Multivariate Analysis, Ranks, and Multivariate Ranks

该项目探索了两个利用排名数据的领域。第一个目标是开发统计程序，这些程序对违反假设的行为具有鲁棒性，同时在假设成立时仍然接近最佳状态。这种基于对数据进行排名的稳健程序有很长的历史，即用值的排名替换数据中的观察值。这个过程有助于改善异常的野生观察的影响，这些观察可能会破坏分析。一些多变量的情况下，以前很少使用排名程序的工作将是这个项目的主要重点。这些包括定义变量关系的某些结构模型，测试随时间推移观察到的多变量数据中的运行，估计方差和协方差，以及测试某些变量是否在给定其他变量的情况下条件独立。一个建议的方法来定义多变量的排名（“迭代排名”），使其分布是独立的基础观测的分布将探讨。第二个领域着眼于直接建模排名数据，其中的数据来自法官排名特定对象的基础上，他们的喜好。一个流行的模型假设，法官和对象可以排列沿着一条线，其中法官位于最近的法官的最喜欢的对象，下一个最接近的第二个最喜欢的对象，等等一个新的模型，也允许法官自己定位最接近的对象，他们最不喜欢将被考虑。还将考虑扩大这些模式，规定一小部分法官完全不按照模式行事。

The project explores two areas that utilize ranked data. The first aims to develop statistical procedures that are robust to violations of assumptions while still working close to optimally when the assumption holds. There is a long history of such robust procedures based on ranking the data, that is, replacing the observed values in the data with the values' ranks. This process helps to ameliorate the effects of unusually wild observations that can ruin an analysis. A number of multivariate situations in which there has previously been little work using rank procedures will be the main focus of this project. These include certain structural models defining the relationship of variables, testing for runs in multivariate data observed over time, estimating variances and covariances, and testing whether certain variables are conditionally independent given some other variables. A proposed method for defining multivariate ranks ("iterated ranks") so that their distribution is independent of the distribution of the underlying observations will be explored.The second area looks at modeling rank data directly, where the data arise from judges ranking particular objects based on their preferences. One popular model posits that judges and objects can be arrayed along a line, where a judge is located nearest the judge's most preferred object, next closest to the second most preferred object, etc. A new model that also allows judges to locate themselves nearest the objects they prefer least will be considered. An extension of these models in which there is provision for a small percentage of judges to act not at all according to the model will also be considered.