GeneralizedPareto: The Generalized Pareto Distribution

Description

Density function, 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.

vector of probabilities.

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:

A Pareto distribution when shape2 == 1;
An Inverse Pareto distribution when shape1 == 1.

The $k$th raw moment of the random variable $X$ is $E[X^k]$, $-\tau < k < \alpha$.

The $k$th limited moment at some limit $d$ is $E[\min(X, d)^k]$, $k > -\tau$ and $\alpha - k$ not a negative integer.

References

Embrechts, P., Kl<U+00FC>ppelberg, C. and Mikisch, T. (1997), Modelling Extremal Events for Insurance and Finance, Springer.

Kleiber, C. and Kotz, S. (2003), Statistical Size Distributions in Economics and Actuarial Sciences, Wiley.

Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2012), Loss Models, From Data to Decisions, Fourth Edition, Wiley.

Examples

Run this code

# NOT RUN {
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)

## variance
mgenpareto(2, 3, 2, 1) - mgenpareto(1, 3, 2, 1)^2

## case with shape1 - order > 0
levgenpareto(10, 3, 3, scale = 1, order = 2)

## case with shape1 - order < 0
levgenpareto(10, 1.5, 3, scale = 1, order = 2)
# }

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