Inference methods for multivariate and high-dimensional data

有了先进的计算资源，统计数据分析和建模的范围为现代多元推理方法提供了新的应用领域。分析多变量数据通常面临几个挑战，部分原因是不同变量之间可能存在复杂的依赖结构。此外，终点通常不在同一尺度上测量，因此特定协方差结构的假设是不充分的。特别困难的是多变量数据的推断，其中一个或多个端点是有序的，因为假设多变量正态性的方法显然不适合这种数据。然而，偏斜或离散数据也不能用多变量正态模型适当地描述。此外，数据通常是在精心设计的阶乘设置中收集的，如果端点的数量大于独立实验单元的数量（高维数据），则复杂性会增加。在这些研究中出现的主要问题是检测终点，组水平，或这些组合，导致统计学意义。为了能够回答这些问题，需要有强大的程序，不做限制性的模型假设。这个项目的中心主题是1的推导。基于半参数位置模型的渐近有效检验，无正态性假设2。使用纯非参数模型框架的基于等级的推理方法3. 1.近似值和调整值2.对于小样本量或高维观测，基于不同的自举、随机化或矩近似技术。4.在执行“全局”测试之后，以及在情况2中，进行多个测试程序以调查“局部”问题。也是5.将上述方法扩展到删失数据和6.上述方法的扩展，以检测替代品的特定模式。在第一个主题中，强大的推理工具，可能是高维多变量数据的基础上，预期值，而第二个主题考虑的Wilcoxon-Mann-Whitney型检验的多变量布局的泛化，使用不同的假设制定。在第三个主题中，近似推理解决方案的开发，使用restaurant和其他技术。第四个主题提供了从全局决策到检测相关变量或因素水平组合的逻辑下一步，这些变量或因素水平组合是导致显著结果的原因。第五个主题涉及由于删失而观察不完整的数据问题，这在真实的数据收集中非常常见。最后，在第六个主题中，推理方法被设计为更强大的检测特定的先验指定的替代品，例如增加或减少的趋势。这些结果有望得到广泛的应用，并广泛增强统计科学的作用。

With greatly advanced computational resources, the scope of statistical data analysis and modeling accommodates pressing new arenas of application for modern multivariate inference methods. Analyzing multivariate data usually faces several challenges, partly due to possibly complex dependence structures between the different variables. Additionally, the endpoints are typically not measured on the same scale, hence assumptions of specific covariance structures are inadequate. Particularly difficult is inference for multivariate data in which one or more endpoints are ordinal, as methods assuming multivariate normality are obviously inappropriate for such data. However, also skewed or discrete data are not appropriately described by a multivariate normal model. Moreover, data are usually collected in elaborate factorial settings, and the complexity increases if the number of endpoint is greater than the number of independent experimental units (high-dimensional data). Among the main questions arising in those studies are the detection of endpoints, group levels, or combinations of these, causing statistical significance. In order to be able to answer these questions, it is desirable to have powerful procedures available that do not make restrictive model assumptions. Central themes of this project are the derivation of 1. asymptotically valid tests based on a semiparametric location model without normality assumption 2. rank-based inference methods using a purely nonparametric model framework 3. approximations and adjustments to 1.-2. for small sample sizes or high dimensional observations, based on different bootstrap, randomization, or moment approximation techniques. 4. multiple testing procedures to investigate "local" questions, after having performed "global" tests, and in case 2. also 5. extensions of the above methods to censored data and 6. extensions of the above methods to detect specific patterns of alternatives. In the first topic, powerful inference tools for possibly high-dimensional multivariate data are derived, based on expected values, while the second topic considers generalizations of Wilcoxon-Mann-Whitney type tests to multivariate layouts, using a different hypothesis formulation. In the third topic, approximative inferential solutions are developed, using resampling and other techniques. The fourth topic provides the logical next step away from global decisions to detecting the relevant variables or factor level combinations that are responsible for significant results. The fifth topic addresses issues of data which are observed incompletely due to censoring, which is quite frequent in real data collection. Finally, in the sixth topic, inference methods are devised to be more powerful for the detection of specific a priori specified alternatives, for example increasing or decreasing trends. The results promise to have wide application and to broadly enhance the role of statistical science.