R: The Generalized Pareto Distribution

GeneralizedPareto {actuar} | R Documentation

## 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.
`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 <= x], otherwise, P[X > x].
`order` | order of the moment.
`limit` | limit of the loss variable.

### Details

The Generalized Pareto distribution with parameters `shape1` = a, `shape2` = b and `scale` = s has density:

f(x) = Gamma(a + b)/(Gamma(a) * Gamma(b)) * (s^a x^(b - 1))/(x + s)^(a + b)

for x > 0, a > 0, b > 0 and s > 0. (Here Γ(α) is the function implemented by R's `gamma()` and defined in its help.)

The Generalized Pareto is the distribution of the random variable

θ (X/(1 - X)),

where X has a beta distribution with parameters α and τ.

The Generalized Pareto distribution has the following special cases:

- A Pareto distribution when `shape2 == 1`;

- An Inverse Pareto distribution when `shape1 == 1`.

The kth raw moment of the random variable X is E[X^k], -shape2 < k < shape1.

The kth limited moment at some limit d is E[min(X, d)^k], k > -shape2 and shape1 - k not a negative integer.

### Value

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

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

### Note

`levgenpareto` computes the limited expected value using `betaint`.

Distribution also known as the Beta of the Second Kind. See also Kleiber and Kotz (2003) for alternative names and parametrizations.

The Generalized Pareto distribution defined here is different from the one in Embrechts et al. (1997) and in Wikipedia; see also Kleiber and Kotz (2003, section 3.12). One may most likely compute quantities for the latter using functions for the Pareto distribution with the appropriate change of parametrization.

The `"distributions"` package vignette provides the interrelations between the continuous size distributions in actuar and the complete formulas underlying the above functions.

### Author(s)

Vincent Goulet vincent.goulet@act.ulaval.ca and Mathieu Pigeon

### References

Embrechts, P., Klüppelberg, 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

    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)

[Package actuar version 3.0-0 Index]