actuar (version 3.3-4)

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)

Value

dgenbeta gives the density,

pgenbeta gives the distribution function,

qgenbeta gives the quantile function,

rgenbeta generates random deviates,

mgenbeta gives the \(k\)th raw moment, and

levgenbeta gives the \(k\)th moment of the limited loss variable.

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

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 \(P[X \le x]\), otherwise, \(P[X > x]\).

order

order of the moment.

limit

limit of the loss variable.

Author

Vincent Goulet

Details

The generalized beta distribution with parameters shape1 \(= \alpha\), shape2 \(= \beta\), shape3 \(= \tau\) and scale \(= \theta\), has density: $$f(x) = \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)\Gamma(\beta)} (x/\theta)^{\alpha \tau} (1 - (x/\theta)^\tau)^{\beta - 1} \frac{\tau}{x}$$ for \(0 < x < \theta\), \(\alpha > 0\), \(\beta > 0\), \(\tau > 0\) and \(\theta > 0\). (Here \(\Gamma(\alpha)\) is the function implemented by R's gamma() and defined in its help.)

The generalized beta is the distribution of the random variable $$\theta X^{1/\tau},$$ where \(X\) has a beta distribution with parameters \(\alpha\) and \(\beta\).

The \(k\)th raw moment of the random variable \(X\) is \(E[X^k]\) and the \(k\)th limited moment at some limit \(d\) is \(E[\min(X, d)]\), \(k > -\alpha\tau\).

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)