# GpdDistribution: Generalized Pareto Distribution In fExtremes: Rmetrics - Modelling Extreme Events in Finance

## Description

A collection and description of functions to compute the generalized Pareto distribution. The functions compute density, distribution function, quantile function and generate random deviates for the GPD. In addition functions to compute the true moments and to display the distribution and random variates changing parameters interactively are available.

The GPD distribution functions are:

 `dgpd` Density of the GPD Distribution, `pgpd` Probability function of the GPD Distribution, `qgpd` Quantile function of the GPD Distribution, `rgpd` random variates from the GPD distribution, `gpdMoments` computes true mean and variance, `gpdSlider` displays density or rvs from a GPD.

## Usage

 ```1 2 3 4 5 6 7``` ```dgpd(x, xi = 1, mu = 0, beta = 1, log = FALSE) pgpd(q, xi = 1, mu = 0, beta = 1, lower.tail = TRUE) qgpd(p, xi = 1, mu = 0, beta = 1, lower.tail = TRUE) rgpd(n, xi = 1, mu = 0, beta = 1) gpdMoments(xi = 1, mu = 0, beta = 1) gpdSlider(method = c("dist", "rvs")) ```

## Arguments

 `log` a logical, if `TRUE`, the log density is returned. `lower.tail` a logical, if `TRUE`, the default, then probabilities are `P[X <= x]`, otherwise, `P[X > x]`. `method` [gpdSlider] - a character string denoting what should be displayed. Either the density and `"dist"` or random variates `"rvs"`. `n` [rgpd][gpdSim\ - the number of observations to be generated. `p` a vector of probability levels, the desired probability for the quantile estimate (e.g. 0.99 for the 99th percentile). `q` [pgpd] - a numeric vector of quantiles. `x` [dgpd] - a numeric vector of quantiles. `xi, mu, beta` `xi` is the shape parameter, `mu` the location parameter, and `beta` is the scale parameter.

## Value

All values are numeric vectors:
`d*` returns the density,
`p*` returns the probability,
`q*` returns the quantiles, and
`r*` generates random deviates.

## Author(s)

Alec Stephenson for the functions from R's `evd` package,
Alec Stephenson for the functions from R's `evir` package,
Alexander McNeil for the EVIS functions underlying the `evir` package,
Diethelm Wuertz for this R-port.

## References

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

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28``` ```## rgpd - par(mfrow = c(2, 2), cex = 0.7) r = rgpd(n = 1000, xi = 1/4) plot(r, type = "l", col = "steelblue", main = "GPD Series") grid() ## dgpd - # Plot empirical density and compare with true density: # Omit values greater than 500 from plot hist(r, n = 50, probability = TRUE, xlab = "r", col = "steelblue", border = "white", xlim = c(-1, 5), ylim = c(0, 1.1), main = "Density") box() x = seq(-5, 5, by = 0.01) lines(x, dgpd(x, xi = 1/4), col = "orange") ## pgpd - # Plot df and compare with true df: plot(sort(r), (1:length(r)/length(r)), xlim = c(-3, 6), ylim = c(0, 1.1), pch = 19, cex = 0.5, ylab = "p", xlab = "q", main = "Probability") grid() q = seq(-5, 5, by = 0.1) lines(q, pgpd(q, xi = 1/4), col = "steelblue") ## qgpd - # Compute quantiles, a test: qgpd(pgpd(seq(-1, 5, 0.25), xi = 1/4 ), xi = 1/4) ```

### Example output

```Loading required package: timeDate

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