GpdGlmFit {fExtremes} | R Documentation

## Modelling the GPD Distribution including GLM

### Description

a collection of functions to model the Generalized Pareto Distribution, GPD, by maximum likelihood approximation based on R's 'ismev' package. In addition to the function 'gpdFit' the parameter estimation allows to include generalized linear modelling, glm, of each parameter.

The functions are:

`gpdglmFit` | 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,
`gpdglmprofPlot` | profile log-likelihoods for return levels,
`gpdglmprofxiPlot` | profile log-likelihoods for shape parameters.

### Usage

    gpdglmFit(x, threshold = min(x), npy = 365, y = NULL, sigl = NULL,
        shl = NULL, siglink = identity, shlink = identity, show = FALSE,
        method = "Nelder-Mead", maxit = 10000, ...)

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

    gpdglmprofPlot(fit, m, xlow, xup, conf = 0.95, nint = 100, ...)
    gpdglmprofxiPlot(fit, xlow, xup, conf = 0.95, nint = 100, ...)

### Arguments

`conf` | [gpdglmprof*Plot] - the confidence coefficient of the plotted profile confidence interval.
`doplot` | a logical. Should the results be plotted?
`fit` | a fitted object either of class `"gpdglm"`.
`m` | [gpdglmprof*Plot] - the return level; i.e. the profile likelihood is for the value that is exceeded with probability 1/`m`.
`maxit` | [gpdglmFit] - the maximum number of iterations.
`method` | [gpdglmFit] - the optimization method; see `optim` for details.
`nint` | [gpdglmprof*Plot] - the number of points at which the profile likelihood is evaluated.
`npy` | [gpdglmFit] - the number of observations per year/block. By default 365.
`object` | [summary] - a fitted object of class `"gpdglmFit"`.
`show` | [gpdglmFit] - a logical; if `TRUE` (the default), print details of the fit.
`sigl, shl` | [gpdglmFit] - numeric vectors of integers, giving the columns of `ydat` that contain covariates for generalized linear modelling of the scale and shape parameters repectively (or `NULL` (the default) if the corresponding parameter is stationary).
`siglink, shlink` | [gpdglmFit] - inverse link functions for generalized linear modelling of the scale and shape parameters repectively.
`threshold` | [gpdglmFit] - the threshold value; a single number or a numeric vector of the same length as `xdat`.
`which` | [plot][summary] - a vector of logicals, one for each plot, denoting which plot should be displayed. By default `c(TRUE, TRUE, TRUE, TRUE, TRUE)`.
`x` | A numeric vector of data to be fitted. [print][plot] - a fitted object of class `"gpdglmFit"`.
`xlow, xup` | [gpdglmprof*Plot] - the least and greatest value at which to evaluate the profile likelihood.
`y` | [gpdglmFit] - a matrix of covariates for generalized linear modelling of the parameters (or `NULL` (the default) for stationary fitting). The number of rows should be the same as the length of `xdat`.
`...` | [gpdglmFit] - other control parameters for the optimization. These are passed to components of the `control` argument of `optim`.

### Details

Simulation:
To simulate a GPD series use the function `gpdSim`.

Parameter Estimation:
`gpdglmFit` fits by the Maximum-likelihood approach the generalized extreme value distribution, including generalized linear modelling of each parameter.

Methods:
`print.gpdglm`, `plot.gpdglm` and `summary.gpdglm` are print, plot, and summary methods for a fitted object of class `gpdglm`.

Nonstationary Models:
For non-stationary fitting it is recommended that the covariates within the generalized linear models are (at least approximately) centered and scaled (i.e. the columns of `ydat` should be approximately centered and scaled).

### Value

A list containing the following components. A subset of these components are printed after the fit. If `show` is `TRUE`, then assuming that successful convergence is indicated, the components `nexc`, `nllh`, `mle`, `rate` and `se` are always printed.

`trans` | An logical indicator for a non-stationary fit.
`model` | A list with components `sigl` and `shl`.
`link` | A character vector giving inverse link functions.
`threshold` | The threshold, or vector of thresholds.
`nexc` | The number of data points above the threshold.
`data` | The data that lie above the threshold. For non-stationary models, the data is standardized.
`conv` | The convergence code, taken from the list returned by `optim`. A zero indicates successful convergence.
`nllh` | The negative logarithm of the likelihood evaluated at the maximum likelihood estimates.
`vals` | A matrix with three columns containing the maximum likelihood estimates of the scale and shape parameters, and the threshold, at each data point.
`mle` | A vector containing the maximum likelihood estimates.
`rate` | The proportion of data points that lie above the threshold.
`cov` | he covariance matrix.
`se` | A vector containing the standard errors.
`n` | The number of data points (i.e. the length of `xdat`).
`npy` | The number of observations per year/block.
`xdata` | The data that has been fitted.

For stationary models four plots are produced; a probability plot, a quantile plot, a return level plot and a histogram of data with fitted density. For non-stationary models two plots are produced; a residual probability plot and a residual quantile plot.

### Author(s)

Alec Stephenson for the code implemented from R's ismev package,
Stuart Scott for the original code, and Diethelm Wuertz for this R-port.

### References

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

### Examples

    ## SOURCE("fExtremes.54B-GpdGlmFit")

    ## Use Rain Data:
       data(rain)

    ## Fit GPD Model:
       xmpExtremes("Start: Parameter Estimation >")
       fit = gpdglmFit(x = rain, threshold = 10)
       print(fit)
       xmpExtremes("Next: Summary Report > ")

    ## Summarize Results:
       xmpExtremes("Next: Profile Likelihood >")
       par(mfrow = c(2, 2), cex = 0.75)
       summary(fit, which = "all")
       # Profile Lielihood:
       par(mfrow = c(2, 1), cex = 0.75)
       gpdglmprofPlot(fit, m = 10, xlow = 55, xup = 75)
       title(main = "Rain")
       gpdglmprofxiPlot(fit, xlow = -0.02, 0.15)
       title(main = "Rain")

[Package fExtremes version 221.10065 Index]