# Why are inf and sup used in the definition of minimax estimators?

An estimator $\hat{\delta}$ is minimax iff $$\sup_\theta R(\theta,\hat{\delta})=\inf_\delta\sup_\theta R(\theta,\delta)$$ or in english iff out of all estimators it has the least maximum risk. For details see e.g. http://en.wikipedia.org/wiki/Minimax_estimator

I am wondering why sup and inf are used in the definition instead of min and max from which the name is derived. According to my understanding both the supremum and the infimum should be in the respective sets and as such minimum und maximum would be more appropriate.

It is possible that the set of values $R(\theta,\hat{\delta})$ is an open set. Therefore, supremum/infimum is more general than maximum/minimum. Suppose, $\{R(\theta,\hat{\delta}): \theta \in \Theta\}=\{x: 0<x<1\}$ where $\Theta$ is the parameter space. This has no maximum, but it has a supremum of 1.
• I just came back to this. For the maximum I understand your argument. But isn't the equal sign requiring that it should be min instead of inf. Otherwise, no estimator could have a maximum risk that is equal to the Infimum of the set. For example if $A:=\{\sup_\theta R(\theta,\delta): \delta \in \Delta \} = \{x:0<x<1\}$, then $\inf(A)=0$ but there does not exist any $\delta$ with $\sup_\theta R(\theta,\delta)=0$. – Julian Karls Nov 16 '15 at 20:13
• ..or is the implication of my comment that for situations in which the set $A$ is open a minimax estimator does not exist and if it exists we can replace $\inf$ by $\min$? – Julian Karls Nov 16 '15 at 20:19