Consistency Of The MLE Under Finite Gamma Mixture Models

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 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.Approaches in literature 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 with respect to the shape parameters to the log-likelihood function.In this paper,we continue to explore other approaches to restore the consistency of the maximum likelihood estimator of the mixing distribution under two-parameter Gamma mixture models.The main contents in the dissertation are as follows:1.In Chapter 4,we show that the constrained maximum likelihood estimator of the mixing distribution under two-parameter Gamma mixture models is consistent by placing a sufficiently small positive lower bound on the scale parameter space when the constrained space contains the true value of scale parameters.2.The maximum likelihood estimator under a special Gamma mixture models is consistent.When the scale or shape parameter in the finite Gamma mixture model is structural,the maximum likelihood estimator of the mixing distribution and structural parameter are well defined and consistent.We provide a general and rigorous proof of these consistency results.In Chapter 5,we show that the maximum likelihood estimator of the mixing distribution and structural parameter under finite Gamma mixture models with structural shape parameter is 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.Compared to the case with structural shape parameter,establishing the consistency when the scale parameter is structural is technically challenging.In Chapter6,we show that the maximum likelihood estimator of the mixing distribution and structural parameter under finite Gamma mixture models with structural scale parameter is strongly consistent.We also 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 parameter provides another flexible yet relatively parsimonious model for observations with intrinsic positive values.3.The consistency results of three different types of penalized maximum likelihood estimators of the mixing distribution under two-parameter Gamma mixture models are established.In Chapter 7,we introduce a new penalty on the ratio of subpopulation shape parameters to obtain a strongly consistent penalized maximum likelihood estimator.The simulation studies support the consistency result on the penalized maximum likelihood estimator.The penalized maximum likelihood estimator obtained is found to perform well,especially when some subpopulation contains large shape parameter.A data example is given to illustrate the application of the finite Gamma mixture model.In Chapter 8,we show that the penalized maximum likelihood estimator of mixing distribution is consistent when a penalty is imposed on the subpopulation scale parameters.In Chapter 9,we show that the penalized maximum likelihood estimator of mixing distribution is consistent when a penalty is imposed on the ratio of subpopulation scale parameters.The simulation studies support the two consistency results on the penalized maximum likelihood estimator.Another data example is given to illustrate the application of the finite Gamma mixture model.