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For observed data $x$, let $F_x$ be the posterior cumulative distribution function. Assuming finite posterior first moment, the posterior risk under asymmetric absolute-error loss isAt points where the posterior has a density, differentiation gives $R_x^{\prime }(a)=(\gamma +\delta )F_x(a)-\gamma$. Risk is convex, so a global minimum is a posterior quantile:More generally any $a$ with $F_x(a^{-})\le \gamma /(\gamma +\delta )\le F_x(a)$ minimizes risk. This is the Bayes quantile under asymmetric absolute-error loss; a larger cost for underestimation moves the chosen quantile upward.

For the uniform sampling model set $M={\max }_i{X}_i$. For a valid positive sample the likelihood is ${\theta }^{-n}{1}_{\{\theta >M\}}$. Multiplying by the prior cancels its power of $\theta$ and gives posterior density proportional to $e^{-\theta }{1}_{\{\theta >M\}}$. Normalization yieldsSolving the quantile equation gives the explicit Bayes estimateThe posterior is the sample maximum plus a unit-rate exponential distribution displacement; it has finite first moment, so the preceding risk minimization is justified here.