TransformedGamma: The Transformed Gamma Distribution

In actuar: Actuarial Functions and Heavy Tailed Distributions

The Transformed Gamma Distribution

Description

Density function, distribution function, quantile function, random generation, raw moments and limited moments for the Transformed Gamma distribution with parameters shape1, shape2 and scale.

Usage

dtrgamma(x, shape1, shape2, rate = 1, scale = 1/rate,
         log = FALSE)
ptrgamma(q, shape1, shape2, rate = 1, scale = 1/rate,
         lower.tail = TRUE, log.p = FALSE)
qtrgamma(p, shape1, shape2, rate = 1, scale = 1/rate,
         lower.tail = TRUE, log.p = FALSE)
rtrgamma(n, shape1, shape2, rate = 1, scale = 1/rate)
mtrgamma(order, shape1, shape2, rate = 1, scale = 1/rate)
levtrgamma(limit, shape1, shape2, rate = 1, scale = 1/rate,
           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.
shape1, shape2, scale	parameters. Must be strictly positive.
rate	an alternative way to specify the scale.
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.

Details

The transformed gamma distribution with parameters shape1 = \alpha, shape2 = \tau and scale = \theta has density:

f(x) = \frac{\tau u^\alpha e^{-u}}{x \Gamma(\alpha)}, % \quad u = (x/\theta)^\tau

for x > 0, \alpha > 0, \tau > 0 and \theta > 0. (Here \Gamma(\alpha) is the function implemented by R's gamma() and defined in its help.)

The transformed gamma is the distribution of the random variable \theta X^{1/\tau}, where X has a gamma distribution with shape parameter \alpha and scale parameter 1 or, equivalently, of the random variable Y^{1/\tau} with Y a gamma distribution with shape parameter \alpha and scale parameter \theta^\tau.

The transformed gamma probability distribution defines a family of distributions with the following special cases:

• A Gamma distribution when shape2 == 1;

• A Weibull distribution when shape1 == 1;

• An Exponential distribution when shape2 == shape1 == 1.

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 > -\alpha\tau.

Value

dtrgamma gives the density, ptrgamma gives the distribution function, qtrgamma gives the quantile function, rtrgamma generates random deviates, mtrgamma gives the kth raw moment, and levtrgamma gives the kth moment of the limited loss variable.

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

Note

Distribution also known as the Generalized Gamma. See also Kleiber and Kotz (2003) for alternative names and parametrizations.

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 [e-mail] and Mathieu Pigeon

References

Kleiber, C. and Kotz, S. (2003), Statistical Size Distributions in Economics and Actuarial Sciences, Wiley.

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

Examples

exp(dtrgamma(2, 3, 4, 5, log = TRUE))
p <- (1:10)/10
ptrgamma(qtrgamma(p, 2, 3, 4), 2, 3, 4)
mtrgamma(2, 3, 4, 5) - mtrgamma(1, 3, 4, 5) ^ 2
levtrgamma(10, 3, 4, 5, order = 2)

actuar documentation built on Aug. 9, 2025, 1:06 a.m.