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In classical statistical decision theory, where we are faced with the problem of estimating a deterministicparameter (vector) ${\bf {x}}$ from observations $y$, an estimator (estimation rule) is called minimax if its maximal risk is minimal among all estimators of $x$. In a sense this is an estimator which performs best in the worst possible case allowed in the problem.

Problem Definition

Consider the problem of estimating a deterministic (not Bayesian) parameter $x$ belonging to some sete ${\mathcal {X}}$ \delta^M(y)</math> is called minimax (or minmax) with respect to a loss function $Insertformulahere$

we have an estimator $\delta$ that is used to estimate a parameter $\theta \in \Theta$. We also assume a risk function $R(\theta ,\delta )$, usually specified as the integral of a loss function. In this framework, ${\tilde {\delta }}$ is called minimax if it satisfies

$\sup _{\theta }R(\theta ,{\tilde {\delta }})=\inf _{\delta }\sup _{\theta }R(\theta ,\delta )$.

An alternative criterion in the decision theoretic framework is the Bayes estimator in the presence of a prior distribution $\Pi$. An estimator is Bayes if it minimizes the average risk

$\int _{\Theta }R(\theta ,\delta )\,d\Pi (\theta )$