Skip to content

IAI Actuarial Core Principles · Risk Modelling and Survival Analysis · Introduction to extreme value theory

A GEV distribution has location mu = 50, scale sigma = 10 and shape xi = -0.5. Its distribution function is F(x) = exp(-[1 + xi(x-mu)/sigma]^(-1/xi)) for 1 + xi(x-mu)/sigma > 0. What is the upper endpoint of the distribution and the value of F(60)?

With xi = -0.5 the upper endpoint is mu - sigma/xi = 70. At x = 60 the bracket equals 0.5, raised to the power 2 gives 0.25, so F(60) = exp(-0.25), about 0.779.

  1. AUpper endpoint 70; F(60) = exp(-0.25)Correct
  2. BUpper endpoint 70; F(60) = exp(-0.5)
  3. CUpper endpoint 50; F(60) = exp(-0.25)
  4. DUpper endpoint 60; F(60) = exp(-0.25)
  5. No finite endpoint; F(60) = exp(-0.25)

Explanation

For xi < 0 the endpoint is mu - sigma/xi = 50 + 10/0.5 = 70. At x = 60: 1 + (-0.5)(10)/10 = 0.5, and the exponent -1/xi = 2, so the bracket is 0.5^2 = 0.25 and F(60) = exp(-0.25). The exp(-0.5) option forgets to square the bracket.

Did you get it right without looking?

One question tells you little. A timed set on Introduction to extreme value theory shows your real accuracy, how long you take and where you lose marks.

Practise Introduction to extreme value theoryFree diagnostic mock

More Introduction to extreme value theory questions