Stein's Unbiased Risk Estimate - Formal Statement

Formal Statement

Let be an unknown parameter and let be a measurement vector whose components are independent and distributed normally with mean and variance . Suppose is an estimator of from, and can be written, where is weakly differentiable. Then, Stein's unbiased risk estimate is given by

where is the th component of the function, and is the Euclidean norm.

The importance of SURE is that it is an unbiased estimate of the mean-squared error (or squared error risk) of, i.e.

with

Thus, minimizing SURE can act as a surrogate for minimizing the MSE. Note that there is no dependence on the unknown parameter in the expression for SURE above. Thus, it can be manipulated (e.g., to determine optimal estimation settings) without knowledge of .