5. Extreme Value Theorem Flashcards
Define Extreme Value theorem.
Extreme value theorem tries to explain the distribution of very rare and large results. It is used to infer the extreme tail behaviour of most statistical distributions.
Describe block maxima models
If we normalize the maximum of a sample, this normalized random variable will act according to a certain distribution.
What are the three distributions of EVT for block maxima model? What is the value of Xi associated with each of them?
Gumbell, Fréchet and Weibull.
Fréchet: Xi > 0
Gumbel: Xi = 0
Weibull: Xi < 0
Describe points over the threshold models
What is the distribution for the points over the threshold model?
Generalized pareto distribution
Describe 2 advantages of using EVT to model tail risk measures of a distibution, compared with a parametric model of the full distribution.
- The full loss distribution will most likely be fitted with the points closer to the mean and give very low weight to the most extreme points. This can lead to a tail that is not fit at all for the extreme losses.
- There is more flexibility when using a different distribution and different parameters for the tail. The typical losses and extreme losses can be modelled to act very differently.
Describe the trade-off in selecting block size for the block maxima approach. Explain how the selection influences the bias and the variance of the estimate.
The trade-off in selecting block size is that we want as many observations as possible in each block so that we get closer and closer to the theoretical maximum and reducing our bias (law of large numbers), but we also want as many blocks to be able to fit our distribution with a maximum likelihood estimation since more blocks will reduce the variance of the estimates of the parameters.
The normal distribution is in the MDA of the Gumbell distribution. Explain in words what this means.
If we had a few samples coming from a normal distribution, and we were interested in the study of the maximum of each sample, we would realize that the distribution of our maximums is Gumbell.
The normal distribution is in the MDA of the Gumbell distribution.
The normal distribution is symetric, but the Gumbell distribution is positively skewed, explain how this is not inconsistent.
Since we are looking at the maximum of the distribution, we can expect that the observations will be mostly higher than the average, giving something that is skewed. In other words, since we are mostly interested in the extreme part of the normal (right side), the fact that it is symmetric when considering the whole distribution is of little importance.
Describe the trade-off in selecting a threshold for the “points over the threshold” approach to estimating Xi. Explain how the selection affects the bias and the variance.
If the threshold is too low, we include more data points and the bias increases since the added data points may not follow the tail distribution. On the other hand, if it is too high, the sample size will be reduced, and the estimates of our parameters will have greater variance. We want a threshold high enough to only include data points that follow the tail distribution, but also low enough that it includes enough data points for our estimations.
