Loggamma: The Loggamma Distribution

In actuar: Actuarial Functions and Heavy Tailed Distributions

The Loggamma Distribution

Description

Density function, distribution function, quantile function, random generation, raw moments and limited moments for the Loggamma distribution with parameters shapelog and ratelog.

Usage

dlgamma(x, shapelog, ratelog, log = FALSE)
plgamma(q, shapelog, ratelog, lower.tail = TRUE, log.p = FALSE)
qlgamma(p, shapelog, ratelog, lower.tail = TRUE, log.p = FALSE)
rlgamma(n, shapelog, ratelog)
mlgamma(order, shapelog, ratelog)
levlgamma(limit, shapelog, ratelog, 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.
shapelog, ratelog	parameters. Must be strictly positive.
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.

Details

The loggamma distribution with parameters shapelog = \alpha and ratelog = \lambda has density:

f(x) = \frac{\lambda^\alpha}{\Gamma(\alpha)}% \frac{(\log x)^{\alpha - 1}}{x^{\lambda + 1}}

for x > 1, \alpha > 0 and \lambda > 0. (Here \Gamma(\alpha) is the function implemented by R's gamma() and defined in its help.)

The loggamma is the distribution of the random variable e^X, where X has a gamma distribution with shape parameter alpha and scale parameter 1/\lambda.

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], k < \lambda.

Value

dlgamma gives the density, plgamma gives the distribution function, qlgamma gives the quantile function, rlgamma generates random deviates, mlgamma gives the kth raw moment, and levlgamma gives the kth moment of the limited loss variable.

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

Note

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 [e-mail] and Mathieu Pigeon

References

Hogg, R. V. and Klugman, S. A. (1984), Loss Distributions, Wiley.

Examples

exp(dlgamma(2, 3, 4, log = TRUE))
p <- (1:10)/10
plgamma(qlgamma(p, 2, 3), 2, 3)
mlgamma(2, 3, 4) - mlgamma(1, 3, 4)^2
levlgamma(10, 3, 4, order = 2)

actuar documentation built on Nov. 8, 2023, 9:06 a.m.