# Generalized Extreme Value Modelling

### Description

A collection and description functions to estimate
the parameters of the GEV distribution. To model
the GEV three types of approaches for parameter
estimation are provided: Maximum likelihood
estimation, probability weighted moment method,
and estimation by the MDA approach. MDA includes
functions for the Pickands, Einmal-Decker-deHaan,
and Hill estimators together with several plot
variants.

The GEV modelling functions are:

`gevrlevelPlot` | k-block return level with confidence intervals. |

### Usage

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### Arguments

`add` |
[gevrlevelPlot] - |

`ci` |
[hillPlot] - |

`kBlocks` |
[gevrlevelPlot] - |

`labels` |
[hillPlot] - |

`object` |
[summary][grlevelPlot] - |

`plottype` |
[hillPlot] - |

`...` |
arguments passed to the plot function. |

### Details

**Parameter Estimation:**

`gevFit`

and `gumbelFit`

estimate the parameters either
by the probability weighted moment method, `method="pwm"`

or
by maximum log likelihood estimation `method="mle"`

. The
summary method produces diagnostic plots for fitted GEV or Gumbel
models.

**Methods:**

`print.gev`

, `plot.gev`

and `summary.gev`

are
print, plot, and summary methods for a fitted object of class
`gev`

. Concerning the summary method, the data are
converted to unit exponentially distributed residuals under null
hypothesis that GEV fits. Two diagnostics for iid exponential data
are offered. The plot method provides two different residual plots
for assessing the fitted GEV model. Two diagnostics for
iid exponential data are offered.

**Return Level Plot:**

`gevrlevelPlot`

calculates and plots the k-block return level
and 95% confidence interval based on a GEV model for block maxima,
where `k`

is specified by the user. The k-block return level
is that level exceeded once every `k`

blocks, on average. The
GEV likelihood is reparameterized in terms of the unknown return
level and profile likelihood arguments are used to construct a
confidence interval.

**Hill Plot:**

The function `hillPlot`

investigates the shape parameter and
plots the Hill estimate of the tail index of heavy-tailed data, or
of an associated quantile estimate. This plot is usually calculated
from the alpha perspective. For a generalized Pareto analysis of
heavy-tailed data using the `gpdFit`

function, it helps to
plot the Hill estimates for `xi`

.

**Shape Parameter Plot:**

The function `shaparmPlot`

investigates the shape parameter and
plots for the upper and lower tails the shape parameter as a function
of the taildepth. Three approaches are considered, the *Pickands*
estimator, the *Hill* estimator, and the
*Decker-Einmal-deHaan* estimator.

### Value

`gevSim`

returns a vector of data points from the simulated series.

`gevFit`

returns an object of class `gev`

describing the fit.

`print.summary`

prints a report of the parameter fit.

`summary`

performs diagnostic analysis. The method provides two different
residual plots for assessing the fitted GEV model.

`gevrlevelPlot`

returns a vector containing the lower 95% bound of the confidence
interval, the estimated return level and the upper 95% bound.

`hillPlot`

displays a plot.

`shaparmPlot`

returns a list with one or two entries, depending on the
selection of the input variable `both.tails`

. The two
entries `upper`

and `lower`

determine the position of
the tail. Each of the two variables is again a list with entries
`pickands`

, `hill`

, and `dehaan`

. If one of the
three methods will be discarded the printout will display zeroes.

### Note

**GEV Parameter Estimation:**

If method `"mle"`

is selected the parameter fitting in `gevFit`

is passed to the internal function `gev.mle`

or `gumbel.mle`

depending on the value of `gumbel`

, `FALSE`

or `TRUE`

.
On the other hand, if method `"pwm"`

is selected the parameter
fitting in `gevFit`

is passed to the internal function
`gev.pwm`

or `gumbel.pwm`

again depending on the value of
`gumbel`

, `FALSE`

or `TRUE`

.

### Author(s)

Alec Stephenson for R's `evd`

and `evir`

package, and

Diethelm Wuertz for this **R**-port.

### References

Coles S. (2001);
*Introduction to Statistical Modelling of Extreme Values*,
Springer.

Embrechts, P., Klueppelberg, C., Mikosch, T. (1997);
*Modelling Extremal Events*,
Springer.

### Examples

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