在高維度 Cox 模式之下,我們考慮柴比雪夫貪婪演算法(Chebyshev’s Greedy
Algorithm, CGA)進行變數過濾(variable screening)。透過逐步選取變數的方式,我們証明了 CGA 的在整個選取過程的一致收斂性,並証明了在 CGA 迭代停止
之後具有確認過濾(Sure screening) 的性質。此外,我們發展了高維度訊息準則
(High-dimensional information criterion, HDIC),並根據該準則設計了三步驟的變數選取流程,証明了該選取流程具有變數選取的一致性。最後,我們將本文提供的模式選取方法套用至漫大性 B 細胞淋巴癌(Diffusion Large B-cell Lymphoma,DLBCL)的實驗資料(Rosenwald et al., 2002)。

Motivated by the conditional sure independence screening [Barut et al. (2016), CSIS], we consider a greedy type method, namely, the Chebyshev Greedy Algorithm [Temlyakov (2015), CGA]. Different to CSIS, which improves the screening performance by conditioning on prior knowledge, CGA sequentially construct the conditioning set to include new variables. In this dissertation, we propose using CGA as a variable screening tool for the high-dimensional survival data and study its convergence rate. We show that the sure screening property can be achieved by CGA with a theoretically justified stopping rule and propose a greedier variant of CGA (gCGA) to enhance its finite sample performance in terms of variable screening. We develop a three-step variable selection procedure which is an essemble of implementing CGA and a high-dimensional information criterion (HDIC), and give the condition under which the variable selection consistency can be achieved by HDIC. The utility of the proposed method is examined by extensive simulation studies and the analysis of a diffuse large B-cell lymphoma (DLBCL) dataset.

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