Confidence Sets for the Date of a Weak Mean Break in Functional Data

We develop confidence sets for the date of a single mean break in functional data when the break may be too weak to be consistently detected. Under each maintained null, segmentwise demeaning removes the unknown mean and break functions, so valid inference does not require consistent detection of the break. We select among invariant tests by maximizing their weighted average local power over alternative break dates and directions. In infinite dimensions, the resulting covariance perturbation can render the null and alternative measures mutually singular, causing the usual likelihood-based derivation of a test that maximizes weighted average power to break down. We characterize the weights under which likelihood-based comparison remains valid and show that covariance-squared weighting yields a simple locally best invariant statistic. Under mild conditions, we establish the asymptotic validity of the procedure and characterize its local power under weak breaks. Simulations show that the resulting confidence sets achieve accurate empirical coverage, whereas a competing interval designed for consistently detectable breaks exhibits severe undercoverage when the break magnitude is small.