對於一個多階段的製造過程，品質或相關突發狀況的根本原因分析是一項重要的議題。而如此大量的機台中，往往是少數的問題機檯導致產能損失。為了能減少損失，應該透過少量的樣品盡快地找出問題機檯，並採取適當的補救措施。由於問題機檯可能導致品質特徵的數值偏移或是不穩定，這引發我們思考關於位置－分散模型 (location-dispersion models) 的模型選擇程序，並證明這套程序具有模型選擇的一致性 (model selection consistency)。簡而言之，我們借用了 Ing and Lai (2011) 中的構想，對位置模型先進行模型選擇，接著介紹一個適用非線性模型且能夠快速篩選變數的演算法－柴比雪夫貪婪演算法 (Chebyshev greedy algorithm)。我們利用它結合位置模型的選擇結果對分散模型進行選模，並利用高維度訊息準則 (high-dimensional information criterion) 汰除偽陽性的變數。模擬結果與關於晶元製造資料的實際應用說明了我們的方法具有實用性。

Root-cause analysis of failures and quality deviations is an important issue for multistage manufacturing processes that often involve a large number of tools. In order to improve yield rates, one should identify the problematic tools as soon as possible. Since there may be only a few problematic tools which either shift the means of the product measurements or inflate their variances, we are led to consider a model selection procedure for high-dimensional sparse location-dispersion models. We propose a two-stage procedure and show it has the variable selection consistency. More specifically, we first borrow the idea from Ing and Lai (2011) to select relevant variables for the location model. We then remove the location model base on the selection results to obtain the dispersion only model and introduce Chebyshev greedy algorithm (CGA) to quickly screen variables and high-dimensional information criterion (HDIC) to achieve model selection consistency for dispersion models. Simulation results and applications to wafer data are provided to shed light on the performance and usefulness of our approach.

Abstract
Acknowledgement
1 Introduction 1
2 Model Selection Procedure for Dispersion Model 2
3 Convergence Rate of CGA for Dispersion Model 5
4 Consistency of CGA+HDIC+Trim under Strong Sparsity 8
5 Simulation Studies 10
5.1 Location-Dispersion Models 10
5.2 Multinomial Logistic Regression Models 13
6 Real Data Analysis 15
References 21
Appendix A 22

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