摘要：
There is currently no existing asymptotic theory for statistical inference on the maximum likelihood estimators of the parameters in a mixture of linear mixed models (MLMMs). Despite this many researchers assume the estimators are asymptotically normally distributed with covariance matrix given by the inverse of the information matrix. Mixture models create new identifiability problems that are not inherited from the underlying linear mixed model (LMM), and this subject has not been investigated for these models. Since identifiability is a prerequisite for the existence of a consistent estimator of the model parameters, then this is an important area of research that has been neglected.
MLMMs are mixture models with random effects, and they are typically used in medical and genetics settings where random heterogeneity in repeated measures data are observed between measurement units (people, genes), but where it is assumed the units belong to one and only one of a finite number of sub-populations or components. This is expressed probabalistically by using a sub-population specific probability distribution function which are often called the component distribution functions. This thesis is motivated by the belief that the use of MLMMs in applied settings such as these is being held back by the lack of development of the statistical inference framework. Specifically this thesis has the following primary objectives;
i To investigate the quality of statistical inference provided by different information matrix based methods of confidence interval construction.
ii To investigate the impact of component distribution function separation on the quality of statistical inference, and to propose a new method to quantify this separation.
iii To determine sufficient conditions for identifiability of MLMMs.