Calibration of thresholding rules for Poisson intensity estimation

TL;DR

This paper develops a data-driven threshold calibration method for Poisson intensity estimation, providing theoretical guarantees and practical guidelines for selecting thresholds to optimize performance without support restrictions.

Contribution

It introduces a novel calibration procedure for thresholding rules in Poisson intensity estimation with proven optimality and practical threshold range recommendations.

Findings

Optimal threshold range identified as [1,12]

Choosing gamma close to 1 yields good performance

Classical methods are sensitive to support restrictions

Abstract

In this paper, we deal with the problem of calibrating thresholding rules in the setting of Poisson intensity estimation. By using sharp concentration inequalities, oracle inequalities are derived and we establish the optimality of our estimate up to a logarithmic term. This result is proved under mild assumptions and we do not impose any condition on the support of the signal to be estimated. Our procedure is based on data-driven thresholds. As usual, they depend on a threshold parameter $\gamma$ whose optimal value is hard to estimate from the data. Our main concern is to provide some theoretical and numerical results to handle this issue. In particular, we establish the existence of a minimal threshold parameter from the theoretical point of view: taking $\gamma<1$ deteriorates oracle performances of our procedure. In the same spirit, we establish the existence of a maximal threshold parameter and our theoretical results point out the optimal range $\gamma\in[1,12]$. Then, we lead a numerical study that shows that choosing $\gamma$ larger than 1 but close to 1 is a fairly good choice. Finally, we compare our procedure with classical ones revealing the harmful role of the support of functions when estimated by classical procedures.