= Bayes quantile under asymmetric absolute-error loss
{c}
{title2=$F(a)=\gamma/(\gamma+\delta)$}
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 $F(a^-)\leq\gamma/(\gamma+\delta)\leq F(a)$. Increasing the cost of underestimation moves the optimal estimate upward.
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