Sometimes They Come Back: Testing Two Simple Hypotheses (In The Realm Of Unlabeled Data)

Abstract

Consider a binary hypothesis where data are independent and identically distributed under the null hypothesis, and known only to be independent under the alternative. The statistician observes an n- vector X <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> (n ≫ 1) and makes a decision using the optimal likelihood ratio test. This seems a widely-known detection problem, but: What if only the set of samples of X <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> are made available to the statistician, while the positions of the individual samples inside the vector are not? Does there exist an optimal test in that case? What is the fundamental performance limit? Are there nicely-performing practical detectors with affordable computational complexity? Answers to these questions are in large part unknown, despite the fact that the problem - which is becoming known under the name of unlabeled detection - is very relevant in modern sensor network applications where the sample positions can be lost due to their means of delivery from the remote units, or because of network attacks.