An Intuitive Explanation
For any particular value of θ the new estimator will improve at least one of the individual mean square errors This is not hard − for instance, if is between −1 and 1, and σ = 1, then an estimator that moves towards 0 by 0.5 (or sets it to zero if its absolute value was less than 0.5) will have a lower mean square error than itself. But there are other values of for which this estimator is worse than itself. The trick of the Stein estimator, and others that yield the Stein paradox, is that they adjust the shift in such a way that there is always (for any θ vector) at least one whose mean square error is improved, and its improvement more than compensates for any degradation in mean square error that might occur for another . The trouble is that, without knowing θ, you don't know which of the n mean square errors are improved, so you can't use the Stein estimator only for those parameters.
Read more about this topic: Stein's Example
Famous quotes containing the words intuitive and/or explanation:
“If mothers are told to do this or that or the other,... they lose touch with their own ability to act.... Only too easily they feel incompetent. If they must look up everything in a book, they are always too late even when they do the right things, because the right things have to be done immediately. It is only possible to act at exactly the right point when the action is intuitive or by instinct, as we say. The mind can be brought to bear on the problem afterwards.”
—D.W. Winnicott (20th century)
“To develop an empiricist account of science is to depict it as involving a search for truth only about the empirical world, about what is actual and observable.... It must involve throughout a resolute rejection of the demand for an explanation of the regularities in the observable course of nature, by means of truths concerning a reality beyond what is actual and observable, as a demand which plays no role in the scientific enterprise.”
—Bas Van Fraassen (b. 1941)