Optimal nonparametric testing of Missing Completely At Random, and its connections to compatibility

TL;DR

This paper develops a nonparametric framework for testing if data are Missing Completely At Random (MCAR), linking it to Fréchet classes and linear programming, and provides consistent tests with optimal detection rates.

Contribution

It characterizes distinguishable alternatives from MCAR, introduces an incompatibility index, and proposes tests that are minimax optimal without needing complete cases.

Findings

Proposes consistent MCAR tests for all detectable alternatives.

Introduces an incompatibility index to measure detectability.

Achieves minimax separation rates up to logarithmic factors.

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 Fr\'echet 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. Our methodology does not require any complete cases to be effective, and is available in the R package MCARtest.