| 摘要: | 高維數據非常態結構的分析是一有趣並且重要的課題。現代統計分析通常包括對高維時間序列之建模和預測。因為它涉及到正確處理時間與空間的因素，故具有一定程度的挑戰。在處理高維狀態空間的相依性上，Copula函數是一個重要的研究工具，因為它允許我們分離邊際分佈和其相依模型。該方法已應用到各個領域，如金融學和生物學。其中一個自然的問題是如何確定其Copula的函數形式。一維參數Copula模型具有許多不同的參數型式，如高斯，克萊頓，弗蘭克，耿貝爾等。但是，此類模型過於嚴格。而另一個極端的非參數模型也產生維度過高的問題。因此，我們引進一較可行的參數化Copula函數：分層阿基米德Copula函數（HAC），它不僅允許較多的參數，並且Copula函數的結構也可有所改變。本研究旨在通過建立一個隱馬可夫模型（HMM）的HAC去解決一問題。此處HAC為處理相依的模型，而HMM是來描述隨時間變化的動態系統。在HMM的框架下，我們證明結構參數和HAC的一致性與漸近正態。並用匯率數據的模擬和實證分析以驗證此一模式。出於對HMM的HAC模型建模和模擬，我們還考慮到在HMM的框架下，信用風險交換和報酬風險變化檢測的問題。並利用馬可夫隨機漫步之理論結果去支持我們的方法。本計劃的研究項目包含高維數據建模與模擬，參數估計，模型選擇，自適MCMC，穩定(最佳)變點檢測，以及相關方面的馬可夫隨機漫步理論。一些尚待解決問題的條列如下： 1) 研究隱藏式馬可夫模型的HAC理論性質和(半)最大概似估計法的效率計算。高效率的計算方法如EM, AECM和Bootstrap方法通過正常Copula的重點取樣。HMM的HAC模型篩選也是一個有趣且重要的議題。 2) 經由通過 Copula方法對信用風險交換的研究，將探討(適應性)隱藏式馬可夫模型的理論與實際課題。對於未知平均平穩序列的區間估計，我們將應用在一般HMM的HAC篩自助法與小偏差事件順序重點取樣。 3) 經由對信用風險交換和報酬風險變化檢測問題的研究，我們將探討均值和波動的同時變化。接下來，我們研究變點統計分割及探索Shiryaev-Roberts法則的貝氏最佳化轉折點偵測問題。研究統計分割上轉折點的問題也考慮將隱藏式馬可夫模型應用於分散快速轉折點偵測上。執行這些問題將涉及期望過量(overshoot)的數值計算。 4）藉由隱藏式馬可夫模型的HAC的參數估計及序慣轉折點偵測的動機，我們將建立自正則極限理論，非線性馬可夫更新理論及馬可夫隨機漫步的修正擴散逼近法，然後將這些結果為最佳預測問題。並利用多維更新理論去處理金融危機蔓延的問題。迭代函數系統與 Copula函數所產生的馬可夫鏈之間的關係也是一個有趣的課題。Understanding the dynamics of high dimensional non-normal dependency structure is an interesting and important task. Modern statistics analysis usually involves modeling and estimation of high dimensional time series, which is a big challenge since it involves proper handling of both temporal and spatial space. In term of high dimensional state space dependency,Copulas is an important tool for this study, as it allows us to separate the marginal distributions and the dependency models. This method has been applied to various fields like finance and biology. The first natural question is how to determine the functional form of Copulas. One parameter Copula models are developed for many different parametric family, like Gaussian, Clayton, Frank, Gumbel, etc.. However, this class of models is too restrictive; while going to another extreme a total nonparametric approach would also run to curse of odeling class for Copulas: Hierarchical Archimedean Copulas (HAC), which allows not only more parameters, but the Copula function’s structure can also be changed. This research project aims at attacking this problem by building up a hidden Markov model (HMM) for HAC, where HAC is a flexible model for high dimensional dependency, and HMM is a classical dynamic technique to describe time varying dynamics. Consistency and asymptotic normality for both parameters and HAC structures are established under the HMM framework. Simulations and empirical analysis for exchange rates data are given to demonstrate our model. Motivated by HMM for HAC modeling and simulations, we also consider the problems of calibrating credit default swap and return-risk change detection under HMM. Theoretical results based on limiting theorems for Markov random walks are also given to support our methodologies. The proposed research project contains high dimensional data modeling and simulation,parameter estimation, model selection, (adaptive) Markov chain Monte Carlo (MCMC), robust and/or optimal change point detection rules, and related probability aspects in Markov random walks. Some open problems are listed as follows: 1) Investigate theoretical properties as well as efficient computation of the (quasi) maximum likelihood estimator (MLE) in HMM for HAC. Study efficient computational methods like EM,AECM, and bootstrap method for normal copula via importance sampling. The issue of model selection for HAC via HMM is also an interesting and important topic. 2) Motivated by the study in Credit-Default-Swap (CDS) via copula method, we will study theoretical and practical issues of (adaptive) MCMC. For the concern of interval estimation for the unknown mean in stationary sequences, we will apply sieve bootstrap in a general state HMM for HAC, and filtering method with sequential importance sampling for moderate deviations events. 3) Motivated by the problem of detecting return-risk tradeoff in risk management, we will investigate simultaneous changes of mean and volatility. Next, we study change points for statistical segmentation and Bayesian optimality of the Shiryaev-Roberts rule for change point detection. Application to decentralized quickest change detection in HMM is also considered. Implementation for these problems will involve numerical computation of the expected overshoot,which also has applications in credit migration. 4) Motivated by parameter estimation and sequential change point detection in HMM for HAC, we will establish self-normalized limiting theorems, multi-dimensional Markov renewal theory, and admissibility of Bayesian estimator under HMM. Then apply these results to optimal prediction problem. A measure of financial contagion will be given via multi-dimensional renewal theory. The relationship between Markov chains induced by iterated function systems and copula functions is an interesting task. 研究期間：10008 ~ 10107 |
