Data-Driven Nonparametric Existence and Association Problems

TL;DR

This paper develops generalized likelihood tests for nonparametric hypothesis testing problems involving unknown distributions, characterizing error exponents using Chernoff information and KL divergence, and highlighting the importance of training-to-testing data ratio.

Contribution

It introduces GL-based tests for nonparametric existence and association problems with unknown distributions, providing error exponent characterizations and insights into data ratio effects.

Findings

Error exponents are characterized by Chernoff information and KL divergence.

Training-to-testing data ratio significantly affects error decay rates.

The proposed tests effectively handle unknown distributions in nonparametric settings.

Abstract

We investigate two closely related nonparametric hypothesis testing problems. In the first problem (i.e., the existence problem), we test whether a testing data stream is generated by one of a set of composite distributions. In the second problem (i.e., the association problem), we test which one of the multiple distributions generates a testing data stream. We assume that some distributions in the set are unknown with only training sequences generated by the corresponding distributions are available. For both problems, we construct the generalized likelihood (GL) tests, and characterize the error exponents of the maximum error probabilities. For the existence problem, we show that the error exponent is mainly captured by the Chernoff information between the set of composite distributions and alternative distributions. For the association problem, we show that the error exponent is captured by the minimum Chernoff information between each pair of distributions as well as the KL divergences between the approximated distributions (via training sequences) and the true distributions. We also show that the ratio between the lengths of training and testing sequences plays an important role in determining the error decay rate.