It is shown in infinite-dimensional Gaussian problems that affine estimators minimax the variance among all estimators of a linear functional subject to a constraint on the bias. Likewise, affine estimators also minimax the square of the bias among all estimates of a linear functional subject to a constraint on the variance.
"Bias-Variance Tradeoffs in Functional Estimation Problems." Ann. Statist. 23 (3) 824 - 835, June, 1995. https://doi.org/10.1214/aos/1176324624