The Sparse Poisson Means Model

TL;DR

This paper investigates the detection of sparse Poisson mixtures, establishing detection boundaries and methods depending on the size of the Poisson means relative to the logarithm of the sample size, supported by numerical experiments.

Contribution

It extends sparse mixture detection theory to Poisson models, identifying regimes where higher criticism or simple multiple testing are optimal.

Findings

Higher criticism achieves detection boundary for larger Poisson means.

Bonferroni correction suffices for smaller Poisson means.

Numerical experiments confirm theoretical detection boundaries.

Abstract

We consider the problem of detecting a sparse Poisson mixture. Our results parallel those for the detection of a sparse normal mixture, pioneered by Ingster (1997) and Donoho and Jin (2004), when the Poisson means are larger than logarithmic in the sample size. In particular, a form of higher criticism achieves the detection boundary in the whole sparse regime. When the Poisson means are smaller than logarithmic in the sample size, a different regime arises in which simple multiple testing with Bonferroni correction is enough in the sparse regime. We present some numerical experiments that confirm our theoretical findings.