Dependence structure modeling: New directions and applications

对条件依赖性和无条件依赖性的建模在统计学中是至关重要的，对依赖性结构的错误描述会导致错误的结论。例如，股票指数收益与交易量之间的关系对于预测和检验一种条件独立形式的格兰杰非因果关系是很重要的。在文献中，联结函数和回归函数是最常用的变量间关系建模方法。在本研究计划中，我们将探讨三个方向的建模条件和无条件依赖。由于格兰杰因果检验是条件独立的一种形式，我研究了几种非因果性检验，当零假设被拒绝时，我基于条件联结和条件分布开发了新的因果性度量。我的提案研究计划的第一个方向是致力于构建格兰杰非因果检验和因果度量，并在金融和医学等领域应用。在接下来的五年里，我将探索和建立一个基于回归中的例外、极值回归和渐近无分布（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，但非常有前途的衡量因果关系的新想法。将在R中为我在这个方向上提出的测试开发一个新的包。第二个方向与我对不完全数据使用copula函数对条件和无条件依赖进行建模的研究有关。在这里，我将考虑由于在横断面调查中常见的右审查和长度偏差抽样而导致的不完整性。在接下来的五年里，我将构建拟合优度检验，以选择适当的右截尾长度偏置数据的参数联结。在许多情况下，两个变量之间的依赖结构受到协变量的影响。因此，我将研究为这种条件依赖结构建模的条件联结，并推导条件依赖的度量。回归函数通常用于对随机变量（或向量）和一组协变量之间的关系进行建模。我的建议的第三个方向是致力于使用非对称误差损失和联结函数的回归估计。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。均值回归模型、分位数回归模型和期望回归模型是我提出的模型的特殊情况。此外，为了结合分位数回归的稳健性和期望回归的效率，我将开发一个新的回归模型。在R中为所有提出的模型提供一个新的包将是一个重要的目标。

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.