Multivariate Analysis, Ranks, and Multivariate Ranks

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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.