gevExpInfo | R Documentation |

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)
```

`scale` , `shape` |
Numeric vectors. Respective values of the GEV parameters
scale parameter |

`eps` |
A numeric scalar. For values of |

`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`

.

`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`

.

```
# 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)
```

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