Change point estimation based on copula-based Markov chain model for binomial time series data

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賴慶杰(Lai, Ching-Chieh) 查詢紙本館藏

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統計研究所

論文名稱

(Change point estimation based on copula-based Markov chain model for binomial time series data)

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摘要(中)

在序列分析中，改變點的偵測與估計是一典型的問題並且在品質管制中扮演重要的角色。本篇考慮了binomial CUSUM管制圖來偵測改變點。當binomial CUSUM管制圖偵測到改變點，樣本獨立的前提下最大概似估計量可用來估計改變點。然而獨立假設在品管上是存疑的。本篇我們建造了新的模型，我們將序列相關性納入考量，並利用copula-based Markov chain來描述此相關性。我們利用最大概似法求得估計量並建造我們的R套件來推廣我們的研究成果。區間估計我們用parametric bootstrap和大樣本近似兩種方法，並將兩者以模擬做比較。本篇比較了我們提出的方法與文獻中的方法並分析實務上的資料來展示我們的研究成果。

摘要(英)

Detection and estimation of a change point is a classical problem in sequential analysis, and is an important practical issue in statistical process control. This paper is concerned about the binomial CUSUM control chart for detecting a change point for attribute data, which is extensively applied to industrial process control, health care, public health surveillance, and other fields. If the binomial CUSUM chart detects a change point, a maximum likelihood estimator can be used to estimate the change point under the assumption that the observations are independent. However, the independence assumption is questionable in many applications of statistical process control. In this paper, we consider a new change point model, where the serial correlation follows a copula-based Markov chain model and the marginal distribution follows the binominal distribution. We develop algorithms for computing the maximum likelihood estimator, and implement them in our original R package. For interval estimation, we propose a parametric bootstrap procedure and an asymptotic normal approximation procedure. We compare the performance of the two interval estimation procedures by simulations. We also compare our proposed method with the existing estimators in terms of mean squared error. We analyze the jewelry manufacturing data for illustration.

關鍵字(中)

★ 馬可夫鏈

關鍵字(英)

★ Binomial time series data
★ copula
★ Markov chain
★ serial dependence
★ attribute control chart
★ parametric bootstrap

論文目次

Chapter 1 Introduction…………….……………………………………………………………………1
Chapter 2 Background………………..…………………………………………………………………3
2.1 Settings and notations……………………………………………………………………………3
2.2. Binomial CUSUM control chart………………………………………………………………...4
2.3. Maximum likelihood estimator (MLE) of change point………………………………………5
2.4. Page’s last zero estimator………………...……………………………………………………...6
2.5. Combined estimator of MLE and last zero………...……………………………………………6
Chapter 3 Proposed method………..…………………………………………………………………...7
3.1 Copula……………………………………………………………………………………………7
3.2. Proposed change point model………………………………………………………………….10
3.3. Proposed estimator for change point………………………………………………………….15
3.4. Numerical maximization and convergence issue………………………………………………17
3.5. Asymptotic normal approximation…………………………………………………………….20
3.5. Confidence set for change point………………………………………………………………22
Chapter 4 Computation and software.…………………………………………………………….......24
4.1 Arguments in the function……………………………………………………………………...24
4.2 Illustration………………………………………………………………………………………26
Chapter 5 Simulation study.……………………………………………………………………….......29
5.1 Simulation designs……………………………………………………………………………...29
5.2 Results for comparing the NR method and the nlm method……………………………………30
5.3 Sensitivity of the proposed estimator under different assumed values of .………………...30
5.4 Results of comparing the Hessian and bootstrap methods……………………………………...31
5.5 Simulation results for comparing existing methods…………………………………………….35
Chapter 6 Data Analysis…...……………………………………………………………………….......41
6.1 Jewelry data…………………………………………………………………………………….41
6.2 Finance data…………………………………………………………………………………….46
Chapter 7 Conclusion…...………………………………………………………………………….......50
Acknowledgements…...…………………………………………………………………………….......50
References.………………………….………...……………………………………………….…….......51
Appendix A……………………………………………...…………………………………….…….......53
Appendix B-1.…………………………………………...…………………………………….…….......56
Appendix B-2.…………………………………………...…………………………………….…….......63
Appendix C.……………...……………………………...…………………………………….…….......64
Appendix D.……………...……………………………...…………………………………….…….......66

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指導教授

江村剛志(Takeshi Emura)

審核日期

2020-7-30

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