Loss function

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2020 Mathematics Subject Classification: Primary: 62C05 [MSN][ZBL]

In a problem of statistical decision making, a non-negative function indicating the loss (cost) to an experimenter given a particular state of the world and a particular decision. Let $X$ be a random variable taking values in a sample space $(\mathfrak{X},\mathcal{B},\mathsf{P}_\theta)$, $\theta \in \Theta$, and let $D = \{d\}$ be the space of all possible decisions that can be taken on the basis of an observed $X$. In the theory of statistical decision functions, any non-negative function $L$ defined on $\Theta \times D$ is called a loss function. The value of a loss function $L$ at an arbitrary point $(\theta,d) \in \Theta \times D$ is interpreted as the cost incurred by taking a decision $d \in D$, when the true parameter is $\theta$, $\theta \in \Theta$.


[1] A. Wald, "Statistical decision functions" , Wiley (1950) Zbl 0040.36402
[2] E.L. Lehmann, "Testing statistical hypotheses" (2nd ed.), Wiley (1986) Zbl 0608.62020



[a1] J.O. Berger, "Statistical decision theory and Bayesian analysis" (2nd ed.) , Springer (1985) Zbl 0572.62008
How to Cite This Entry:
Loss function. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article