In many applications of two-component mixture models such as zero-inflated models and cure rate models, it is often of interest to conduct inferences for the mixing weights. Score and Wald tests have been particularly useful for this purpose. But the existing testing procedures often rely on restrictive assumptions such as the constancy of the mixing weights under the alternative. In this thesis, we suggest an extension to covariate dependent mixing weights, which is very typical in real applications of these finite mixture models. We discuss these tests under the two representations of the mixture model, the marginal and the hierarchical representations.Under the marginal representation of the mixture model, we propose a technique based on a decomposition of the mixing weights into terms that have an obvious statistical interpretation. We make good use of this decomposition to lay the foundation of the test. We apply these techniques to both the score and Wald tests. An important feature of the score tests is that they do not require the alternative model to be fitted. However, the advent of computer software with robust functions and procedures has made simple a routine fitting of the alternative model in practice. We make good use of this opportunity by using results generated from these fits to develop a Wald test statistic to evaluate homogeneity in this class of models. We show how the proposed Wald test can be performed with a minimal programming effort for the practicing statistician. Simulation results show that the proposed score test and Wald test statistics can greatly improve efficiently over test statistics based on constant mixing weights. A real life example in dental caries research is used to illustrate the methodology.The tests of homogeneity developed under the marginal representation of the mixture model are usually based on the two-sided alternatives with potentially negative mixing weights. These tests, however, are not valid for two-component models which maintain the hierarchical representation of the mixture model. But in practice, zero-inflated models that maintain this representation are usually fit. To assess the inclusion of mixing weights in two-component models under the hierarchical representation, we adopt a general formulation where the mixing weight is allowed to depend on covariates. A complication is that some nuisance parameters are unidentified under the null. One possible solution to the identifiability problem is to fix these nuisance parameters conditional upon which the pivotal function results in processes of these parameters. We extend this idea by deriving a score-based test statistic from these processes. We establish the limiting null distribution of this statistic as a functional of chi-squared processes which is rigorously approximated by a simple resampling procedure. We apply these tests to zero-inflated models for discrete data. Our simulation results also show that the proposed tests can greatly improve efficiently over tests based on constant mixing weights. The practical utility of the methodology is illustrated using a real life data examples, the dental caries data.
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