The least favorable noise

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.