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External Seminar by William Pouliot (University of Birmingham)
Abstract: We consider the problem of kernel regression estimation in the presence of Not-Missing-At-Random (NMAR), or nonignorable, response variables. Our proposed approach involves two steps: In the first step, we construct a family of models (possibly infinite dimensional) indexed by the unknown nonignorability component of the missing probability mechanism. In the second step, a search is carried out to find the empirically optimal member of an appropriate cover (or subclass) of the underlying family in the sense of minimizing the mean squared prediction error. Our methods use a data-splitting approach which is quite easy to implement. We also derive exponential bounds on the performance of the resulting estimators in terms of their deviations from the true regression curve in general L_p norms, where we also allow the size of the cover or subclass to diverge as the sample size n increases. These bounds together with the Borel-Cantelli lemma immediately yield various strong convergence results for the proposed estimators. As an application of our findings, we consider the problem of nonparametric statistical classification based on the proposed regression estimators and also look into their rates of convergence in different settings.