The default definition as part of the PD: informative withdrawals and thin grades

A bank that sets the PD of a grade from the published statistics of rating agencies inherits two choices it seldom shows: which events count as a default, and what becomes of a rating withdrawn during the year. The European repository sets no common definition of a default event. We propose a protocol that treats the definition and the treatment of withdrawals as parameters of the PD: the PD of each grade under each page with its capital, the bias of each treatment on paths of known truth, transition-test rejection rates, and the exact coverage of interval estimators. On S&P's cohorts of 2000 to 2025 the long-run PD of CCC to C is 26.64% on the default page and 8.61% on the transition page (gap index 0.323, 95% interval 0.259 to 0.391), and the IRB risk weight falls from 58.0% to 47.5%. With withdrawal ten times as likely in the year before default, removing the withdrawn ratings, following them to the year end and a duration estimator understate the PD of grade B by -35.1±0.3, -15.6±0.3 and -37.2±0.3 percent. The tests of time homogeneity and of the last matrix reject in 5.8% and 1.4% of repetitions at the 5% level, the last-matrix test in 16.3% when withdrawal is twenty times as likely. The first-order conjecture of the theory is refuted for CCC to C. With 50 obligors the Wald interval contains the PD of AAA in 0.33% of samples. The protocol measures these biases and repairs none. Repository and reproducible pipeline (MIT): https://github.com/fsantibanezleal/CAOS_Contraste .