Li-Shan Huang (Chair)

Abstract

Given a set of incomplete observations, we study the  nonparametric problem of testing whether data are Missing Completely At  Random (MCAR). Our first contribution is to characterise precisely the set of alternatives that can be distinguished from the MCAR null hypothesis.  This reveals interesting and novel links to the theory of Fréchet classes (in particular, compatible distributions) and linear programming, that  allow us to propose MCAR tests that are consistent against all detectable  alternatives. We define an incompatibility index as a natural measure of  ease of detectability, establish its key properties, and show how it can  be computed exactly in some cases and bounded in others. Moreover, we  prove that our tests can attain the minimax separation rate according to  this measure, up to logarithmic factors.