Skip to content

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

X1, ..., Xn are independent Exponential variables with rate 1, and Mn is their maximum. Which statement about the limiting behaviour of Mn - ln n as n tends to infinity is correct?

The normalised maximum converges to the standard Gumbel distribution, because (1 - e^{-x}/n)^n tends to exp(-e^{-x}). The exponential distribution has a light, exponentially decaying tail and so belongs to the Gumbel domain of attraction.

  1. AIt converges to the standard Gumbel distribution with F(x) = exp(-exp(-x))Correct
  2. BIt converges to a standard Fréchet distribution with shape 1
  3. CIt converges to a Weibull distribution with a finite upper endpoint
  4. DIt converges to a normal distribution by the central limit theorem
  5. It degenerates to a constant with no non-trivial limit

Explanation

P(Mn - ln n <= x) = (1 - e^{-x}/n)^n, which tends to exp(-e^{-x}). This is the Gumbel distribution. The exponential has an exponentially decaying tail and so lies in the Gumbel domain of attraction, not Fréchet.

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