing the Erlang distribution to approximate the Gamma distribution efficiently for a given parameter. We provide a rigorous verification of the approximation method and validate its effectiveness using several examples.