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Loggamma: The Loggamma Distribution

Description

Density, 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 $k$ limited moment at some limit $d$ is $E[\min(X, d)]$.

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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