View source: R/family.extremes.R

gev | R Documentation |

Maximum likelihood estimation of the 3-parameter generalized extreme value (GEV) distribution.

```
gev(llocation = "identitylink", lscale = "loglink",
lshape = logofflink(offset = 0.5), percentiles = c(95, 99),
ilocation = NULL, iscale = NULL, ishape = NULL, imethod = 1,
gprobs.y = (1:9)/10, gscale.mux = exp((-5:5)/6),
gshape = (-5:5) / 11 + 0.01,
iprobs.y = NULL, tolshape0 = 0.001,
type.fitted = c("percentiles", "mean"),
zero = c("scale", "shape"))
gevff(llocation = "identitylink", lscale = "loglink",
lshape = logofflink(offset = 0.5), percentiles = c(95, 99),
ilocation = NULL, iscale = NULL, ishape = NULL, imethod = 1,
gprobs.y = (1:9)/10, gscale.mux = exp((-5:5)/6),
gshape = (-5:5) / 11 + 0.01,
iprobs.y = NULL, tolshape0 = 0.001,
type.fitted = c("percentiles", "mean"), zero = c("scale", "shape"))
```

`llocation, lscale, lshape` |
Parameter link functions for For the shape parameter,
the default |

`percentiles` |
Numeric vector of percentiles used for the fitted values.
Values should be between 0 and 100.
This argument is ignored if |

`type.fitted` |
See |

`ilocation, iscale, ishape` |
Numeric. Initial value for the location parameter, |

`imethod` |
Initialization method. Either the value 1 or 2.
If both methods fail then try using |

`gshape` |
Numeric vector.
The values are used for a grid search for an initial value
for |

`gprobs.y, gscale.mux, iprobs.y` |
Numeric vectors, used for the initial values.
See |

`tolshape0` |
Passed into |

`zero` |
A specifying which
linear/additive predictors are modelled as intercepts only.
The values can be from the set {1,2,3} corresponding
respectively to |

The GEV distribution function can be written

`G(y) = \exp( -[ (y-\mu)/ \sigma ]_{+}^{- 1/ \xi}) `

where `\sigma > 0`

,
`-\infty < \mu < \infty`

,
and `1 + \xi(y-\mu)/\sigma > 0`

.
Here, `x_+ = \max(x,0)`

.
The `\mu`

, `\sigma`

, `\xi`

are known as the
*location*, *scale* and *shape* parameters respectively.
The cases
`\xi>0`

,
`\xi<0`

,
`\xi = 0`

correspond to the Frechet,
reverse
Weibull, and Gumbel types respectively.
It can be noted that the Gumbel (or Type I) distribution accommodates
many commonly-used distributions such as the normal, lognormal,
logistic, gamma, exponential and Weibull.

For the GEV distribution, the `k`

th moment about the mean exists
if `\xi < 1/k`

.
Provided they exist, the mean and variance are given by
`\mu+\sigma\{ \Gamma(1-\xi)-1\}/ \xi`

and
`\sigma^2 \{ \Gamma(1-2\xi) - \Gamma^2(1-\xi) \} / \xi^2`

respectively,
where `\Gamma`

is the gamma function.

Smith (1985) established that when `\xi > -0.5`

,
the maximum likelihood estimators are completely regular.
To have some control over the estimated `\xi`

try
using `lshape = logofflink(offset = 0.5)`

, say,
or `lshape = extlogitlink(min = -0.5, max = 0.5)`

, say.

An object of class `"vglmff"`

(see `vglmff-class`

).
The object is used by modelling functions such as `vglm`

,
and `vgam`

.

Currently, if an estimate of `\xi`

is too close to 0 then
an error may occur for `gev()`

with multivariate responses.
In general, `gevff()`

is more reliable than `gev()`

.

Fitting the GEV by maximum likelihood estimation can be numerically
fraught. If `1 + \xi (y-\mu)/ \sigma \leq 0`

then some crude evasive action is taken but the estimation process
can still fail. This is particularly the case if `vgam`

with `s`

is used; then smoothing is best done with
`vglm`

with regression splines (`bs`

or `ns`

) because `vglm`

implements
half-stepsizing whereas `vgam`

doesn't (half-stepsizing
helps handle the problem of straying outside the parameter space.)

The VGAM family function `gev`

can handle a multivariate
(matrix) response, cf. multiple responses.
If so, each row of the matrix is sorted into
descending order and `NA`

s are put last.
With a vector or one-column matrix response using
`gevff`

will give the same result but be faster and it handles
the `\xi = 0`

case.
The function `gev`

implements Tawn (1988) while
`gevff`

implements Prescott and Walden (1980).

Function `egev()`

has been replaced by the
new family function `gevff()`

. It now
conforms to the usual VGAM philosophy of
having `M1`

linear predictors per (independent) response.
This is the usual way multiple responses are handled.
Hence `vglm(cbind(y1, y2)..., gevff, ...)`

will have
6 linear predictors and it is possible to constrain the
linear predictors so that the answer is similar to `gev()`

.
Missing values in the response of `gevff()`

will be deleted;
this behaviour is the same as with almost every other
VGAM family function.

The shape parameter `\xi`

is difficult to estimate
accurately unless there is a lot of data.
Convergence is slow when `\xi`

is near `-0.5`

.
Given many explanatory variables, it is often a good idea
to make sure `zero = 3`

.
The range restrictions of the parameter `\xi`

are not
enforced; thus it is possible for a violation to occur.

Successful convergence often depends on having a reasonably good initial
value for `\xi`

. If failure occurs try various values for the
argument `ishape`

, and if there are covariates,
having `zero = 3`

is advised.

T. W. Yee

Yee, T. W. and Stephenson, A. G. (2007).
Vector generalized linear and additive extreme value models.
*Extremes*, **10**, 1–19.

Tawn, J. A. (1988).
An extreme-value theory model for dependent observations.
*Journal of Hydrology*, **101**, 227–250.

Prescott, P. and Walden, A. T. (1980).
Maximum likelihood estimation of the parameters of the
generalized extreme-value distribution.
*Biometrika*, **67**, 723–724.

Smith, R. L. (1985).
Maximum likelihood estimation in a class of nonregular cases.
*Biometrika*, **72**, 67–90.

`rgev`

,
`gumbel`

,
`gumbelff`

,
`guplot`

,
`rlplot.gevff`

,
`gpd`

,
`weibullR`

,
`frechet`

,
`extlogitlink`

,
`oxtemp`

,
`venice`

,
`CommonVGAMffArguments`

.

```
## Not run:
# Multivariate example
fit1 <- vgam(cbind(r1, r2) ~ s(year, df = 3), gev(zero = 2:3),
data = venice, trace = TRUE)
coef(fit1, matrix = TRUE)
head(fitted(fit1))
par(mfrow = c(1, 2), las = 1)
plot(fit1, se = TRUE, lcol = "blue", scol = "forestgreen",
main = "Fitted mu(year) function (centered)", cex.main = 0.8)
with(venice, matplot(year, depvar(fit1)[, 1:2], ylab = "Sea level (cm)",
col = 1:2, main = "Highest 2 annual sea levels", cex.main = 0.8))
with(venice, lines(year, fitted(fit1)[,1], lty = "dashed", col = "blue"))
legend("topleft", lty = "dashed", col = "blue", "Fitted 95 percentile")
# Univariate example
(fit <- vglm(maxtemp ~ 1, gevff, data = oxtemp, trace = TRUE))
head(fitted(fit))
coef(fit, matrix = TRUE)
Coef(fit)
vcov(fit)
vcov(fit, untransform = TRUE)
sqrt(diag(vcov(fit))) # Approximate standard errors
rlplot(fit)
## End(Not run)
```

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