Absolute-error loss ($\ell (a,\theta )=|a-\theta |$)

The loss $|a-\theta |$ for estimating a real parameter $\theta$ by $a$. Its risk of a decision rule is the corresponding expected value. Unlike squared-error loss, its pointwise size is a metric, so the triangle inequality gives a direct overlap version of the Le Cam two-point lemma.

This loss assigns different positive costs to underestimation and overestimation of a scalar parameter. Here $\gamma$ penalizes estimates below the parameter and $\delta$ estimates above it. Equal costs give a multiple of absolute-error loss.

For a posterior distribution with finite first moment, the posterior risk derivative at continuity points is $(\gamma +\delta )F(a)-\gamma$. Convexity therefore makes a posterior $\gamma /(\gamma +\delta )$-quantile a Bayes estimator. With atoms, the condition is $F(a^{-})\le \gamma /(\gamma +\delta )\le F(a)$. Increasing the cost of underestimation moves the optimal estimate upward.