---
tag: 9D47D1Q89E
kind: definition
name: Minimax estimator
msc: [62C20]
defines: 8EG1M1WZ9C
uses: [N26TSVRH2S, 0SSS6ERQ0S, 0WCWA1KT3M]
sources:
  - url: https://en.wikipedia.org/wiki/Minimax_estimator
---

## Definition

Let $\set{P_\theta : \theta \in \Theta}$ be a statistical model with a loss function, and let $R(\theta, \delta)$ be the [[risk function|N26TSVRH2S]] of an [[estimator|0SSS6ERQ0S]] $\delta$.
The *minimax risk* of the problem is
$$R^* = \inf_{\delta} \sup_{\theta \in \Theta} R(\theta, \delta),$$
where the infimum is over all estimators.
An estimator $\delta^*$ is *minimax* if
$$\sup_{\theta \in \Theta} R(\theta, \delta^*) = R^*.$$

## Remarks

A minimax estimator need not exist, and it need not be unique.
A Bayes estimator whose risk function is constant in $\theta$ is minimax.

Example: let $X$ be [[binomial|0WCWA1KT3M]] with parameters $n$ and $\theta \in [0, 1]$, with squared error loss.
Let $\delta^*(X) = (X + \sqrt{n}/2)/(n + \sqrt{n})$.
A direct computation of variance plus squared bias gives $R(\theta, \delta^*) = \frac{1}{4(1 + \sqrt{n})^2}$ for every $\theta$.
This $\delta^*$ is the Bayes estimator for the beta prior with parameters $\sqrt{n}/2$ and $\sqrt{n}/2$, and its risk is constant, so it is minimax.

Non-example: $\delta(X) = X/n$ in the same problem.
Its risk is $\theta(1 - \theta)/n$, with maximum $\frac{1}{4n}$ at $\theta = \frac{1}{2}$.
Since $\frac{1}{4n} > \frac{1}{4(1 + \sqrt{n})^2}$, its worst-case risk exceeds $R^*$, so it is not minimax.