actuar: GeneralizedBeta – R documentation

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GeneralizedBeta

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)

actuar

Actuarial Functions and Heavy Tailed Distributions

v3.1-2

GPL (>= 2)

Authors

Vincent Goulet [cre, aut], Sébastien Auclair [ctb], Christophe Dutang [aut], Nicholas Langevin [ctb], Xavier Milhaud [ctb], Tommy Ouellet [ctb], Alexandre Parent [ctb], Mathieu Pigeon [aut], Louis-Philippe Pouliot [ctb], Jeffrey A. Ryan [aut] (Package API), Robert Gentleman [aut] (Parts of the R to C interface), Ross Ihaka [aut] (Parts of the R to C interface), R Core Team [aut] (Parts of the R to C interface), R Foundation [aut] (Parts of the R to C interface)

Initial release

2021-03-30