Robust Estimation and Inference

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Errors and perturbations which must be filtered to obtain useful inferences and predictions arise from several sources, including: (1) random fluctuations, e.g. observations are affected by measurement errors, natural fluctuations and sampling variability, (2) data contamination, e.g. data often include measurements of uneven quality, outliers, gross errors and cases from populations other than the target one, and (3) missing data. Most traditional statistical procedures deal with (1) and there are many papers dealing with (2) and (3) separately. However, there are few papers dealing with(1), (2) and (3) simultaneously. Some of my proposed research will aim at filling this gap. I wish to develop procedures able to deal with all the above mentioned sources of uncertainty, using computational efficient and scalable algorithms. Consider a data table with n rows -- one for each case -- and p columns -- one for each variable or feature. With the advent of cheap computing and storage, many modern datasets are variables-rich and cases-poor. This is referred to as "small n-- large p problem" in the literature. This phenomenon is also related to the so called curse of dimensionality problem in Statistics. Given a certain goal (e.g. prediction of future values for some response variable (s) in the data table, it is common to find that a large number of variables (which I call noise variables) hurt instead of helping this task. Hence noise variables constitute a fourth type of perturbation which needs to be filtered to better extract the information contained in the remaining signal variables. In addition, signal variables themselves may be partially redundant and subsets of signal variables (which we call phalanxes) may have better predictive power than the full set of signal variables. Phalanxes can be used to construct statistical models which results can then be ensembled to provide a single prediction/classification. The problem of selecting phalanxes (phalanx formation) is a generalization of model selection where we allow for different groups of variables to form cooperating models to perform a single task. There are many practical and theoretical questions regarding this model building approach which I would like to address. Our former PhD student Jabed Tomal did some ground breaking work on this topic in the context of drug discovery. Prof. Welch and I now wish to enroll a new PhD student to expand this work which has potential for application in many areas of industry and science. The classical robustness model is based on the paradigm that the vast majority of cases (rows in the data table) are free of contamination and useful to perform the given task. Hence, only a minority of contaminated cases may need to be identified and filtered (downweighted). Unfortunately this paradigm is not fully satisfactory in the case of very high dimensional data tables. If there is a small and independent probability, d, that a cell (individual entry in the data table) is contaminated then the probability that a case (a row in the data table) is contaminated is e=1- (1-d)^{p} which can quickly become larger than 0.5. For example, if d=0.01 and p=100 we have e=0.63397. Alqallaf, Van Aelst, Yohai and Zamar (2009) brings attention to this problem called "propagation of outliers" and propose some possible approaches to address it. I wish to further study this problem. My former Ph.D. student Mike Danilov constructed robust S-estimates of multivariate location and scatter that can efficiently deal with missing at random cells. This was an important building block for constructing robust estimates against outliers propagation. My current PhD student Andy Leung is pursuing this research direction.