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 $\epsilon$.

翻译：设感兴趣的随机变量$X$被观测到。本文研究“最不利噪声”$\hat{Y}_{\epsilon}$，它是在$Y$与$X$独立且$\mathrm{var} Y \leq \epsilon^2$的类中，使预测误差$E [X - E[X|X+Y]]^2$最大（或使$E[X| X+Y]$的方差最小）的噪声。该问题最早由Ernst、Kagan和Rogers研究（[3]）。在本文中，我们证明最不利噪声$\hat{Y}_{\epsilon}$必然存在，且其方差必为$\epsilon^2$。存在性的证明依赖于我们为条件期望的方差建立的收敛性结果。此外，我们证明函数$\inf_{\mathrm{var} Y \leq \epsilon^2} \, \mathrm{var} \, E[X|X+Y]$关于$\epsilon$既是严格递减的，也是右连续的。