Statistical methodology for rank based sampling design and finite mixture models

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In many research studies, the relationship between variables is investigated to classify and explain better the homogeneities and heterogeneities in datasets. In fishery surveys, for example, this classification can be translated into the age determination of fish from length-frequency data. In Endocrinology, the classification can be used to determine the Thyroid disorder status of patients. In Neurobiology, this classification can be used to understand the hippocampal neural activities that are linked with the ability to remember the order of life events. Classification and cluster analysis are fundamental elements of statistical science and many applications. Mixture probabilistic models are flexible and powerful statistical tools for model-based classification and  clustering. The power of mixture models allows the use of likelihood-based methods, which are preferred for various statistical inferences, including hypothesis testing, classification, clustering and predictive density estimation, to name a few. The flexibility of mixture models allows the analysis of complex datasets. This includes the situations where 1) the data structure is itself complex and challenging or 2) a different sampling design is more appropriate for the target population. As technology advances, we are witnessing dramatic progress in research and experiments producing complex and high-volume datasets, e.g., in Neuroimaging such as  functional magnetic resonance imaging (fMRI) data. I plan to investigate how we can deal with high dimensional mixture models  and handle them with low-dimensional models. This leads to more efficient classification and cluster analysis and enables us to make better inferences about the heterogeneity in data. In many statistical surveys, there are two critical challenges that researchers often face: 1) the cost of taking measurements and 2) obtaining more representative samples from the population of interest. In Osteoporosis research, for example, the diagnosis of  bone disorder requires measuring  bone mineral density (BMD)  through dual X-ray absorptiometry imaging from skeletal sites such as femoral neck and lumber spine. This procedure is costly and time-consuming; however, practitioners have access to easy-to-measure characteristics such as age, weight or baseline BMD from patients. Rank-based sampling designs present a  solution to the above challenges. These sampling designs enable us to use  this easy-to-measure information (e.g., age) to improve the data collection process leading to  more representative samples from the population of interest (e.g., bone disorder).  I plan to investigate how we can exploit these informative rank-based samples to make better inferences about the  mixture models.