Calculates the expected information matrix for the GEV distribution.
gev11e(scale, shape)
gev22e(scale, shape, eps = 0.003)
gev33e(shape, eps = 0.003)
gev12e(scale, shape, eps = 0.003)
gev13e(scale, shape, eps = 0.003)
gev23e(scale, shape, eps = 0.003)
gevExpInfo(scale, shape, eps = 0.003)Numeric vectors. Respective values of the GEV parameters
scale parameter \(\sigma\) and shape parameter \(\xi\). For
gevExpInfo, scale and shape must have length 1.
A numeric scalar. For values of \(\xi\) in shape that lie in
(-eps, eps) an approximation is used instead of a direct calculation.
See Details. If eps is a vector then only the first element is used.
gevExpInfo returns a 3 by 3 numeric matrix with row and column
named loc, scale, shape. The other functions return a numeric vector of
length equal to the maximum of the lengths of the arguments, excluding
eps.
gevExpInfo calculates, for 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.
# Expected information matrices for ...
# ... scale = 2 and shape = -0.4
gevExpInfo(2, -0.4)
#> loc scale shape
#> loc 0.4131759 0.4744927 1.768649
#> scale 0.4744927 1.3526141 3.932333
#> shape 1.7686493 3.9323334 16.905985
# ... scale = 3 and shape = 0.001
gevExpInfo(3, 0.001)
#> loc scale shape
#> loc 0.11120536 -0.04722437 0.1371977
#> scale -0.04722437 0.20255785 0.1095938
#> shape 0.13719772 0.10959382 2.4181684
# ... scale = 3 and shape = 0
gevExpInfo(3, 0)
#> loc scale shape
#> loc 0.11111111 -0.04697604 0.1372801
#> scale -0.04697604 0.20263118 0.1108283
#> shape 0.13728011 0.11082830 2.4236061
# ... scale = 1 and shape = 0.1
gevExpInfo(1, 0.1)
#> loc scale shape
#> loc 1.1109842 -0.64498331 0.41951015
#> scale -0.6449833 1.80124846 0.01231181
#> shape 0.4195102 0.01231181 1.99698389
# The individual components of the latter matrix
gev11e(1, 0.1)
#> [1] 1.110984
gev12e(1, 0.1)
#> [1] -0.6449833
gev13e(1, 0.1)
#> [1] 0.4195102
gev22e(1, 0.1)
#> [1] 1.801248
gev23e(1, 0.1)
#> [1] 0.01231181
gev33e(0.1)
#> [1] 1.996984