Strong consistency of the MLE under two-parameter Gamma mixture models with a structural scale parameter

We study the strong consistency of the maximum likelihood estimator under a special finite mixture of two-parameter Gamma distributions. Somewhat surprisingly, the likelihood function under Gamma mixture with a set of independent and identically distributed observations is unbounded. There exist many sets of nonsensical parameter values at which the likelihood value is arbitrarily large. This leads to an inconsistent, or arguably undefined, maximum likelihood estimator. Interestingly, when the scale or shape parameter in the finite Gamma mixture model is structural, the maximum likelihood estimator of the mixing distribution is well defined and strongly consistent. Establishing the consistency when the shape parameter is structural is technically less challenging and already given in the literature. In this paper, we prove the consistency when the scale parameter is structural and provide some illustrative simulation experiments. We further include an application example of the model with a structural scale parameter to salary potential data. We conclude that the Gamma mixture distribution with a structural scale parameter provides another flexible yet relatively parsimonious model for observations with intrinsic positive values.