Spectral backtests unbounded and folded

In the spectral backtesting framework of Gordy and McNeil (JBF, 2020) a probability measure on the unit interval is used to weight the quantiles of greatest interest in the validation of forecast models using probability-integral transform (PIT) data. We extend this framework to allow general Lebesgue-Stieltjes kernel measures with unbounded distribution functions, which brings powerful new tests based on truncated location-scale families into the spectral class. Moreover, by considering uniform distribution preserving transformations of PIT values the test framework is generalized to allow tests that are focused on both tails of the forecast distribution.