Specification:

In order to construct the measure we define the set of consecutive loss exceedence events $\Psi^{\pm_{(k,P)}}(c)$ so that:  

$$\Psi^{\pm_{(k,P)}}(c) = \{t\vert\{t,t-1\}\subset\Phi^{\pm_{(k,P)}}(c),t\in[3,1001]\}.$$ (32)

Using sets $\Psi^{\pm_{(k,P)}}(c)$ we can finally specify the serial exceedence ratio measure

$$s^{\pm_{(k,P)}}(c) = {{\char93 }(\Psi^{\pm_{(k,P)}}(c))\over1000(1-{c\over100})^2}$$ (33)

Finally, as specified by expressions  and , we construct the univariate average exceedence ratio

$$s_u(c) = {1\over20}\left(\sum_{k=1}^{10}s^{+_{(k)}}(c) + \sum_{k=1}^{10}s^{-_{(k)}}(c)\right)$$ (34)

and the multivariate average exceedence ratio

$$s_m(c) = {1\over2}\left(s^{+_{(P)}}(c) + s^{-_{(P)}}(c)\right)$$ (35)

plotted in figures 3u and 3m respectively for all candidate models.