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 study, the intercept is not identifiable, contrary to prospective studies. In combined case-control studies, however, the intercepts are proved to be identifiable under mild conditions. The resulting maximum likelihood estimates of the intercepts and slopes are proved to be consistent and asymptotically normal, with asymptotic variances achieving the semiparametric efficiency lower bound.當代科技發展中，如何識別高維數據中的重要變量識別具有巨大挑戰性，尤其是當預測變量之間存在復雜關系時。在第壹個章節中，沒有任何回歸模型的假設明，我們引入了基於分位數關聯的統計量來確定重要的預測變量，該變量足夠靈活以尋找廣泛意義上的變量依賴性。我們還開發了逐步過程以進壹步識別重要變量。由於不涉及優化或重采樣，因此計算效率很高。這壹章節還證明了其理論性質，並證明了該方法漸近地控制了在給定了顯著性水平的錯誤發現率。數值研究（包括模擬研究和實際數據分析）包含支持性證據，表明該方法在實際數據方面表現合理。在第二個章節中，我們考慮了分位數回歸模型。分位數回歸中現有的方法假設先驗概率均等地對待所有預測變量，但是與現有方法不同的是，這個方法利用預測變量之間的圖形結構來改善稀疏分位數回歸中參數估計，模型選擇和預測的性能。我們還提出了壹種采用線性化技術的乘數交替方向算法(ADMM) 以數值方式實現該方法，並證明了其收斂性。第三部分，我們考慮了邏輯回歸模型的有效估計。如果兩個或兩個以上分別進行控制采樣的數據集合並起來計算，那將會極大改善分析結果。即使是完全無關的研究，也可以合並分析，從而產生更好的統計結果，例如更有效的參數估計。在這壹部分，我們展示了當使用邏輯回歸模型時，合並分析將產生更準確的估計所有參數包括截距項。眾所周知，在單個病例對照研究中，與先驗性分析不同的是，截距無法識別。但是，在結合的病例對照研究中，截距被證明是在溫和條件下可識別。截距和斜率的最大似然估計被證明是有壹致性的的，並且是漸近的，漸近方差達到了半參有效估計的下界。Tang, Wenlu.Thesis Ph.D. Chinese University of Hong Kong 2020.Includes bibliographical references (leaves 121-135).Abstracts also in Chinese.Title from PDF title page (viewed on 17, September 2021)