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
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