# Is the shrinkage techique all about finite samples?

With the right choice of the penalty, the ridge estimator in the linear regression model $$y=X \beta +\varepsilon, \quad E[X \varepsilon]=0, \quad \beta \in \mathbb{R}^k$$ will have a smaller MSE that the OLS estimator. The asymptotic distributions of the ridge estimator and the OLS estimator (that is, the distribution limits of $\sqrt{n}(\widehat{\beta}_{OLS}-\beta)$ and $\sqrt{n}(\widehat{\beta}_{ridge}-\beta)$) are the same.

Is shrinkage always just improving the properties of the estimator in a finite sample?

Does it make sense to use shrinkage in large samples (in the case of a finite-dimensional parameter of interest)?

• Like Cliff AB, I have never encountered an infinite sample. Not even theoretically. – whuber Jan 13 '17 at 19:03