R: The Generalized Pareto Distribution

gpdUC {VGAM} | R Documentation

## The Generalized Pareto Distribution

### Description

Density, distribution function, quantile function and random generation for the generalized Pareto distribution (GPD) with location parameter `location`, scale parameter `scale` and shape parameter `shape`.

### Usage

    dgpd(x, location = 0, scale = 1, shape = 0, log = FALSE,
         tolshape0 = sqrt(.Machine$double.eps))
    pgpd(q, location = 0, scale = 1, shape = 0,
         lower.tail = TRUE, log.p = FALSE)
    qgpd(p, location = 0, scale = 1, shape = 0,
         lower.tail = TRUE, log.p = FALSE)
    rgpd(n, location = 0, scale = 1, shape = 0)

### Arguments

`x, q` | vector of quantiles.
`p` | vector of probabilities.
`n` | number of observations. If `length(n) > 1` then the length is taken to be the number required.
`location` | the location parameter mu.
`scale` | the (positive) scale parameter sigma.
`shape` | the shape parameter xi.
`log` | Logical. If `log = TRUE` then the logarithm of the density is returned.
`lower.tail, log.p` | Same meaning as in `punif` or `qunif`.
`tolshape0` | Positive numeric. Threshold/tolerance value for resting whether xi is zero. If the absolute value of the estimate of xi is less than this value then it will be assumed zero and an exponential distribution will be used.

### Details

See `gpd`, the VGAM family function for estimating the two parameters by maximum likelihood estimation, for formulae and other details. Apart from `n`, all the above arguments may be vectors and are recyled to the appropriate length if necessary.

### Value

`dgpd` gives the density, `pgpd` gives the distribution function, `qgpd` gives the quantile function, and `rgpd` generates random deviates.

### Note

The default values of all three parameters, especially xi = 0, means the default distribution is the exponential.

Currently, these functions have different argument names compared with those in the evd package.

### Author(s)

T. W. Yee and Kai Huang

### References

Coles, S. (2001) An Introduction to Statistical Modeling of Extreme Values. London: Springer-Verlag.

### See Also

`gpd`, `Exponential`.

### Examples

    ## Not run:  loc <- 2; sigma <- 1; xi <- -0.4
    x <- seq(loc - 0.2, loc + 3, by = 0.01)
    plot(x, dgpd(x, loc, sigma, xi), type = "l", col = "blue", ylim = c(0, 1),
         main = "Blue is density, red is cumulative distribution function",
         sub = "Purple are 5,10,...,95 percentiles", ylab = "", las = 1)
    abline(h = 0, col = "blue", lty = 2)
    lines(qgpd(seq(0.05, 0.95, by = 0.05), loc, sigma, xi),
          dgpd(qgpd(seq(0.05, 0.95, by = 0.05), loc, sigma, xi), loc, sigma, xi),
          col = "purple", lty = 3, type = "h")
    lines(x, pgpd(x, loc, sigma, xi), type = "l", col = "red")
    abline(h = 0, lty = 2)

    pgpd(qgpd(seq(0.05, 0.95, by = 0.05), loc, sigma, xi), loc, sigma, xi)

    ## End(Not run)

[Package VGAM version 1.0-3 Index]