The existence of the least favorable noise
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Suppose that a random variable X of interest is observed. This paper concerns "the least favorable noise" Ŷ_ϵ, 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 var Y ≤ϵ^2. This problem was first studied by Ernst, Kagan, and Rogers ([3]). In the present manuscript, we show that the least favorable noise Ŷ_ϵ must exist and that its variance must be ϵ^2. The proof of existence relies on a convergence result we develop for variances of conditional expectations. Further, we show that the function inf_var Y ≤ϵ^2 var E[X|X+Y] is both strictly decreasing and right continuous in ϵ.
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