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.

有限 Gamma 混合模型通常用于描述收入数据、保险数据和响应值本质上为正的应用中的数据的随机性。然而，用于模型拟合的流行似然方法不适用于该模型，因为它的似然函数是无界的。因此，最大似然估计量没有明确定义。已经开发了其他方法来实现对混合分布的一致估计，例如在形状参数上设置上限或对对数似然函数添加惩罚。在本文中，我们表明，如果有限 Gamma 混合模型中的形状参数是结构性的，那么混合分布的直接最大似然估计量是明确定义的并且是强一致的。我们还提供了模拟结果，证明了估计量的一致性。我们用结构形状参数说明模型在家庭收入数据中的应用。拟合的混合分布导致了关于可支配收入水平的几种可能的亚群结构。