Change Point Estimation Based on Copula-based Markov Chain Model for Normal Time Series

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劉蓮希(Lien-Hsi Liu) 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 Normal Time Series)

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摘要(中)

變更點檢測是時間序列分析的重要組成部分，因為變更點的存在表明數據生成過程中發生了突然而重大的變更。檢測變更點可以幫助我們事前預警和事後分析，其被應用在許多的領域，如工業質量控制、金融市場分析、網絡流量分析等等。在傳統方法中，假設觀測值是獨立的情況下，可以使用最大概似估計器来估計變化點。然而，在許多實際應用中，觀測值通常是相依的，所以獨立假設的最大概似估計器方法通常
是低效的。在本文中，我們擴展最大概似估計器方法，將其應用在觀測值相依的情況中，我們提出一個新的變化點模型，其中序列相關遵循基於 copula 的馬爾可夫鏈模型，邊際分佈遵循常態分佈，然後我們得到其對應的概似函數，而為了解決最大概似估計量的問題，我們應用了牛頓-拉弗森方法。在實證研究中，我們分析了股票報酬數據來說明。

摘要(英)

Change point detection is an important part of time series analysis because the existence of change points indicates that there is a sudden and significant change in the process of data generation. Detecting change points can help us with pre-warning and post analysis. It is widely used in many fields, such as industrial quality control, financial market analysis, network traffic analysis, and so on. In the literature review, the maximum likelihood estimator can be used to estimate the change point under the assumption that the observations are independent. However, in many practical applications, the observations usually have dependent structure, so the
maximum likelihood estimator method with independent hypothesis is usually inefficient. In this paper, we extend the maximum likelihood estimator method to the case of dependent observations. We propose a new change point model, which the serial correlation follows the copulabased Markov chain model, and the marginal distribution follows the normal distribution and
then obtain its corresponding likelihood function. The Newton Raphson method is applied to solve the maximum likelihood estimators. In the empirical study, we analyze the stock return data for illustration.

關鍵字(中)

★ 變更點
★ 耦合
★ 時間序列數據
★ 序列依賴
★ 序列分析
★ 常態分佈
★ 馬爾可夫鏈模型
★ 牛頓-拉弗森

關鍵字(英)

★ change point
★ copula
★ time series data
★ serial dependence
★ sequential analysis
★ normal distribution
★ Markov chain model
★ Newton-Raphson

論文目次

CONTENTS
摘要 I
Abstract II
List of Tables V
List of Figures VII
Chapter 1: INTRODUCTION 1
Chapter 2: BACKGROUND 3
2.1 Introduction to Copulas 3
2.2 Review Maximum Likelihood Estimator (MLE) Method for Change Point 6
Chapter 3: METHODOLOGY 8
3.1 Change Point Model 8
3.2 Proposed Estimator Method for Change Point 11
3.3 Reparameterization and Convergence Issues 13
3.4 Asymptotic Normal Approximation 14
3.5 Confidence Interval for Change Point 17
Chapter 4: SIMULATION STUDY 18
4.1 Simulation Setup 18
4.2 Simulation Results 19
4.2.1 Results for the Newton-Raphson Method 19
4.2.2 Sensitivity Analysis 19
4.2.3 Results for 95% Confidence Interval 20
4.2.4 Results for Fixed Parameters and Heavy-tailed Data 20
4.2.5 Results for Sensitive Analysis of the Heavy-tailed Data 20
4.2.6 Results for Comparing with the MLE Method 20
4.2.7 Results for Comparing with the Dependent Method 21
Chapter 5: EMPIRICAL STUDY 31
5.1 Data Description 31
5.2 Empirical Results 34
Chapter 6: CONCLUSION 38
6.1 Conslusion and Discussion 38
References 40
Appendices 42
A Derivation of Clayton copula density and transition density 42
B Derivation of log-likelihood function 43
C The first and second derivatives of log-likelihood function 44
D Derivation of the log-likelihood function of the autoregressive (AR) model 54

參考文獻

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指導教授

孫立憲(Li-Hsien Sun)

審核日期

2021-7-23

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