# Decision function

Let $X$ be a random variable that takes values in a sample space $\left({\mathfrak{X},\mathcal{B},\mathbf{P}_\theta}\right)$, $\theta \in \Theta$, and let $D = \{ d \}$ be the set of all possible decisions $d$ that can be taken relative to the parameter $\theta$ with respect to a realization of $X$. According to the accepted terminology in mathematical statistics and the theory of games, any $\mathcal{B}$-measurable transformation $\delta : \mathfrak{X} \rightarrow D$ of the space of realizations $\mathfrak{X}$ of $X$ into the set of possible decisions $D$ is called a decision function. For example, in the statistical estimation of the parameter $d$ any point estimator $\hat\theta = \hat\theta(x)$ is a decision function. A basic problem in statistics in obtaining statistical conclusions is the choice of a decision function $\delta(\cdot)$ that minimizes the risk $$R(\theta,\delta) = \mathbf{E}_\theta[L(\theta,\delta_X)]$$ relative to the loss function $L(\cdot,\cdot)$ used.