GpdFit {fExtremes} | R Documentation

## Modelling the Generalized Pareto Distribution

### Description

A collection and description of functions to model the Generalized Pareto Distribution, GPD, based on R's 'evir' package. Two approaches for parameter estimation are provided: Maximum likelihood estimation and the probability weighted moment method.

The functions are:

`gpdSim` | generates data from the GPD,
`gpdFit` | fits empirical or simulated data to the distribution,
`print` | print method for a fitted GPD object of class ...,
`plot` | plot method for a fitted GPD object,
`summary` | summary method for a fitted GPD object,
`gpdqPlot` | estimation of high quantiles,
`gpdquantPlot` | variation of high quantiles with threshold,
`gpdriskmeasures` | prescribed quantiles and expected shortfalls,
`gpdsfallPlot` | expected shortfall with confidence intervals,
`gpdshapePlot` | variation of shape with threshold,
`gpdtailPlot` | plot of the tail.

### Usage

    gpdSim(model = list(shape = 0.25, location = 0, scale = 1), n = 1000)
    gpdFit(x, threshold = NA, nextremes = NA, type = c("mle", "pwm"),
        information = c("observed", "expected"), ...)

    ## S3 method for class 'gpdFit':
    print(x, ...)
    ## S3 method for class 'gpdFit':
    plot(x, which = "all", ...)
    ## S3 method for class 'gpdFit':
    summary(object, doplot = TRUE, which = "all", ...)

    gpdqPlot(x, pp = 0.99, ci.type = c("likelihood", "wald"), ci.p = 0.95,
        like.num = 50)
    gpdquantPlot(data, p = 0.99, models = 30, start = 15, end = 500,
        reverse = TRUE, ci = 0.95, autoscale = TRUE, labels = TRUE, ...)
    gpdriskmeasures(x, plevels = c(0.99, 0.995, 0.999, 0.9995, 0.9999))
    gpdsfallPlot(x, pp = 0.99, ci.p = 0.95, like.num = 50)
    gpdshapePlot(data, models = 30, start = 15, end = 500, reverse = TRUE,
        ci = 0.95, autoscale = TRUE, labels = TRUE, ...)
    gpdtailPlot(fit, optlog = NA, extend = 1.5, labels = TRUE, ...)

### Arguments

`autoscale` | whether or not plot should be automatically scaled; if not, xlim and ylim graphical parameters may be entered.
`ci` | the probability for asymptotic confidence band; for no confidence band set to zero.
`ci.p` | the probability for confidence interval (must be less than 0.999).
`ci.type` | the method for calculating a confidence interval: `"likelihood"` or `"wald"`.
`data` | a numeric vector of data.
`doplot` | a logical. Should the results be plotted?
`extend` | optional argument for plots 1 and 2 expressing how far x-axis should extend as a multiple of the largest data value. This argument must take values greater than 1 and is useful for showing estimated quantiles beyond data.
`fit` | [print][plot][summary] - print method, a fitted object of class `"gpd"`.
`information` | whether standard errors should be calculated with `"observed"` or `"expected"` information. This only applies to the maximum likelihood method; for the probability-weighted moments method `"expected"` information is used if possible.
`labels` | optional argument for plots 1 and 2 specifying whether or not axes should be labelled.
`like.num` | the number of times to evaluate profile likelihood.
`model` | [gpdsim] - a list with components `shape`, `location` and `scale` giving the parameters of the GPD distribution. By default the shape parameter has the value 0.25, the location is zero and the scale is one.
`models` | the number of consecutive gpd models to be fitted.
`n` | [gpdsim] - lnumber of generated data points, an integer value.
`nextremes` | [gpdFit] - the number of upper extremes to be used (either this or `threshold` must be given but not both).
`object` | [summary] - a fitted object of class `"gpdFit"`.
`optlog` | optional argument for plots 1 and 2 giving a particular choice of logarithmic axes: `"x"` x-axis only; `"y"` y-axis only; `"xy"` both axes; `""` neither axis.
`plevels, p, pp` | a vector of probability levels, the desired probability for the quantile estimate (e.g. 0.99 for the 99th percentile).
`reverse` | should plot be by increasing threshold (`TRUE`) or number of extremes (`FALSE`).
`start, end` | the lowest and maximum number of exceedances to be considered.
`threshold` | a threshold value (either this or `nextremes` must be given but not both).
`type` | a character string selecting the desired estimation mehtod, either `"mle"` for the maximum likelihood mehtod or `"pwm"` for the probability weighted moment method. By default, the first will be selected. Note, the function `gpd` uses `"ml"`.
`which` | if `which` is set to `"ask"` the function will interactively ask which plot should be displayed. By default this value is set to `FALSE` and then those plots will be displayed for which the elements in the logical vector `which` ar set to `TRUE`; by default all four elements are set to `"all"`.
`x` | [gpdFit] - the data vector. Note, there are two different names for the first argument `x` and `data` depending which function name is used, either `gpdFit` or the EVIS synonyme `gpd`. [print][plot] - a fitted object of class `"gpdFit"`.
`...` | control parameters and plot parameters optionally passed to the optimization and/or plot function. Parameters for the optimization function are passed to components of the `control` argument of `optim`.

### Details

Simulation:
`gpdSim` simulates data from a Generalized Pareto distribution.

Parameter Estimation:
`gpdFit` fits the model parameters either by the probability weighted moment method or the maxim log likelihood method. The function returns an object of class `"gpd"` representing the fit of a generalized Pareto model to excesses over a high threshold. The fitting functions use the probability weighted moment method, if method `method="pwm"` was selected, and the the general purpose optimization function `optim` when the maximum likelihood estimation, `method="mle"` or `method="ml"` is chosen.

Methods:
`print.gpd`, `plot.gpd` and `summary.gpd` are print, plot, and summary methods for a fitted object of class `gpdFit`. The plot method provides four different plots for assessing fitted GPD model.

gpd* Functions:
`gpdqPlot` calculates quantile estimates and confidence intervals for high quantiles above the threshold in a GPD analysis, and adds a graphical representation to an existing plot. The GPD approximation in the tail is used to estimate quantile. The `"wald"` method uses the observed Fisher information matrix to calculate confidence interval. The `"likelihood"` method reparametrizes the likelihood in terms of the unknown quantile and uses profile likelihood arguments to construct a confidence interval.

`gpdquantPlot` creates a plot showing how the estimate of a high quantile in the tail of a dataset based on the GPD approximation varies with threshold or number of extremes. For every model `gpdFit` is called. Evaluation may be slow. Confidence intervals by the Wald method may be fastest.

`gpdriskmeasures` makes a rapid calculation of point estimates of prescribed quantiles and expected shortfalls using the output of the function `gpdFit`. This function simply calculates point estimates and (at present) makes no attempt to calculate confidence intervals for the risk measures. If confidence levels are required use `gpdqPlot` and `gpdsfallPlot` which interact with graphs of the tail of a loss distribution and are much slower.

`gpdsfallPlot` calculates expected shortfall estimates, in other words tail conditional expectation and confidence intervals for high quantiles above the threshold in a GPD analysis. A graphical representation to an existing plot is added. Expected shortfall is the expected size of the loss, given that a particular quantile of the loss distribution is exceeded. The GPD approximation in the tail is used to estimate expected shortfall. The likelihood is reparametrised in terms of the unknown expected shortfall and profile likelihood arguments are used to construct a confidence interval.

`gpdshapePlot` creates a plot showing how the estimate of shape varies with threshold or number of extremes. For every model `gpdFit` is called. Evaluation may be slow.

`gpdtailPlot` produces a plot of the tail of the underlying distribution of the data.

### Value

`gpdSim`
returns a vector of datapoints from the simulated series.
`gpdFit`
returns an object of class `"gpd"` describing the fit including parameter estimates and standard errors.
`gpdquantPlot`
returns invisible a table of results.
`gpdshapePlot`
returns invisible a table of results.
`gpdtailPlot`
returns invisible a list object containing details of the plot is returned invisibly. This object should be used as the first argument of `gpdqPlot` or `gpdsfallPlot` to add quantile estimates or expected shortfall estimates to the plot.

### Author(s)

This function is based on Alec Stephenson's R-package `evir` ported from the `EVIS` library, Extreme Values in S, written by Alexander McNeil. The `fExtremes` port and the change and addition of some functions were done by Diethelm Wuertz.

### References

Hosking J.R.M., Wallis J.R., (1987); Parameter and quantile estimation for the generalized Pareto distribution, Technometrics 29, 339–349.

### Examples

    ## SOURCE("fExtremes.53B-GpdFit")

    ## Load Data:
       data(danish)

    ## gpdSim -
       # Simulate GPD Data:
       xmpExtremes("\nStart: Simulate a GPD Distributed Sample > ")
       x = gpdSim(model = list(shape = 0.25, location = 0, scale = 1), n = 1000)

    ## gpdFit -
       xmpExtremes("\nNext: Fit Simulated Data to GPD using PWM > ")
       fit = gpdFit(x, nextremes = length(x), type = "pwm")
       print(fit)
       par(mfcol = c(4, 2), cex = 0.7)
       summary(fit)

    ## gpdFit -
       xmpExtremes("\nNext: Fit Simulated Data to GPD using MLE > ")
       fit = gpdFit(x, nextremes = length(x), type = "mle")
       print(fit)
       summary(fit)

    ## gpdFit -
       xmpExtremes("\nNext: Fit Danish Fire Data to Excess Losses over 10 > ")
       fit = gpdFit(danish, 10, type = "mle")
       print(fit)
       par(mfrow = c(2, 2), cex = 0.7)
       summary(fit)

    ## gpdqPlot -
       xmpExtremes("\nNext: 99.5th Percentiles for Danish Fire Data > ")
       fit = gpdFit(danish, threshold = 10, type = "mle")
       par(mfrow = c(1, 1))
       tail = gpdtailPlot(fit)
       gpdqPlot(tail, 0.995)
       title(main = "Danish Data: 99.5th Percentile")

    ## gpdquantPlot -
       xmpExtremes("\nNext: 99.9th Percentiles for Danish Fire Data > ")
       par(mfrow = c(1, 1))
       gpdquantPlot(danish, p = 0.999)
       title(sub = "Danish Fire: GPD High Quantile")

    ## gpdsfallPlot -
       xmpExtremes("\nNext: Expected Shortfall for Danish Fire Data > ")
       fit = gpdFit(danish, nextremes = floor(length(danish)/10), type = "mle")
       par(mfrow = c(1, 1))
       tp = gpdtailPlot(fit)
       gpdsfallPlot(tp, 0.999)
       title(main = "Danish Fire: Expected Shortfall")

    ## gpdriskmeasures -
       xmpExtremes("\nNext: Quantiles and Expected Shortfalls > ")
       # Give estimates of 0.999 and 0.9999 quantiles - Danish Fire Date:
       fit = gpdFit(danish, threshold = 10, type = "mle")
       par(mfrow = c(1, 1))
       gpdriskmeasures(fit, c(0.99, 0.995, 0.999, 0.9995, 0.9999))

    ## gpdshapePlot -
       xmpExtremes("\nNext: Shape Plot of Heavy-Tailed Simulated Data > ")
       set.seed(4711)
       par(mfrow = c(1, 1))
       gpdshapePlot(gpdSim(n = 1000))
       title(sub = "Simulated GPD", cex.sub = 0.7)

    ## gpdshapePlot -
       xmpExtremes("\nNext: Shape Plot of Heavy-Tailed Danish Fire Data > ")
       par(mfrow = c(1, 1))
       gpdshapePlot(danish)
       title(sub = "Danish Fire", cex.sub = 0.7)

    ## gpdtailPlot -
       xmpExtremes("\nNext: Plot Tail Estimate of Danish Fire Data >")
       fit = gpdFit(danish, threshold = 10, type = "mle")
       par(mfrow = c(1, 1))
       gpdtailPlot(fit, main = "Danish Fire: GPD Tail Estimate", col = "steelblue4")

[Package fExtremes version 221.10065 Index]