gevExpInfo calculates, for a single pair of values
\((\sigma, \xi) = \) (scale, shape), the expected information matrix for a
single observation from a GEV distribution with distribution function
$$F(x) = P(X \leq x) = \exp\left\{ -\left[ 1+\xi\left(\frac{x-\mu}{\sigma}\right)
\right]_+^{-1/\xi} \right\},$$
where \(x_+ = \max(x, 0)\).
The GEV expected information is defined only for \(\xi > -0.5\) and does
not depend on the value of \(\mu\).
The other functions are vectorized and calculate the individual
contributions to the expected information matrix. For example, gev11e
calculates the expectation \(i_{\mu\mu}\) of the negated second
derivative of the GEV log-density with respect to \(\mu\), that is, each
1 indicates one derivative with respect to \(\mu\). Similarly, 2
denotes one derivative with respect to \(\sigma\) and 3 one derivative
with respect to \(\xi\), so that, for example, gev23e calculates the
expectation \(i_{\sigma\xi}\) of the negated GEV log-density after one
taking one derivative with respect to \(\sigma\) and one derivative with
respect to \(\xi\). Note that \(i_{\xi\xi}\), calculated using
gev33e, depends only on \(\xi\).
The expectation in gev11e can be calculated in a direct way for all
\(\xi > -0.5\). For the other components, direct calculation of the
expectation is unstable when \(\xi\) is close to 0. Instead, we use
a quadratic approximation over (-eps, eps), from a Lagrangian
interpolation of the values from the direct calculation for \(\xi = \)
-eps, \(0\) and eps.