The existence of the least favorable noise

Abstract

Suppose that a random variable $X$ of interest is observed. This paper concerns "the least favorable noise" $\hat{Y}_{\epsilon}$, which maximizes the prediction error $E [X - E[X|X+Y]]^2 $ (or minimizes the variance of $E[X| X+Y]$) in the class of $Y$ with $Y$ independent of $X$ and $\mathrm{var} Y \leq \epsilon^2$. This problem was first studied by Ernst, Kagan, and Rogers ([3]). In the present manuscript, we show that the least favorable noise $\hat{Y}_{\epsilon}$ must exist and that its variance must be $\epsilon^2$. The proof of existence relies on a convergence result we develop for variances of conditional expectations. Further, we show that the function $\inf_{\mathrm{var} Y \leq \epsilon^2} \, \mathrm{var} \, E[X|X+Y]$ is both strictly decreasing and right continuous in $\epsilon$.