The least favorable noise

Philip A. Ernst; Abram M. Kagan; L.C.G. Rogers

doi:10.1214/22-ECP467

2022 The least favorable noise

Philip A. Ernst, Abram M. Kagan, L.C.G. Rogers

Author Affiliations +

Electron. Commun. Probab. 27: 1-11 (2022). DOI: 10.1214/22-ECP467

Abstract

Suppose that a random variable X of interest is observed perturbed by independent additive noise Y. This paper concerns the “the least favorable perturbation” ${\hat{Y}_{\varepsilon }}$ , which maximizes the prediction error $E{(X-E(X|X+Y))^{2}}$ in the class of Y with $var(Y)\le \varepsilon$ . We find a characterization of the answer to this question, and show by example that it can be surprisingly complicated. However, in the special case where X is infinitely divisible, the solution is complete and simple. We also explore the conjecture that noisier Y makes prediction worse.

Dedication

We dedicate this work to our colleague, mentor, and friend, Professor Larry Shepp (1936–2013)

Acknowledgments

We thank Professor Dan Crisan (Imperial College London) and Dongzhou Huang (Rice University) for helpful discussions. We also wish to thank an anonymous referee whose helpful and detailed comments have greatly improved the quality of this manuscript. The first-named author gratefully acknowledges the support of ARO-YIP-71636-MA, NSF DMS-1811936, ONR N00014-18-1-2192, and ONR N00014-21-1-2672.

Citation

Download Citation

Philip A. Ernst. Abram M. Kagan. L.C.G. Rogers. "The least favorable noise." Electron. Commun. Probab. 27 1 - 11, 2022. https://doi.org/10.1214/22-ECP467