GEV Distribution Expected Information — gevExpInfo • gamlssx

Calculates the expected information matrix for the GEV distribution.

Usage

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)

Arguments

scale, shape: Numeric vectors. Respective values of the GEV parameters scale parameter $\sigma$ and shape parameter $\xi$. For gevExpInfo, scale and shape must have length 1.
eps: 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.

Value

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.

Details

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.

Examples

# 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