Consistency of the MLE under a two-parameter Gamma mixture model with a structural shape parameter

Finite Gamma mixture models are often used to describe randomness in income data, insurance data, and data in applications where the response values are intrinsically positive. The popular likelihood approach for model fitting, however, does not work for this model because its likelihood function is unbounded. Because of this, the maximum likelihood estimator is not well-defined. Other approaches have been developed to achieve consistent estimation of the mixing distribution, such as placing an upper bound on the shape parameter or adding a penalty to the log-likelihood function. In this paper, we show that if the shape parameter in the finite Gamma mixture model is structural, then the direct maximum likelihood estimator of the mixing distribution is well-defined and strongly consistent. We also present simulation results demonstrating the consistency of the estimator. We illustrate the application of the model with a structural shape parameter to household income data. The fitted mixture distribution leads to several possible subpopulation structures with regard to the level of disposable income.