  Quantum metrology derives its capabilities from the careful employ of quantum
resources for carrying out measurements. This advantage, however, relies on
refined data postprocessing, assessed based on the variance of the estimated
parameter. When Bayesian techniques are adopted, more elements become available
for assessing the quality of the estimation. Here we adopt generalized
classical Cram\'er-Rao bounds for looking in detail into a phase-estimation
experiment performed with quantum light. In particular, we show that the
third-order absolute moment can give a superior capability in revealing biases
in the estimation, compared to standard approaches. Our studies point to the
identification of an alternative strategy that brings a possible advantage in
monitoring the correct operation of high-precision sensors.
