A very brief introduction
Climate
Finance
Industry
Hazard \(\approx\) probability: the chance of an event at least as severe as some value happening within in a given space-time window.
\[\Pr(X > x) = 1 - F_X(x).\]
Risk \(\approx\) cost: the potential economic, social and environmental consequences of perilous events that may occur in a specified period of time or space.
\[ \text{VaR}_\alpha (X) = F_X^{-1}(\alpha) \quad or \quad \text{ES}_\alpha(X) = \mathbb{E}[X | X < \text{VaR}_\alpha(X)].\]
Depending on the peril we are considering, the definition of a “bad” outcome differs:
Without loss of generality, we can focus on modelling large positive values, by transforming our data and results as appropriate.
\[g(X_i) \quad \text{e.g.} \quad -X \quad \text{or} \quad X^{-1}.\]
An issue with risk / hazard modelling is that we are by definition interested in the rare events, which make up only a very small proportion of our data.
Robust regression:
Generalises to quantile regression:
What if we need to do beyond the historical record?
e.g. estimate a 1 in 1000-year flood from 50 years of data.
Extreme Value Theory allows principled extrapolation beyond the range of the observed data.
Focuses on the most extreme observations.
If we care about
\[M_n = \max \{X_1, \ldots, X_n\}\]
How can we model
\[\begin{align*} F_{M_n}(x) &= \Pr(X_1 \leq x,\ \ldots, \ X_n \leq x) \\ &= \Pr(X \leq x) ^n \\ &= F_X(x)^n? \end{align*}\]
Analogue of CLT for Sample Maxima. Let’s revisit the CLT:
Suppose \(X_1, X_2, X_3, \ldots,\) is a sequence of i.i.d. random variables with \(\mathbb{E}[X_i] = \mu\) and \(\text{Var}[X_i] = \sigma^2 < \infty\).
As \(n \rightarrow \infty\), the random variables \(\frac{\sqrt{n} (\bar{X}_n - \mu)}{\sigma}\) converge in distribution to a standard normal distribution.
\[ \frac{\sqrt{n} (\bar{X}_n - \mu)}{\sigma} \overset{d}{\longrightarrow} \mathcal{N}(0,1).\]
Rephrasing this as the partial sums rather than partial means:
\[\frac{S_n}{\sigma\sqrt{n}} - \frac{\mu}{\sigma / \sqrt{n}} \overset{d}{\longrightarrow} \mathcal{N}(0,1).\]
Analogue of CLT for Sample Maxima. Let’s revisit the CLT:
Under weak conditions on \(F_X\) and where appropriate sequences of constants \(\{a_n\}\) and \(\{b_n\}\) exist:
\[a_n S_n - b_n \overset{d}{\longrightarrow} \mathcal{N}(0,1).\]
If suitable sequences of normalising constants exist, then as \(n \rightarrow \infty\):
\[\begin{equation} \label{eqn:lit_extremes_normalising} \Pr\left(\frac{M_n - b_n}{a_n} \leq x \right) \rightarrow G(x), \end{equation}\]
where \(G\) is distribution function of a Fréchet, Gumbel or negative Weibull random variable.
This links to the concept of Maximal Domain of Attraction: if we know \(F_X(x)\) then we can identify \(G(x)\).
But we don’t know \(F\)!
These distributional forms are united in a single parameterisation by the Unified Extremal Types Theorem.
The resulting generalised extreme value (GEV) family of distribution functions has the form
\[\begin{equation} \label{eqn:lit_extremes_GEV} G(x) = \exp\left\{ -\left[ 1 + \xi \frac{x - \mu}{\sigma} \right]_{+}^{-1/\xi}\right\}, \end{equation}\]
where \(x_+ = \max(x,0)\), \(\sigma \in \mathbb{R}^+\) and \(\mu , \xi \in \mathbb{R}\). The parameters \(\mu, \sigma\) and \(\xi\) have respective interpretations as location, scale and shape parameters.
CLT and UETT are asymptotic results, we use them as approximations for finite \(n\).
Split the data into \(m\) blocks of length \(k\) and model the \(M_k\).
How to pick the block size? Trade-off between bias and variance.
Annual blocks often uses as a pragmatic choice to handle seasonal trends.