高維度共變異矩陣之推估及其應用

Estimation of high-dimensional covariance matrices and their applications

吳嘉馨(Chia-hsin Wu)

指導教授 ： 銀慶剛

國立中央大學/理學院/統計研究所/碩士(2008年)

摘要

有鑑於高維度資料的普遍性與重要性，本篇論文的研究重點是在適當的稀疏性的假設下估計高維度共變異矩陣及其反矩陣，順帶提及估計結果在Markowitz模型上的應用。本篇論文提出的貪婪訊息法是利用修正的Cholesky分解法及兩種貪婪演算法與數種訊息準則間的配合去做推估，模擬結果顯示，相較於Bickel和Levina所提出的截段估計法及門檻估計法而言，貪婪訊息法可在大部份的情況下得到令人滿意的估計結果。

關鍵字

高維度共變異矩陣 ； Cholesky分解法 ； 貪婪訊息法

並列摘要

In consideration of its growing importance in various applications, this thesis will focus on estimation of high-dimensional covariance matrices and their inverses under proper sparseness assumptions. We proposed a so-called greedy information procedure which combined modified Cholesky decompositions for the population covariance matrices and greedy algorithms with corrected information criterions as their stopping rules. We will also apply the proposed procedure to Markowitz models and compare its performance with those of banding and thresholding methods given by Bickel and Levina. Simulation results show that our method performs favorably in most cases.

並列關鍵字

Greedy algorithm ； Information criterion ； Covariance

參考文獻

[1] Bickel, P.J. and Levina, E.(2008a). Regularized estimation of large covariance matrices, The Annals of Statistics, Vol.36, pp.199-227.

[2] Bickel, P.J. and Levina, E.(2008b). Covariance regularization by thresholding, working paper.

[3] Buhlmann, P.(2006). Boosting for high-dimensional linear models, The Annals of Statistics, Vol.34, pp.559-583.

[6] Frankfurter, G.M., Phillips, H.E. and Seagle, J.P.(1971).Portfolio selection: The effects of uncertain means, variances and covariances, The Journal of Financial and Quantitative Analysis, Vol.6, pp.1251-1262.

[7] Ing, C.K., Lai, T.L. and Chen, Z.(2008). Model selection criterion and its consistency in boosting for high-dimensional sparse linear models, working paper.