Sharp Fréchet--Hoeffding Bounds under Selectively Incomplete Observation

Incomplete records can sharpen bounds on joint outcome probabilities even when no pair of outcomes is jointly observed. I derive sharp pointwise Fréchet--Hoeffding bounds combining known population marginals with selectively observed records, without restricting the observation mechanism. I establish an exact reduction of the identification problem to finite-dimensional optimization, yielding computable characterizations of both bounds. Illustrations involving income mobility, potential outcomes, and split questionnaires demonstrate how incomplete records can improve identification relative to the classical bounds.