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

在统计应用的许多领域中，具有数千个相互依赖的测量值的多元数据是新的规范。因此，多变量设置中的建模依赖已成为现代统计数据分析的关键组成部分。copula是一种灵活有效的工具，可以对随机变量之间的依赖结构进行建模，在许多研究领域都有应用。目前的建议旨在从几个方向推进联结的理论和应用。假设一个联结函数C属于一个以参数为索引的参数族。通常有两种方法来估算？：伪极大似然和矩量估计方法。这两种方法都有严重的缺陷，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的方法，然后推导出对？作为经验矩和理论矩方程组的解。然而，对于大量的联轴，理论矩的显式形式是不可用的，这在很大程度上限制了方法的范围。我建议用一个模拟理论力矩的版本。预期这种办法是有效、灵活的，原则上适用于任何联结模式；（II）同样，我将开发一个模拟的最大似然估计器；（III）此外，我将研究模拟成对似然类型估计器。在处理了估计copula参数的问题之后，我将把注意力转向假设检验，如拟合优度检验，检验两个copula之间的等式，copula模型中的改变点以及任意维度的互换性检验。所有提议的检验都将基于伯恩斯坦经验联结公式、其相关密度或最近引入的贝塔经验联结公式。逻辑回归是将事件概率作为协变量函数建模的有效工具。最近，我在二元回归模型中开发了一种基于copula的链接函数。我将通过使用vine copula或分层阿基米德copula将此方法扩展到高维和混合协变量的多项回归和多元结果。此外，我将在计数数据的分位数回归的背景下研究这种方法。所提出的方法将用于开发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经过审查和修复的数据。在使用联结函数的参数和半参数回归模型中，一个主要的障碍是联结函数的错配。为了克服这个问题，我建议使用基于非参数copula的方法来估计接触数据的分位数和期望回归函数。将特别注意经过审查的数据。

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.