Theoretical Background

Generalized Extreme Value Distribution

The GEV distribution unifies the three types of extreme value distributions. The cumulative distribution function (CDF) of a GEV random variable \(X\) with location \(\mu\), scale \(\sigma > 0\), and shape \(\xi\) is:

CDF:

\[F(x; \mu, \sigma, \xi) = \exp\left\{-\left[1 - \xi\left(\frac{x - \mu}{\sigma}\right)\right]^{1/\xi}\right\} \quad \text{for } \xi \neq 0\]

\[F(x; \mu, \sigma, 0) = \exp\left\{-\exp\left[-\left(\frac{x - \mu}{\sigma}\right)\right]\right\} \quad \text{for } \xi = 0\]

Generalized Pareto Distribution

The CDF of exceedances \(Y = X - \mu > 0\) over threshold \(\mu\) following a GPD, with scale \(\sigma > 0\) and shape \(\xi\):

CDF:

\[F(y; \mu, \sigma, \xi) = 1 - \left(1 + \xi\frac{y - \mu}{\sigma}\right)^{-1/\xi} \quad \text{for } \xi \neq 0\]

\[F(y; \mu, \sigma, 0) = 1 - \exp\left(-\frac{y - \mu}{\sigma}\right) \quad \text{for } \xi = 0\]

Non-stationary Framework

In a non-stationary framework, parameters are modeled as functions of covariates:

Location (linear):

\[\mu(t) = \beta_0 + \beta_1 Z_1(t) + \beta_2 Z_2(t) + \dots\]

Scale (exponential):

\[\sigma(t) = \exp(\alpha_0 + \alpha_1 Z_1(t) + \alpha_2 Z_2(t) + \dots)\]

Shape (linear):

\[\xi(t) = \kappa_0 + \kappa_1 Z_1(t) + \kappa_2 Z_2(t) + \dots\]

Where \(Z(t)\) is a dynamic covariate that changes with time and affects the extreme value distributions.

Non-Stationarity Configuration via Config Vector

In nsEVDx, non-stationarity is controlled via a configuration vector:

\[\text{config} = [a, b, c]\]

Each element in the configuration specifies the number of covariates for the location (\(\mu\)), scale (\(\sigma\)), and shape (\(\xi\)) parameters:

• A value of 0 indicates stationarity.

• Values > 0 indicate non-stationary modeling using the corresponding number of covariates.

This framework allows flexible, parsimonious modeling of non-stationary extreme value distributions, including covariates only where supported by data.