This dissertation research is concentrated on the Contaminated Beta(CB) model and its application in micro-array data analysis. Modified Likelihood Ratio Test (MLRT) introduced by [Chen et al., 2001] is used for testing the omnibus null hypothesis of no contamination of Beta(1,1)([Dai and Charnigo, 2008]). We design constraints for two-component CB model, which put the mode toward the left end of the distribution to reflect the abundance of small p-values of micro-array data, to increase the

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... t power. A three-component CB model might be useful when distinguishing high differentially expressed genes and moderate differentially expressed genes. If the null hypothesis above is rejected, we considered developing a method of testing the hypothesis of two-component vs three-component CB model. We first study CB model with one-parameter kernel distribution by fixing the other shape parameter across all the components. Using MLRT introduced by [Chen et al., 2004], we find the feasibility of this model after investigation. Then we consider a three-component CB model and designed constraints to guarantee the identifiability. We also study model selection and use sBIC introduced by [Drton and Plummer, 2017] to determine the number of components. We applied our tests and model to a toddler Down Syndrome data sets.