Exponentially Consistent Statistical Classification of Continuous Sequences with Distribution Uncertainty

TL;DR

This paper introduces distribution-free classification tests for continuous sequences with distribution uncertainty, demonstrating exponential decay of error probabilities across fixed-length, sequential, and two-phase methods, even under distribution deviations.

Contribution

It proposes novel distribution-free tests for continuous sequence classification that handle distribution uncertainty and prove exponential error decay for multiple test designs.

Findings

Error probabilities decay exponentially for all test types.

Tests are effective even with distribution deviations.

Results extend to cases with and without the null hypothesis.

Abstract

In multiple classification, one aims to determine whether a testing sequence is generated from the same distribution as one of the M training sequences or not. Unlike most of existing studies that focus on discrete-valued sequences with perfect distribution match, we study multiple classification for continuous sequences with distribution uncertainty, where the generating distributions of the testing and training sequences deviate even under the true hypothesis. In particular, we propose distribution free tests and prove that the error probabilities of our tests decay exponentially fast for three different test designs: fixed-length, sequential, and two-phase tests. We first consider the simple case without the null hypothesis, where the testing sequence is known to be generated from a distribution close to the generating distribution of one of the training sequences. Subsequently, we generalize our results to a more general case with the null hypothesis by allowing the testing sequence to be generated from a distribution that is vastly different from the generating distributions of all training sequences.