actuar (version 3.3-4)

GammaSupp: Moments and Moment Generating Function of the Gamma Distribution

Description

Raw moments, limited moments and moment generating function for the Gamma distribution with parameters shape and scale.

Usage

mgamma(order, shape, rate = 1, scale = 1/rate)
levgamma(limit, shape, rate = 1, scale = 1/rate, order = 1)
mgfgamma(t, shape, rate = 1, scale = 1/rate, log = FALSE)

Value

mgamma gives the \(k\)th raw moment,

levgamma gives the \(k\)th moment of the limited loss variable, and

mgfgamma gives the moment generating function in t.

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

Arguments

order

order of the moment.

limit

limit of the loss variable.

rate

an alternative way to specify the scale.

shape, scale

shape and scale parameters. Must be strictly positive.

t

numeric vector.

log

logical; if TRUE, the cumulant generating function is returned.

Author

Vincent Goulet , Christophe Dutang and Mathieu Pigeon

Details

The \(k\)th raw moment of the random variable \(X\) is \(E[X^k]\), the \(k\)th limited moment at some limit \(d\) is \(E[\min(X, d)^k]\) and the moment generating function is \(E[e^{tX}]\), \(k > -\alpha\).

References

Johnson, N. L. and Kotz, S. (1970), Continuous Univariate Distributions, Volume 1, Wiley.

Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2012), Loss Models, From Data to Decisions, Fourth Edition, Wiley.

See Also

GammaDist

Examples

mgamma(2, 3, 4) - mgamma(1, 3, 4)^2
levgamma(10, 3, 4, order = 2)
mgfgamma(1,3,2)