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)

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.

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.

Author

Vincent Goulet vincent.goulet@act.ulaval.ca and Mathieu Pigeon

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

Run this code

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)

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