Introduction:

Instead of computing the mean log-likelihood over all events above a certain confidence level - we now compute the mean log-likelihood over all events wherein the predicted movement exceeds a certain absolute size - parametrized by the percentile level of the event size. (The nomenclature volatility percentile is motivated by the popular identification of the movement size |x_t^(k,P)| with volatility.) In this case the X-axis is independent of the model - so the points plotted for every model which correspond to the same volatility percentile - refer to the same extreme events. This measure tests the conformity of the model predictions to the tail of the empirical unconditional distribution instead of the tail of the forecast conditional distribution (presented in the last section).