Shrinkage estimator

Remarks

The weight $w$ may be a constant or may depend on the data. The term is also used more broadly for any estimator that pulls a basic estimate toward a target, for example by a penalty in a regression. The plain James–Stein estimator has a weight that can be negative; its positive-part version fits the definition above.

Example: let $X$ be normal with mean $\theta$ and known variance $\sigma^2$, and give $\theta$ the normal prior with mean $0$ and variance $\tau^2$. The posterior mean is $\frac{\tau^2}{\tau^2 + \sigma^2} X$. This is a shrinkage estimator with $\delta_0 = X$, $\nu = 0$ and constant weight $w = \tau^2/(\tau^2 + \sigma^2) \in (0, 1)$.

Non-example: $\delta(X) = 2X$ in the same model. It has the form above with $\nu = 0$ and $w = 2$, which is not in $[0, 1]$, so it pushes the estimate away from the target rather than toward it.