Source Erasure, Record Coarsening and a Sharp Boundary for Exact Unbiased Estimation in Fixed Budget Double Sampling

Background Outcome ascertainment in veterinary and medical follow-up can combine initially observed and subsequently recovered outcomes. We examine how discarding the provenance of recovered outcomes affects exact unbiased estimation. The practical question is whether a follow-up study must record, for every outcome, the stage at which that outcome was obtained.Methods We studied an iid binary double-sampling experiment with uniform, error-free recovery of missing outcomes under a fixed budget and a known finite response-odds bound. Polynomial coefficient comparisons established the unbiasedness boundary. Low-order moments and total variation characterized identification and the fixed-one single-cohort limit. Exact conditional risk calculations used paired human ascertainment data and a veterinary reconstruction from inpatient and later records. All budget-limited uniform recovery was specified by the evaluation design, and unresolved outcomes remained unassigned.Results For every known finite response-odds bound greater than one, a parameter-independent real-valued rule based on merged counts is exactly unbiased throughout the admissible domain if and only if the recovery budget is at least the cohort size minus one. The argument extends to parameter-free auxiliary randomization and yields a complete classification of deterministic recovery sizes. In the fixed-one experiment, low-order population moments identify interior parameters, yet the laws at distinct missing-outcome probabilities become indistinguishable from one coarsened record as cohort size grows. Retaining provenance recovers a classical Hansen–Hurwitz estimator, whose exact unbiasedness need not minimize mean squared error. The clinical reference calculations illustrate this distinction.Conclusions These results provide a finite-sample benchmark linking recovery design, record coarsening and exact unbiased estimability, and yield an operational criterion: an exactly unbiased merged-record rule exists if and only if the recovery budget reaches the cohort size minus one. The clinical examples show how the choice of recorded fields and the recovery budget together determine the bias and risk of routinely reported proportions at a fixed reference composition.