actuar (version 0.9-3)

GeneralizedPareto: The Generalized Pareto Distribution

Description

Density, distribution function, quantile function, random generation, raw moments and limited moments for the Generalized Pareto distribution with parameters shape1, shape2 and scale.

Usage

dgenpareto(x, shape1, shape2, rate = 1, scale = 1/rate,
             log = FALSE)
  pgenpareto(q, shape1, shape2, rate = 1, scale = 1/rate,
             lower.tail = TRUE, log.p = FALSE)
  qgenpareto(p, shape1, shape2, rate = 1, scale = 1/rate,
             lower.tail = TRUE, log.p = FALSE)
  rgenpareto(n, shape1, shape2, rate = 1, scale = 1/rate)
  mgenpareto(order, shape1, shape2, rate = 1, scale = 1/rate)
  levgenpareto(limit, shape1, shape2, rate = 1, scale = 1/rate,
               order = 1)

Arguments

x, q

vector of quantiles.

p

vector of probabilities.

n

number of observations. If length(n) > 1, the length is taken to be the number required.

shape1, shape2, scale

parameters. Must be strictly positive.

rate

an alternative way to specify the scale.

log, log.p

logical; if TRUE, probabilities/densities $p$ are returned as $\log(p)$.

lower.tail

logical; if TRUE (default), probabilities are $P[X \le x]$, otherwise, $P[X > x]$.

order

order of the moment.

limit

limit of the loss variable.

Value

• dgenpareto gives the density, pgenpareto gives the distribution function, qgenpareto gives the quantile function, rgenpareto generates random deviates, mgenpareto gives the $k$th raw moment, and levgenpareto gives the $k$th moment of the limited loss variable.

  Invalid arguments will result in return value NaN, with a warning.

Details

The Generalized Pareto distribution with parameters shape1 $= \alpha$, shape2 $= \tau$ and scale $= \theta$ has density: $$f(x) = \frac{\Gamma(\alpha + \tau)}{\Gamma(\alpha)\Gamma(\tau)} \frac{\theta^\alpha x^{\tau - 1}}{ (x + \theta)^{\alpha + \tau}}$$ for $x > 0$, $\alpha > 0$, $\tau > 0$ and $\theta > 0$. (Here $\Gamma(\alpha)$ is the function implemented by R's gamma() and defined in its help.)

The Generalized Pareto is the distribution of the random variable $$\theta \left(\frac{X}{1 - X}\right),$$ where $X$ has a Beta distribution with parameters $\alpha$ and $\tau$.

The Generalized Pareto distribution has the following special cases:

• AParetodistribution whenshape2 == 1;
• AnInverse Paretodistribution whenshape1 == 1.

References

Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2004), Loss Models, From Data to Decisions, Second Edition, Wiley.

Examples

exp(dgenpareto(3, 3, 4, 4, log = TRUE))
p <- (1:10)/10
pgenpareto(qgenpareto(p, 3, 3, 1), 3, 3, 1)
qgenpareto(.3, 3, 4, 4, lower.tail = FALSE)
mgenpareto(1, 3, 2, 1) ^ 2
levgenpareto(10, 3, 3, 3, order = 2)