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

An analyst fits a generalised Pareto distribution (GPD) to exceedances over a threshold u using maximum likelihood. Which of the following best describes the effect of raising the threshold u substantially?

Raising the threshold reduces bias because the GPD approximation improves in the far tail, but it leaves fewer exceedances, so the variance of the fitted parameters increases. Choosing the threshold is therefore a trade-off between bias and variance.

  1. ABias from the GPD approximation tends to fall, but fewer exceedances are available so the variance of parameter estimates risesCorrect
  2. BBoth bias and variance of the estimates necessarily fall because only the most extreme data are used
  3. CBias rises and variance falls because the GPD approximation becomes worse in the far tail
  4. DThe shape parameter is forced to equal zero for any sufficiently high threshold
  5. The estimates are unchanged because the GPD is threshold-stable only for the Gumbel case

Explanation

The threshold choice involves a bias-variance trade-off. A higher threshold makes the GPD limit result more accurate, reducing bias, but leaves fewer exceedances, so estimates become more variable. The option claiming both fall ignores the loss of data.

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