actuar (version 3.0-0)

Loggamma: 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.

Value

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

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

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 \(k\)th raw moment of the random variable \(X\) is \(E[X^k]\) and the \(k\)th limited moment at some limit \(d\) is \(E[\min(X, d)^k]\), \(k < \lambda\).

References

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

Examples

# NOT RUN {
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)
# }