New Developments of Nonlinear Dependent Models, with Applications in Genetics, Finance and the Environment

在环境、金融市场、信号和图像处理等领域中，高维和复杂的数据被常规地收集起来。一个主要的挑战是找到方法来分析这些数据集的结构，使模型具有所需的依赖属性，并识别和验证模式。该提案的主要目标是为重要和具有挑战性的低样本和高维统计推断问题做出重要的方法和理论贡献，例如生物信息学中的降维问题，以及来自环境和金融市场的极端依赖问题。该建议包括四个重要步骤。首先，该提案追求一系列新的非线性依赖度量的发展。研究了相关测度（商相关系数）的极限分布，并分析了相关结构给定时的渐近幂。在DNA微阵列数据分析中，利用商相关性选择基因的最佳特征子集，然后使用所选择的子集来预测所有样本数据的类别。其次，引入了可变阈值的尾相关度量（尾商相关系数）。该测量与多元极值的统计研究有关，并用于评估环境变量的渐近（in）依赖性。第三，建议包括渐近（in）相关的多元极大值和移动极大值过程的统计估计方法的发展。这使得人们可以有效地研究聚类时空极端观测。第四，研究残差为m的GARCH（r,s）模型。首先，该建议的智力优点源于使用商相关概念的有效降维方法。在基因表达谱中有成千上万个变量（基因）的DNA微阵列数据分析中，分类预测是一个重要问题。由于所调查数据的高维特征和小样本量，确定要使用的基因子集是很重要的。最终目标是选择有助于分类和预测的最小基因子集。在现有的基因选择方法中，很难找到一种方法在应用于不同的数据集时总是比其他方法表现更好。该建议的具体目的是为这一问题找到解决办法。除了方法上的优点和具体应用之外，该建议还具有相当广泛的影响。在不同领域（如上所述）的应用中，极端风险扮演着重要的科学、社会以及（可能）政治角色。传播新的统计工具以更好地了解联合极值的发生是非常重要的。这可以在新的研究生课程、面向广大读者的期刊上的出版物以及与其他领域的科学家的讨论中很好地实现。仅举一个提案可能产生巨大影响的例子，让我们考虑一下金融风险管理。由于建立了新的银行偿付能力监管规则（所谓的巴塞尔协议II提案），银行不得不提出（例如，在他们对信贷风险的分析中）压力测试程序，这些程序可以立即根据极端联合运动制定。同样，在多险种保险中，承保人必须承担许多不同险种的共同巨额损失。正是针对这些类型的应用程序，该建议产生了新的工具。

High-dimensional and complex data are now collected routinely in the fields of environment, financial markets, and signal and image processing. A major challenge is to find methods to analyze the structure of such data sets, to fit models with desired dependence properties, and to identify and validate patterns. A major goal of the proposal is to make significant methodological and theoretical contributions to the important and challenging low-sample and high-dimension statistical inference problems such as dimension reduction in bio-informatics, and extreme dependence problems arising from environmental, and financial markets. The proposal consists of four important steps. First, the proposal pursues a series of developments of new measures for nonlinear dependencies. The investigator studies the limiting distributions of dependence measures (quotient correlation coefficients) and analyzes asymptotic powers when the dependence structures are specified. In DNA microarray data analysis, the quotient correlation is used to select the best feature subset of genes, and then the selected subset is used to predict classes for all sample data. Second, a tail dependent measure (a tail quotient correlation coefficient) with varying threshold is introduced. This measure is related to the study of statistics of multivariate extremes, and is used to assess asymptotic (in)dependencies in environmental variables. Third, the proposal includes the development of statistical estimation methods for asymptotically (in)dependent multivariate maxima and moving maxima processes. This allows one to efficiently study clustered spatial-temporal extreme observations. Fourth, the proposal studies GARCH(r,s) models with m-dependent residuals.The intellectual merit of the proposal in a first instance stems from an efficient dimension reduction approach using the quotient correlation concept. In DNA microarray data analysis, in which there are thousands of variables (genes) in gene expression profiles, and class prediction is an important problem. It is important to identify subsets of genes to work with, due to the high dimensional feature and small sample size of the data under investigation. The ultimate goal is to select the smallest subset of genes which contribute toward the classifications and predictions. Among existing gene selection methods, it is hard to find one which always performs better than the rest when applying them to different data sets. The proposal specifically aims at finding a solution for this. Beyond methodological merits and specific applications, the proposal also has a considerable broad impact. Throughout applications in diverse fields (like above), extreme risks play an important scientific, societal, as well as (possibly) political role. The dissemination of new statistical tools leading to a better understanding of the occurrence of joint extremes is of great importance. This can be well achieved at the level of new graduate courses, publications in journals aimed at a broad audience and in discussion with scientists from other fields. To name just one potential example where the proposal has great impact, let us consider financial risk management. Due to the establishment of new regulatory-rules for banking solvency (so-called Basel II proposal), banks have to come up (in their analysis of credit risk, for instance) with stress testing procedures which can immediately be formulated in terms of extremal co-movements. Similarly, in multi-line insurance, underwriters have to take care of joint large losses in many different lines of business. It is exactly for these kinds of applications that the proposal yields new tools.