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

多元数据的半参数和非参数推理：理论与应用

• 批准号: RGPIN-2020-05496
• 负责人: Belalia, Mohamed
• 金额: $ 1.31万
• 依托单位: University of Windsor
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=717199
• 关键词: Semi; non; parametric; inference; multivariate

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 kth order moments of the random variable TC(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.

具有数千个相互依赖的测量值的多变量数据是许多统计应用领域的新规范。因此，在多变量设置中建模依赖性已成为现代统计数据分析中的关键组成部分。Copula函数是一种灵活有效的随机变量相关结构建模工具，在众多研究领域有着广泛的应用。 假设一个Copula函数C属于一个由参数索引的参数族。两种方法是常见的估计：伪最大似然法和矩估计法。这两种方法都有严重的缺点，限制了它们在实际环境中的适用性。或者，我提议：（I）模拟矩法，通过构造随机变量TC（U）的k阶矩的无偏非参数估计，进而导出作为经验矩和理论矩方程组的解的估计。然而，对于大量的copula，一个明确的形式的理论的时刻是不可用的，这大大限制了该方法的范围。我建议一个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，适用于任何模型的Copula;（II）同样，我将开发一个模拟的最大似然估计;（III）此外，我将研究一个模拟的成对似然型估计。 在处理了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，两个Copula之间的相等性检验，Copula模型中的变点检验和任意维中的交换检验。所有提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。 逻辑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开发了一个基于copula的链接函数。我将通过使用藤copula或分层阿基米德copula将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。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。所提出的方法将被用来开发程序的分类和变量选择高维数据。我相信这些方法可以适用于审查和修复数据的情况。 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的错误设定。为了克服这个问题，我建议使用一个非参数copula为基础的方法来估计分位数和预期的回归函数的可调数据。将特别注意删失数据。