Parametric APUB for Decision Making under Parameter Uncertainty

Parametric optimization can exploit a known distribution family but usually treats its fitted parameters as certain. We develop parametric Average Percentile Upper Bound (APUB) optimization to adjust a decision and certify its expected cost against this epistemic uncertainty. APUB refits the parametric model on bootstrap datasets and minimizes an average of upper percentiles of the resulting expected-cost functions. We study generative parametric bootstrap and weighted-likelihood bootstrap as two prior-free constructions in one framework. The theory establishes decision consistency under construction-specific conditions, regular fixed-decision coverage, and finite-ensemble error control; an adaptive algorithm grows and reuses the fitted ensemble. In paired, fixed-truth mixed-normal and mixed-log-normal newsvendor experiments, positive-level PB- and WLB-APUB have lower mean regret than SAA and fitted-model optimization at small samples, while the adjustment contracts as the sample size increases. Only APUB displays approximately level-tracking coverage curves, giving its nominal level an interpretable one-sided confidence meaning. APUB avoids this prior-specification risk through a data-driven resampling adjustment.