Stein's unbiased risk estimate

In statistics, Stein's unbiased risk estimate (SURE) is an unbiased estimator of the mean-squared error of "a nearly arbitrary, nonlinear biased estimator." In other words, it provides an indication of the accuracy of a given estimator. This is important since the true mean-squared error of an estimator is a function of the unknown parameter to be estimated, and thus cannot be determined exactly.

The technique is named after its discoverer, Charles Stein.

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 .