James–Stein estimator  The James–Stein estimator is a biased estimator of the mean of Gaussian random vectors. It can be shown that the James–Stein estimator dominates the "ordinary" least squares approach, i.e., it has lower mean squared error. It is the best-known example of Stein's phenomenon. An earlier version of the estimator was developed by Charles Stein in 1956, and is sometimes referred to as Stein's estimator. The result was improved by Willard James and Charles Stein in 1961. Suppose the vector formula_1 is the unknown mean of a formula_2-variate normally distributed (with known covariance matrix formula_3) random variable formula_4:  We are interested in obtaining an estimate formula_6 of formula_1, based on a single observation, formula_8, of formula_9. This is an everyday situation in which a set of parameters is measured, and the measurements are corrupted by independent Gaussian noise. Since the noise has zero mean, it is very reasonable to use the measurements themselves as an estimate of the parameters. This is the approach of the least squares estimator, which is formula_10. As a result, there was considerable shock and disbelief when Stein demonstrated that, in terms of mean squared error formula_11, this approach is suboptimal. The result became known as Stein's phenomenon. If formula_12 is known, the James–Stein estimator is given by  James and Stein showed that the above estimator dominates formula_14 for any formula_15, meaning that the James–Stein estimator always achieves lower mean squared error (MSE) than the maximum likelihood estimator. By definition, this makes the least squares estimator inadmissible when formula_15. Notice that if formula_17 then this estimator simply takes the natural estimator formula_18 and shrinks it towards the origin 0. In fact this is not the only direction of shrinkage that works. Let ν be an arbitrary fixed vector of length formula_2. Then there exists an estimator of the James-Stein type that shrinks toward ν, namely  The James–Stein estimator dominates the usual estimator for any ν. A natural question to ask is whether the improvement over the usual estimator is independent of the choice of ν. The answer is no. The improvement is small if formula_21 is large. Thus to get a very great improvement some knowledge of the location of θ is necessary.