The Newsvendor with Distributional Advice

We study the single-period newsvendor problem when a learned demand model supplies a full predictive distribution $\Fhat$ of unknown accuracy, alongside $n$ historical samples. An exact identity organizes the analysis: the excess cost of acting on the advice equals $(b+h)$ times the CDF area between the two critical fractiles, which yields the sharp bound $(b+h)\,\W(F,\Fhat)$, while advice with the correct critical fractile incurs no excess cost regardless of its global error. Hedging with a Wasserstein ball around $\Fhat$ cannot help when $b\ge h$, a known insensitivity that we complement by characterizing the case $b