---
tag: 709T196XCT
kind: concept
name: James–Stein estimator
msc: [62J07, 62C15, 62F10]
specializes: [GPBSS1V3SS]
sources:
  - url: https://en.wikipedia.org/wiki/James–Stein_estimator
same_as:
  wikidata: Q6146297
  wikipedia: James–Stein estimator
---

## Summary

The James–Stein estimator estimates the mean vector of a multivariate normal distribution in dimension $p \ge 3$ by shrinking the observation toward a fixed point.
Under squared error loss its risk is strictly smaller than that of the observation itself, for every value of the mean.
It is the standard illustration of [[Stein's example|7SC9E65XDN]] and the prototype of [[shrinkage estimators|GPBSS1V3SS]].

## Explanations

### L1 human

Suppose one observes a noisy vector $X$ whose average is the unknown vector $\theta$.
The James–Stein estimator multiplies $X$ by a factor a bit smaller than $1$, pulling it toward the origin.
The factor is close to $1$ when $X$ is far from the origin, and smaller when $X$ is close to it.
For example, with $p = 3$, unit noise and observation $X = (1, 1, 1)$, the factor is $1 - 1/3$, and the estimate is $(2/3, 2/3, 2/3)$.