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arXiv · 2103.09794

The least favorable noise

Philip A. Ernst·Abram M. Kagan·L. C. G. Rogers
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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" $\hat Y_\ep$, which maximizes the prediction error $E(X-E(X|X+Y))^2$ in the class of $Y$ with $ \var (Y)\leq \ep$. 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.

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BibTeXRIS

Philip A. Ernst, Abram M. Kagan, L. C. G. Rogers. 2021-03-17. The least favorable noise. https://arxiv.org/abs/2103.09794

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