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 .
Read more about this topic: Stein's Unbiased Risk Estimate
Famous quotes containing the words formal and/or statement:
“The formal Washington dinner party has all the spontaneity of a Japanese imperial funeral.”
—Simon Hoggart (b. 1946)
“Truth is that concordance of an abstract statement with the ideal limit towards which endless investigation would tend to bring scientific belief, which concordance the abstract statement may possess by virtue of the confession of its inaccuracy and one-sidedness, and this confession is an essential ingredient of truth.”
—Charles Sanders Peirce (1839–1914)