The least favorable noise

03/17/2021

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by   Philip A. Ernst, et al.

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Suppose that a random variable X of interest is observed perturbed by independent additive noise Y. This paper concerns the "the least favorable perturbation" Ŷ_, which maximizes the prediction error E(X-E(X|X+Y))^2 in the class of Y with (Y)≤. 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.