臺灣博碩士論文加值系統

近年來，利用偏斜t分配的多變量混合建模方法對於穩健性的基於模式分群與分類上已被視為一個有效且具彈性的工具。研究學者實際上普遍會遇到資料含有遺失值的問題。在這篇論文中，當資料含有遺失值時，對於多變量偏斜t 模型的最大概似函數的估計我們提供一個計算上便利的EM-type 程序。此外，我們提出一個基於共同訊息的方法，使用分數的外積來求取最大概似估計值近似的共變異數矩陣。在我們的演算法中，為了協助推導與執行上的便利，我們利用兩個輔助
的排列矩陣快速判斷每筆觀察項目中可觀察到與遺失的部分。最後，我們藉由不同遺失比例下的模擬資料與含有遺失值的實例來闡述所提出方法的實用性。

Multivariate mixture modeling approach using the skew-t distribution has been recently examined as a powerful and flexible tool for robust model-based clustering and classification. Missing data are a ubiquitous problem for researchers encountered in practice. In this thesis, we offer a computationally flexible EM-type procedure for maximum likelihood estimation of multivariate skew-t mixture models when missing values occur in data. Further, we present a common information-based approach to approximating the asymptotic covariance matrix of the ML estimator using the outer product of the scores. To assist the development and ease the implementation of our algorithm, we make use of two auxiliary permutation matrices for fast determining
the observed and missing parts of each observation. The practical usefulness of the proposed methodology is illustrated through simulations with varying proportions of artificial missing values and a real data example with genuine missing values.

1. 導論 1
2. 背景知識 3
2.1. 受限的多變量偏斜常態分配 3
2.2. 受限的多變量偏斜t 分配 4
3. 具遺失訊息的多變量混合偏斜t 模型7
3.1. 模型架構 7
3.2. 用ECM 演算法計算最大概似估計 9
3.3. 估計的標準誤差 12
4. 計算策略 14
4.1. 初始值的設定 14
4.2. 收斂性評估 14
4.3. 模型選擇 15
4.4. 效能評估 16
5. 模擬研究 17
5.1. 模擬一 17
5.2. 模擬二 19
6. 實例分析 22
7. 結論 27
A. 具遺失訊息的MSTMIX 模型邊際分配推導 28
B. ECM 演算法之E 步驟的證明 30
C. 印地安人糖尿病相關資料分析 38

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