| 摘要: | 研究期間：10208~10307;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. |
