Minimax Acquisition Under a Shared Bounded Bias: Critical Windows and Sparse Portfolios

Cheap model predictions can reduce the cost of estimating population means, but collecting more predictions does not remove a shared calibration bias. We study how to allocate a limited budget among trusted labels, predictions, and correlated paired observations when this bias lies in a known interval. Existing prediction-powered methods combine predictions with trusted labels to obtain valid inference, while recent minimax allocation theory optimizes acquisition under unrestricted prediction bias. These results leave unresolved how a finite bias bound changes which auxiliary observations are worth buying and how spending on them should scale with the budget. For Gaussian observations and continuous allocations, we reduce the exact fixed-design minimax risk over all measurable estimators to a bounded-normal-mean problem and derive two cost-normalized information criteria that distinguish regimes in which auxiliary data offer no improvement, optimal auxiliary spending remains bounded, or its share of the budget approaches one. We also derive sharp second-order risk expansions in the bounded-spending and critical regimes, characterize a 1/B-wide transition window for total budget B, and prove that an exact optimum can be attained using at most three observation types.