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 conviction that the best way to prepare children for a harsh, rapidly changing world is to introduce formal instruction at an early age is wrong. There is simply no evidence to support it, and considerable evidence against it. Starting children early academically has not worked in the past and is not working now.”
—David Elkind (20th century)
“If we do take statements to be the primary bearers of truth, there seems to be a very simple answer to the question, what is it for them to be true: for a statement to be true is for things to be as they are stated to be.”
—J.L. (John Langshaw)