What is an estimate of the tail parameter of the combined dataset?

For a hypotherical UoM, the number of losses in two non-overlapping datasets is 24 and 32 respectively. The Pareto tail parameters for the two datasets calculated using the maximum likelihood estimation method are 2 and 3 .

What is an estimate of the tail parameter of the combined dataset?
A . 2.57
B . 2.23
C . 3
D . Cannot be determined

View Answer

Answer: A

Explanation:

For a number of processes, including many in finance, while a distribution such as the normal distribution is a good approximation of the distribution near the modal value of the variable, the same normal distribution may not be a good estimate of the tails. For this reason, the Pareto distribution is one of the distributions that is often used to model the tails of another distribution. Generally, if you have a set of observations, and you discard all observations below a threshold, you are left with what are called ‘exceedances’. The threshold needs to be reasonably far out in the tail. If from each value of the exceedances you subtract the threshold value, the resulting dataset is estimated by the generalized Pareto distribution.

The Pareto distribution has a ‘shape parameter’. The average of two Pareto distributions with tail parameters 1 and 2 (is a Greek character, pronounced as ‘sai’ (saa-eee)), is the weighted average of 1 and 2 with weights proportional to the number of observations in the datasets underlying the distributions.