在實務應用，線性迴歸是一個常用來建立解釋變數與反應變數關係的方法。本論文探討具有時
間序列干擾項的迴歸模型的統計推論，主要包含以下兩部分。論文的第一部分，我們探討長記
憶過程的逆自我共變異數矩陣的估計問題。首先我們利用修正的喬萊斯基分解與一個遞增階次
的自我迴歸模型，提出一個估計長記憶過程的逆自我共變異數矩陣的方法，並得證此逆矩陣估
計量具有一致性。接著，我們考慮具有長記憶時間序列干擾項的線性迴歸模型。由於干擾項是
無法觀測的，我們利用最小平方的殘差估計干擾項，進而估計其逆自我共變異數矩陣。我們證
明此方法所獲得的逆矩陣估計量依然具有一致性，並應用到具有長記憶時間序列干擾項迴歸模
型的係數估計。最後我們以模擬研究檢驗所推導的理論性質。

本論文的第二部分，我們探討高維度稀疏迴歸模型的選模議題。當迴歸模型的干擾項為獨立且
同分佈時，文獻中已有多種具有一致性的選模方法。然而，鮮少研究是探討迴歸模型的干擾項
同時具有異質變異與時間相關性的選模推論。本研究主要的目標是對此類模型，提供具有一致
性的選模方法。我們考慮一個具有時間序列干擾項(包括短記憶過程與長記憶過程)的高維度稀
疏迴歸模型，此模型包含位置擴散模型。我們利用正交貪婪演算法來依序選取解釋變數，並使
用高維度訊息準則移除無相關的解釋變數，以達到選模的一致性。我們利用模擬研究來測試所
提出選模方法之可行性。在實證分析方面，我們將此選模方法應用到晶圓測試的資料，尋找造
成晶圓品質不良的問題機台。

Linear regression is a well-known method to establish relationship between responses
and explanatory variables, and has been used extensively in practical applications. This
dissertation consists of two parts focus on statistical inference for linear regression models
with time series errors. The first part concerns the problem of estimating inverse autocovariance
matrices of long-memory processes admitting a linear representation. A modified
Cholesky decomposition and an increasing order autoregressive model are adopted to construct
the inverse autocovariance matrix estimate. We show that the proposed estimate is
consistent in spectral norm. We further extend the result to linear regression models with
long-memory time series errors. In particular, the same approach still works well based
on the estimated least squares errors when our goal is to consistently estimate the inverse
autocovariance matrix of the error process. Applications of this result to estimating
unknown parameters in the aforementioned regression model are also given. Simulation
studies are performed to confirm the theoretical results.

In the second study of this dissertation, we consider model selection in sparse high-dimensional
regression. High-dimensional model selection with independent and identically
distributed errors is a much studied problem. However, little attention has been
focused on heteroscedasticity and time series errors. This work aims at providing a consistent
model selection procedure for high-dimensional sparse regression models with time
series errors. We propose a high-dimensional sparse regression model with short- or long-
range dependent errors. Moreover, our proposed model includes the location-dispersion
model. The first step in our model selection procedure is to sequentially select predictors
via an orthogonal greedy algorithm (OGA). To achieve consistent selection, we use a
high-dimensional information criterion (HDIC) to remove irrelevant predictors. Simulation
studies are conducted to illustrate our theoretical findings. In addition, we apply the
approach to wafer acceptance test (WAT) data, and investigate and identify problematic tools.

論文審定書 i
誌謝 ii
摘要 iii
Abstract iv
1 Introduction 1
2 Estimation of inverse autocovariance matrices for long-memory processes 4
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Long-memory model and our proposed estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6
2.1 Long-memory model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Our proposed estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.1 Bias analysis of banded Cholesky factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10
3.2 Estimation analysis of banded Cholesky factors . . . . . . . . . . . . . . . . . . . . . . . . . . . .13
4 Some extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.1 The proposed estimate based on the least squares residuals . . . . . . . . . . . . . . . . . 17
4.2 The finite predictor coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19
4.3 The rate of convergence of the FGLSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5 Simulation study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21
5.1 Selection of the banding parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5.2 Finite sample performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22
6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29
3 Model selection for high-dimensional sparse regression model with time series errors 30
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2 Model and methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.1 Regression model with time series errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3 Theoretical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1 Selection consistency for our proposed method . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Extension to location-dispersion model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37
4 Simulation study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40
4.1 Parameter setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2 Simulation examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40
5 Real example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .47
6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .48
4 Future work 49
References 53
Appendices 56
A Proofs in Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .56
B Proofs in Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .72

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