Stein's unbiased risk estimate

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In statistics, Stein's unbiased risk estimate (SURE) is an unbiased estimator of the mean-squared error of "a nearly arbitrary, nonlinear biased estimator."[1] 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.[2]

Formal statement[edit]

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[3]

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 .

Proof[edit]

We wish to show that

We start by expanding the MSE as

Now we use integration by parts to rewrite the last term:

Substituting this into the expression for the MSE, we arrive at

Applications[edit]

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.[2] The technique has also been used by Donoho and Johnstone to determine the optimal shrinkage factor in a wavelet denoising setting.[1]

References[edit]

  1. ^ a b Donoho, 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. JSTOR 2291512.
  2. ^ a b Stein, Charles M. (November 1981). "Estimation of the Mean of a Multivariate Normal Distribution". The Annals of Statistics. 9 (6): 1135–1151. doi:10.1214/aos/1176345632. JSTOR 2240405.
  3. ^ Wasserman, Larry (2005). All of Nonparametric Statistics.