Moment Distribution of Hyper-Erlang Distribution and Its Approximation Application

Moment distributions can be used to study many important problems in the field of social sciences, such as the Gini coefficient of residents’ income, Lorenz curve, etc. The set of Hyper-Erlang distributions is a dense subset of the set of non-negative probability distribution functions, which has a wide applicability, so Hyper-Erlang distributions are chosen as the object of this study. In this paper, we give the analytical expression of the moment distribution of HyperErlang distribution and prove that the moment distribution function of Hyper-Erlang distribution is closed within the set of Hyper-Erlang distribution. Combined with the HyperErlang distribution parameter estimation algorithm, eight commonly used distributions are fitted, and the practicality and effectiveness of the Hyper-Erlang moment distribution method proposed in this paper are verified through the comparison of the original moment distribution of the commonly used distributions and the approximation effect of the Hyper-Erlang moment distribution. The approach proposed in this paper can be used to give a moment distribution expression for any data, and valid approximate moment distribution expressions for all types of probability distributions for which no analytical moment distribution expression can be derived.