Introduction:

The main drawback of the exceedence ratio as a performance measure is that it is only sensitive to the frequency and not the degree with which the loss exceeds that predicted at a certain confidence level. A natural measure for the degree of exceedence is the log-likelihood contribution  

$${\cal L}_t^{_{(k,P)}} = \ln(p_t^{_{(k,P)}}(x_{t+1}^{_{(k,P)}}))$$ (36)

which is the logarithm of the probability density of the realized event in terms of the forecast probability distribution p_t^(k,P). While we have used the mean log-likelihood as a criteria for optimization in the in-sample, we now extend it to measure performance out-of-sample. Although the expression  refers only to a Gaussian conditional distribution - it does qualitatively show through its third term that it is appropriate to regard the log-likelihood contribution ${\cal L}_t^{_{(k,P)}}$ of a single event as a comparative (between models) measure of the degree of exceedence.