R: The Generalized Beta Distribution

GeneralizedBeta {actuar} | R Documentation

## The Generalized Beta Distribution

### Description

Density function, distribution function, quantile function, random generation, raw moments and limited moments for the Generalized Beta distribution with parameters `shape1`, `shape2`, `shape3` and `scale`.

### Usage

    dgenbeta(x, shape1, shape2, shape3, rate = 1, scale = 1/rate,
             log = FALSE)
    pgenbeta(q, shape1, shape2, shape3, rate = 1, scale = 1/rate,
             lower.tail = TRUE, log.p = FALSE)
    qgenbeta(p, shape1, shape2, shape3, rate = 1, scale = 1/rate,
             lower.tail = TRUE, log.p = FALSE)
    rgenbeta(n, shape1, shape2, shape3, rate = 1, scale = 1/rate)
    mgenbeta(order, shape1, shape2, shape3, rate = 1, scale = 1/rate)
    levgenbeta(limit, shape1, shape2, shape3, 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, shape3, 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 beta distribution with parameters `shape1` = a, `shape2` = b, `shape3` = c and `scale` = s, has density:

f(x) = Gamma(a + b)/(Gamma(a) * Gamma(b)) (c (x/s)^(ac) [1 - (x/s)^c]^(b - 1))/x

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

The generalized beta is the distribution of the random variable

s X^(1/c),

where X has a beta distribution with parameters a and b.

The kth raw moment of the random variable X is E[X^k] and the kth limited moment at some limit d is E[min(X, d)], k > -shape1 * shape3.

### Value

`dgenbeta` gives the density, `pgenbeta` gives the distribution function, `qgenbeta` gives the quantile function, `rgenbeta` generates random deviates, `mgenbeta` gives the kth raw moment, and `levgenbeta` gives the kth moment of the limited loss variable.

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

### Note

This is not the generalized three-parameter beta distribution defined on page 251 of Johnson et al, 1995.

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

### References

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, Volume 2, Wiley.

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

### Examples

    exp(dgenbeta(2, 2, 3, 4, 0.2, log = TRUE))
    p <- (1:10)/10
    pgenbeta(qgenbeta(p, 2, 3, 4, 0.2), 2, 3, 4, 0.2)
    mgenbeta(2, 1, 2, 3, 0.25) - mgenbeta(1, 1, 2, 3, 0.25) ^ 2
    levgenbeta(10, 1, 2, 3, 0.25, order = 2)

[Package actuar version 3.0-0 Index]