Dependence structure modeling: New directions and applications

在统计学中，条件依赖和无条件依赖的建模是至关重要的，而对依赖结构的错误描述会导致错误的结论。例如，股指收益率和成交量之间的关系对于预测和检验格兰杰非因果关系这一条件独立性形式具有重要意义。在文献中，Copula和回归函数是最流行的建模变量之间关系的方法。在这个研究项目中，我们将探索条件依赖和无条件依赖建模的三个方向。*由于格兰杰因果关系检验是条件独立性的一种形式，我研究了几种非因果关系检验，当零假设被拒绝时，我发展了基于条件Copula和条件分布的新的因果关系度量。我提议的研究计划的第一个方向是致力于构建格兰杰非因果关系检验和因果关系度量，并将其应用于金融和医学等领域。在接下来的五年里，我将探索和建立一个基于经验检验、极端回归检验和回归中的渐近无分布(ADF)检验的新闻非因果检验。此外，我还将研究一些具有挑战性但非常有前景的衡量因果关系的新想法。将在R中开发一个新的包，用于我在此方向上提议的测试。*第二个方向与我使用Copula函数对不完全数据的条件和无条件依赖进行建模的研究有关。在这里，我将考虑由于右审查和长度偏差抽样的不完整性，这在横断面调查中很常见。在接下来的五年里，我将构建拟合优度检验，以便为右删失的长度偏差数据选择适当的参数Copula。在许多情况下，两个变量之间的相关性结构会受到一个协变量的影响。因此，我将研究用于对这种条件依赖结构进行建模的条件Copula，并导出条件依赖的度量。*回归函数通常用于建模随机变量(或向量)与一组协变量之间的关系。我建议的第三个方向是使用非对称误差损失和Copula函数进行回归估计。在接下来的五年里，我将研究基于(A)对称误差损失和Copula的回归函数的更一般的模型。均值、分位数和期望回归模型是我提出的模型的特例。此外，为了结合分位数回归的稳健性和预期回归的效率，我将开发一个回归的新闻模型。要实现的一个重要目标是在R中为所有提议的模型制定新的一揽子计划。*****

Modeling the conditional and unconditional dependence is crucial in statistics and a misspecification of the dependence structure leads to a wrong conclusions. For example, the relationship between stock index returns and trading volume is important for prediction and testing Granger non-causality which a form of conditional independence. In the literature, copula and regression functions are the most popular approches for modeling the relationship between variables. In this research program we will explore three directions of modeling the conditional and unconditional dependence. ******Since Granger causality tests is a form of conditional independence, I have studied several tests of non-causality and when the null hypothesis is rejected, I have developed new measures of causality based on conditional copula and conditional distributions. The first direction of my proposal research program is devoted to the construction of tests of Granger non-causality and measures of causality with applications in finance and medicine among other fields. Over the next five years, I will explore and build a news non-causality tests based on exepctile, extremile regression and Asymptotically Distribution-Free (ADF) tests in regressions. Also, I will investigate some challenging, but very promising, novel ideas for measuring causality. A new package in R for my proposed tests in this direction will be developed. ******The second direction is related to my research on modeling of conditional and unconditional dependence using copula functions for incomplete data. Here, I will consider the incompleteness due to the right-censoring and length-biased sampling which is common in cross-sectional surveys. Over the next five years, I will construct goodness-of-fit tests in order to select the adequate parametric copula of the right-censored length-biased data. In many situations, the dependence structure between two variables is influenced by a covariate. I will thus investigate the conditional copula for modeling this conditional dependence structure and derive measures of the conditional dependence.******Regression functions are usually used for modeling the relationship between a random variable (or vector) and a set of covariates. The third direction of my proposal is devoted to regression estimation using asymmetric error loss and copula functions. During the next five years, I will investigate a more general models for the regression function based on (a)symmetric error loss and copula. The mean, quantile and expectile regression models are a special cases of my proposed models. Also, to combine the robustness of the quantile regression and the efficiency of the expectile regression, I will develop a news models for regression. A new package in R for all the proposed models will be an important objective to achieve. *****