Semi- and non-parametric inference for multivariate data: theory and application

具有数千个相互依赖的量度的多变量数据是许多统计应用领域的新规范。因此，多变量设置中的相关性建模已成为现代统计数据分析中的一个重要组成部分。Copula是一种灵活有效的建模随机变量间相关性结构的工具，在众多研究领域有着广泛的应用。假设Copula函数C属于以参数？为指标的参数族。常用的估计方法有两种：伪极大似然法和矩估计法。这两种方法都存在严重的缺陷，限制了它们在实际环境中的适用性。或者，我提出：(I)通过构造随机变量T~C(U)的k阶矩的无偏非参数估计来模拟矩方法，然后推导出？作为经验矩和理论矩方程组的解。然而，对于大量的Copula，理论力矩的显式形式是不可用的，这大大限制了该方法的适用范围。我建议采用一种模拟理论力矩的版本。这种方法预计是有效的、灵活的，原则上适用于任何Copula模型；(Ii)类似地，我将开发一个模拟的最大似然估计量；(Iii)我还将研究一个模拟的成对似然类型估计量。在讨论了Copula参数的估计问题之后，我将把注意力转向假设检验，例如拟合优度检验、检验两个Copula之间的相等性、Copula模型中的变点以及在任意维度上的互换性检验。所有拟议的测试都将基于伯恩斯坦经验Copula、其相关密度或最近引入的Beta经验Copula。Logistic回归是将事件概率建模为协变量函数的有效工具。最近，我在二元回归模型中发展了一种基于Copula的连接函数。我将把这种方法扩展到高维和混合协变量的多项回归和多变量结果，使用Vine Copula或Hierarchy Archimedian Copula。此外，我将在计数数据的分位数回归的上下文中研究这种方法。拟议的方法将用于制定高维数据的分类和变量选择程序。我相信，这些方法可以适用于审查和治愈数据的情况。在使用Copula的参数和半参数回归模型中，一个主要的障碍是Copula的错误指定。为了克服这个问题，我建议使用一种基于非参数Copula的方法来估计CONT数据的分位数和期望回归函数。将特别关注被审查的数据。

Multivariate data with thousands of interdependent measurements are the new norm in many fields of statistical applications. As a consequence, modeling dependence in multivariate setups has become a critical component in modern statistical data analysis. Copulas are flexible and efficient tools to model dependence structures among random variables with applications in numerous areas of research.The current proposal aims at advancing theory and applications of copula in several directions. Assume that a copula function C belongs to a parametric family indexed by parameters ?. Two approaches are common to estimate ?: Pseudo-maximum likelihood and Method of Moments' estimators. Both methods suffer from serious drawbacks that limit their applicability in practical settings. Alternatively, I propose: (I) a simulated method of moments by constructing unbiased nonparametric estimators for the k-th order moments of the random variable T~C(U), and then deduce an estimate of ? as a solution of the system of equations of empirical and theoretical moments. However, for a large number of copulas, an explicit form for the theoretical moments is not available, which considerably limits the scope of the method. I suggest a version where the theoretical moments are simulated. This approach is expected to be efficient, flexible and, in principle, applicable to any model of copula; (II) Similarly, I will develop a simulated maximum likelihood estimator; (III) Also, I will investigate a simulated pairwise likelihood type estimator. After treating the problem of estimating the copula parameter, I will turn my attention to hypothesis testing such as a goodness-of-fit test, test the equality between two copula, change points in copula models and test for exchangeability in an arbitrary dimension. All the proposed tests will be based on the Bernstein empirical copula, its associated density or the more recently  introduced beta empirical copula. Logistic regressions are efficient tools in modeling probabilities of events as functions of covariates. Recently, I have developed a copula based link functions in binary regression models. I will extend this method to multinomial regression and multivariate outcomes for high dimensional and mixed covariates by using vine copula or hierarchical Archimedean copulas. Furthermore, I will investigate this approach in the context of quantile regressions for count data. The proposed methodology will be used to develop procedures for classification and variables selection for high dimensional data. I believe that these approaches can be adapted to the case of censored and cured data. In parametric and semiparametric regression models that use copulas, one major obstacle is the copula misspecification. To overcome this issue, I propose using a nonparametric copula based approach to estimate the quantile and expectile regression functions for cont data. Particular attention will be given to censored data.