The Two Component Generalized Finite Erlang Mixture and its Properties and Special Cases

This research proposes the construction of a two-component generalized finite Erlang mixture with four parameters. Three distinct cases of this mixture are presented, differing in their mixing weights and corresponding component probability (Erlang) distributions. Special cases of these mixtures, including the one-, two-, and three-parameter finite Erlang mixtures, have also been derived. The statistical properties examined for these mixtures include the distribution function, survival function, hazard function, moment generating function, raw and central moments, mean, variance, coeffcient of skewness, coeffcient of kurtosis, and order statistics. Parameter estimation for the finite Erlang mixtures was conducted using both the method of moments and maximum likelihood estimation. Furthermore, the 4-parameter mixed Erlang distributions were applied to a real dataset on the relief times of patients receiving an analgesic, to evaluate their goodness of fit. The results demonstrate the potential of these mixtures to provide robust modeling for empirical data, suggesting their applicability in various statistical and practical contexts.