Open Access
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


Suppose that a random variable X of interest is observed perturbed by independent additive noise Y. This paper concerns the “the least favorable perturbation” Yˆε, which maximizes the prediction error E(XE(X|X+Y))2 in the class of Y with var(Y)ε. 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.


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


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.


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Philip A. Ernst. Abram M. Kagan. L.C.G. Rogers. "The least favorable noise." Electron. Commun. Probab. 27 1 - 11, 2022.


Received: 18 March 2021; Accepted: 16 April 2022; Published: 2022
First available in Project Euclid: 17 May 2022

MathSciNet: MR4368695
zbMATH: 1498.60070
Digital Object Identifier: 10.1214/22-ECP467

Primary: 60E07 , 60E10
Secondary: 60E05

Keywords: Infinitely divisible distributions , least favorable perturbation , self-decomposable random variable

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