高維度共變異矩陣之推估及其應用

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吳嘉馨(Chia-hsin Wu) 查詢紙本館藏

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統計研究所

論文名稱

高維度共變異矩陣之推估及其應用
(Estimation of high-dimensional covariance matrices and their applications)

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摘要(中)

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。

摘要(英)

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.

關鍵字(中)

★ 高維度共變異矩陣
★ Cholesky分解法
★ 貪婪訊息法

關鍵字(英)

★ Greedy algorithm
★ Information criterion
★ Covariance

論文目次

第一章緒論..................................... 1
第二章文獻回顧................................. 3
2.1 Markowitz 相關問題探討.................................... 3
2.2 共變異矩陣之推估-調整(Regularized)估計法.................. 5
第三章高維度共變異矩陣推估..................... 7
3.1 修正的Cholesky 分解法(Modified Cholesky decomposition).... 8
3.2 貪婪演算法(Greedy algorithm) ............................. 9
3.2.1 純貪婪演算法(Pure greedy algorithm: PGA) ....................... 10
3.2.2 正交貪婪演算法(Orthogonal greedy algorithm: OGA) ............... 11
3.3 估計方法及過程.......................................... 12
3.4 相關理論基礎............................................ 14
3.4.1 符號定義....................................................... 14
3.4.2 貪婪訊息法之一致性............................................. 16
3.4.3 共變異矩陣在operator norm 之收斂性............................. 17
第四章共變異矩陣推估之結果與比較.............. 19
4.1 貪婪演算法 vs 訊息準則.................................. 19
4.2 模擬與比較.............................................. 21
4.2.1 一階自我迴歸: AR(1) ............................................ 22
4.2.2 一階移動平均: MA(1) ............................................ 30
第五章結論與建議.............................. 38
5.1 結論.................................................... 38
5.2 建議.................................................... 39
參考文獻....................................... 40

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.
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[9] Lai, T.L. and Xing, H.(2008). Statistical Models and Methods for Financial Markets, Springer, New York.
[10] Levina, E., Rothman, A. and Zhu, J.(2008). Sparse estimation of large covariance matrices via a nested lasso penalty, The Annals of Applied Statistics, Vol.2, pp.245-263.
[11] Markowitz, H.(1952). Portfolio selection, The Journal of Finance, Vol.7, pp.77-91.
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指導教授

銀慶剛(Ching-kang Ing)

審核日期

2008-6-19

推文