In statistics, Stein's unbiased risk estimate (SURE) is an unbiasedestimator 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.
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.
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 .
A standard application of SURE is to choose a parametric form for an estimator, and then optimize the values of the parameters to minimize the risk estimate. This technique has been applied in several settings. For example, a variant of the James–Stein estimator can be derived by finding the optimal shrinkage estimator. The technique has also been used by Donoho and Johnstone to determine the optimal shrinkage factor in a waveletdenoising setting.
^ abDonoho, David L.; Iain M. Johnstone (December 1995). "Adapting to Unknown Smoothness via Wavelet Shrinkage". Journal of the American Statistical Association. Journal of the American Statistical Association, Vol. 90, No. 432. 90 (432): 1200–1244. doi:10.2307/2291512. JSTOR2291512.