gevExpInfo {gamlssx}

GEV Distribution Expected Information

Description

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.

Details

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.

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.

Examples

# Expected information matrices for ...
# ... scale = 2 and shape = -0.4
gevExpInfo(2, -0.4)
# ... scale = 3 and shape = 0.001
gevExpInfo(3, 0.001)
# ... scale = 3 and shape = 0
gevExpInfo(3, 0)
# ... scale = 1 and shape = 0.1
gevExpInfo(1, 0.1)

# The individual components of the latter matrix
gev11e(1, 0.1)
gev12e(1, 0.1)
gev13e(1, 0.1)
gev22e(1, 0.1)
gev23e(1, 0.1)
gev33e(0.1)