Inferences for Multivariate Semiparametric and Nonparametric Models with Applications to Risk Management

本建议的目的是发展广泛适用的半参数和非参数估计和推断的新方法，研究这些新方法的理论性质，并评价它们在数据分析中的有效性。这一建议不仅引入了一些创新技术，还提供了对统计基础的各种新的和深入的见解。这将对未来统计方法、计算和理论的研究产生重大影响。特别是，提出了三个相互关联的领域进行研究。首先，介绍了一类柔性半参数和非参数模型。这使得人们能够研究反应变量与其协变量的关联程度。提出了广义似然比统计量，用于检验多元半参数和非参数模型中的各种假设。其次，提出了新的半参数和非参数模型来理解利率动态和股票价格波动。此外，为了提高债券波动率估计的效率，并更准确地估计投资组合的市场风险，引入了状态域信息。第三，利用非凹惩罚似然方法，提出了存在大量变量时变量选择的新方法。其创新之处在于，他们在估计参数的同时选择变量。上述技术广泛适用于许多科学和工程问题。多变量非参数、半参数和大型参数模型得到了广泛的应用。统计问题经常出现，例如某些变量或因素是否对公众健康很重要；一些风险因素是否对患者的生存时间有重大影响；利率动态或股票价格过程是否依赖于时间或遵循某些著名的假设等。然而，在多变量半参数和非饱和非参数模型中，还没有普遍适用的工具来回答这些问题。这里提出的技术允许人们在没有限制性模型假设的情况下客观地检验科学假说。这些技术可以更好地为金融衍生品定价和管理投资风险，在大型流行病学研究的分析中识别重要的风险变量及其可能的相互作用，并仔细审查著名的股票价格假说。

AbstractPI: Jianqing FanDMS-0204329The objectives of this proposal are to develop new and widely applicable approaches for semiparametric and nonparametric estimation and inferences, to study theoretical properties of these new approaches, and to evaluate their efficacy in data analyses. This proposal not only introduces a number of innovative techniques, but also provides various new and deep insights into statistical foundation. It will have significant impact on the future research of statistical methodologies, computation and theories. In particular, three inter-related areas are proposed for study. Firstly, a family of flexible semiparametric and nonparametric models is introduced. This allows one to study the extent to which response variables are associated with their covariates. The generalized likelihood ratio statistics is proposed for testing various hypotheses in multivariate semiparametric and nonparametric models. Secondly, new semiparametric and nonparametric models are proposed for understanding interest-rate dynamics, and stock price volatilities. Furthermore, the information on state-domain is incorporated to improve the efficiency of volatility estimation for bonds and to more accurately estimate the market risks of a portofolio. Thirdly, new techniques for variable selection, in the presence of a large number of variables, are proposed via nonconcave penalized likelihood. The innovation is that they estimate parameters and select variables simultaneously. The above techniques are widely applicable to many scientific and engineering problems. Multivariate nonparametric, semiparametric and large parametric models have been widely used. Statistical questions often arise such as if certain variables or factors are important to public health; if some risk factors contribute significantly to the survival time of patients; and if interest-rate dynamics or stock price processes are time-dependent or follow certain famous hypotheses, among others. Yet, there are no generally applicable tools available to answer these questions in multivariate semiparametric and non-saturated nonparametric models. The techniques proposed here permit one to objectively test scientific hypotheses without restrictive model assumptions. The techniques allow to better price financial derivatives and manage investment risk, to identify important risk variables and their possible interactions in the analysis of large epidemiological studies and to scrutinize famous hypotheses on stock prices