- Stein's unbiased risk estimate
statistics, Stein's unbiased risk estimate (SURE) is an unbiased estimatorof the mean-squared errorof a given estimator, in a deterministic estimation scenario. In other words, it provides an indication of the accuracy of a given estimator. This is important since, in deterministic estimation, the true mean-squared error of an estimator generally depends on the value of the unknown parameter, and thus cannot be determined completely.
The technique is named after its discoverer,
Charles Stein. cite journal|title=Estimation of the Mean of a Multivariate Normal Distribution|journal=The Annals of Statistics|date=Nov. 1981|first=Charles M.|last=Stein|coauthors=|volume=9|issue=6|pages=1135–1151|id= |url=http://links.jstor.org/sici?sici=0090-5364%28198111%299%3A6%3C1135%3AEOTMOA%3E2.0.CO%3B2-5|accessdate=2008-03-30|month=Nov|year=1981|doi=10.1214/aos/1176345632 ]
Let be an unknown deterministic parameter and let be a measurement vector which is distributed normally with mean and covariance . Suppose is an estimator of from . Then, Stein's unbiased risk estimate is given by:where is the th component of the estimate, and is the
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 be expected to minimize the MSE. Except for the first term in SURE, which is identical for all estimators, 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 estimatorcan 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 wavelet denoisingsetting. [ cite journal|title=Adapting to Unknown Smoothness via Wavelet Shrinkage|journal=Journal of the American Statistical Association|date=Dec. 1995|first=David L.|last=Donoho|coauthors=Iain M. Johnstone|volume=90|issue=432|pages=1200–1244|id= |url=http://links.jstor.org/sici?sici=0162-1459%28199512%2990%3A432%3C1200%3AATUSVW%3E2.0.CO%3B2-K|accessdate=2008-03-30|month=Dec|year=1995|doi=10.2307/2291512 ]
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