
This function basically involves you setting a value for u that is your ālimitā. The function then only considers points above the limit, and gives you the average of those values. So for example, letās assume the following empirical distribution with the following n points:
20, 15, 4, 12, 17, 2, 45, 20,19, 22, 14
Let me set u to a few different values to determine the empirical mean excess loss function:
u=5āe(u)=((205)Ć1+(155)Ć1+(45)Ć0+(125)Ć1+(175)Ć1+āÆ)/(1+1+0+1+1+āÆ)=15.444
Note that simply boils down to taking the average excess value above u for those points that are above u.
u=20āe(u)=(0+0+0+0+0+0+25+0+0+2+0)/2=13.5

Similar to your previous question 3, the choice of u has to be done in a way that we capture enough tail data to be able to have some level of confidence in the distribution we fit, but also we donāt want to capture too much data as we arenāt interested in modelling the body of the distribution > only the tail. One way of determining this u value is by studying the graph of e(u). If we choose a value of u that is too high, we wonāt have a linear relationship > we will have an unpredictable function. That is because if u is very high, a small number of data points will be included in the calculation of e(u) and the value will likely jump around. For example, in the above example, e(20)=13.5 and only 2 points were used in the calculation. Moving to e(22) would give us: e(22)=23/1=23. We see an extreme jump as small number of data points are used and then fall out of the calculation as u increases. At the same time, a u that is too small will have the majority of the points used in the calculation which causes the denominator to be large. However, many of these points (on the lefthand side of the distribution) will have small excesses over u because they are not the righthand extreme values. So as u increases and some of these points drop out of the calculation, the result will be that the numerator will decrease a small amount but the denominator will also drop rapidly as many points in the body of the distribution fall out of the calculation. This will likely see e(u) decrease until a ābalanceā is found between only considering extremeenough values, but not considering too few. That is where you see the linear increase in the e(u) functions in the graphs. The extreme points after the solid line indicate that there are too few data points at those high levels of u to have a stable relationship. Therefore, the chosen value of u should probably be around the area where the linearlyincreasing pattern begins (at the start of the solid line).