Shrinkage estimation of Efron and Morris (1972)

Question

I read this article: Article1 and which was refined by the second article Article2 that was considered as a generalization of the James-Stein estimator.

In article 1 for example, they considered the following estimator:

$\sum' = [a S^{-1} + (b/trace(S))I]^{-1}$

where $a = k-p-1$ and $b = p(p+1) -2$ and $k>p+1$.

After I read the two articles, I became a little bit confused.

(1) Do they always consider that the true covariance matrix $\sum$ is proportional to the identity matrix (i.e., $\sum= \gamma I$), where $\gamma$ is a constant? And finally they give us the new shrinkage estimator based only on this assumption?

(2) If the answer of question (1) is yes, their proposed shrinkage estimator will not become accurate if we assume that the true covariance matrix $\sum$ is not proportional to the identity matrix. Am I wrong?

Answer 1

I never looked at any of this stuff before, but it is interesting. In light of advances in nonlinear optimization, I think there are likely to be better approaches available nowadays, especially with regard to how to more directly do eigenvalue optimization, rather than using ad hoc adjustments and constraints.

They are NOT considering the true covariance matrix to be proportional to the identity matrix. What they are doing is adjusting the best unbiased estimator of the inverse of the covariance in the direction toward a scalar multiple of the identity matrix, in which the scaling is based on trace of best unbiased estimator of covariance. In other words, they are adjusting the inverse covariance estimate in the direction of a perfectly conditioned matrix, i.e., a matrix having 2-norm condition number = 1 (which is the "best" possible conditioning). That is why this is called eigenvalue shrinkage. The eigenvalues are being compressed in the direction toward being equal, i.e., they are downward adjusting the 2-norm condition number of the estimated inverse covariance matrix, which therefore is downward adjusting the condition number of the estimated covariance matrix.

So rather than calling this eigenvalue shrinkage, it would be more informative and less confusing to call it (2-norm) condition number shrinkage.