R: The Loggamma Distribution

Loggamma {actuar} | R Documentation

## 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 <= x], otherwise, P[X > x].
`order` | order of the moment.
`limit` | limit of the loss variable.

### Details

The loggamma distribution with parameters `shapelog` = a and `ratelog` = b has density:

f(x) = (b^a (log(x))^(a - 1))/(Gamma(a) * x^(b + 1))

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

The loggamma is the distribution of the random variable exp(X), where X has a gamma distribution with shape parameter a and scale parameter 1/b.

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 < ratelog.

### 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 vincent.goulet@act.ulaval.ca 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)

[Package actuar version 3.0-0 Index]