Comment:

This relative measure, $\ell_{u,m}$, is maximized when the distribution of events in $\Phi^{\pm_{(k,P)}}(c)$ are predicted with the correct probability by the model. Of course this applies strictly only when ${\char93 }(\Phi^{\pm_{(k,P)}}(c))$is large. In figures 4u and 4m we have plotted data in steps of $1\%$starting from $50\%$ up to $99\%$ - which is for increasing confidence levels but decreasing population of contributing events. It is precisely because the robustness of the statistic is in question for very high levels of confidence, that we have taken the approach of visualizing the measure as a function of the confidence level. When one model shows consistently superior performance than another in the entire high confidence level range - can we be sure that it is a superior model in the context of risk management.

The figures 4u and 4m inform us about the relative performance of the models for events deemed improbable by the model itself. As already discussed in section  the sets $\Phi^{\pm_{(k,P)}}(c)$ for different models do not refer to the same subset of $t\in[2,1001]$. The motivation of the next measure is to evaluate how the model performs explicitly for large price moves and to allow comparison of models over exactly the same data.