Sharp Estimand-Specific Precision Envelopes for Blockwise Missing Data Under Missing at Random

Abstract

An overall missingness rate does not determine how much precision survives: at 25% missing, an exact calculation raises the standard error by 1.2% under one admissible mechanism and 21.4% under another. For variable or all-or-none block missingness at random, we characterize the precision attainable across mechanisms sharing an observation rate and a lower bound on observation probabilities. The envelope is sharp: it contains every attainable efficiency bound. The favorable endpoint is explicit. In atomless spaces the adverse endpoint integrates the quantile function of the residual influence variance, so it is evaluated, not optimized; in mixed atomic and nonatomic spaces it has an exact attained variational characterization. Crossings with a precision requirement divide missingness into guaranteed, allocation-sensitive, and inadmissible regions. The construction also handles absolute variance and interval width requirements and returns the minimum-cost recovery as a priority ordering. For a partially observed continuous outcome, cross-fitted one-step estimators and a fail-closed bootstrap give asymptotic simultaneous inference on a prespecified grid under stated regularity conditions, and inversion yields an ordered confidence set for both thresholds. Without an overlap restriction, no positive observation rate guarantees nontrivial retention. Public clinical illustrations are reported, with Monte Carlo joint coverage below nominal in the smallest samples.