Quantile-related high dimensional variable selection and combining case-control studies

Ph.D. ; In modern scientific discoveries, important variables identification in analyzing high dimensional data is intrinsically challenging, especially when there are complex relationships among predictors. ; In the first part of the thesis, without any specification of a regression model, we introduce a quantile association-based statistic to identify influential predictors, which is flexible enough to capture a wide range of dependence. A stepwise procedure is developed for further identifying important variables. It is computationally efficient as no optimization or resampling is involved. This chapter also proves its theoretical properties and justifies the proposal asymptotically controls the false discovery rate at a given significance level. Numerical studies including simulation studies and real data analysis contain supporting evidence that the proposal performs reasonably well in practical settings. ; In the second part, we consider the quantile regression model. But different from existing methods in quantile regression which treat all the predictors equally with the same priori, we take advantage of the graphical structure among predictors to improve the performance of parameter estimation, model selection, and prediction in sparse quantile regression. An alternating direction method of multipliers (ADMM) algorithm with a linearization technique is proposed to implement the proposed method numerically, and its convergence is justified. ; In the third part, an efficient estimation for logistic regression model is considered. It would be hugely beneficial if two or more separately conducted case-control studies, even for entirely irrelevant purposes, can be merged together with a unified analysis that produces better statistical properties, e.g., a more accurate estimation of parameters. In this part, we show that, when using the popular logistic regression model, the combined analysis produce a more accurate estimation of the slope parameters. It is known that, in a single logistic case-control ...