TransformedGamma {actuar}R Documentation

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 pp are returned as log(p)\log(p).

lower.tail

logical; if TRUE (default), probabilities are P[Xx]P[X \le x], otherwise, P[X>x]P[X > x].

order

order of the moment.

limit

limit of the loss variable.

Details

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

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

for x>0x > 0, α>0\alpha > 0, τ>0\tau > 0 and θ>0\theta > 0. (Here Γ(α)\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 θX1/τ,\theta X^{1/\tau}, where XX has a gamma distribution with shape parameter α\alpha and scale parameter 11 or, equivalently, of the random variable Y1/τY^{1/\tau} with YY a gamma distribution with shape parameter α\alpha and scale parameter θτ\theta^\tau.

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

The kkth raw moment of the random variable XX is E[Xk]E[X^k] and the kkth limited moment at some limit dd is E[min(X,d)k]E[\min(X, d)^k], k>ατk > -\alpha\tau.

Value

dtrgamma gives the density, ptrgamma gives the distribution function, qtrgamma gives the quantile function, rtrgamma generates random deviates, mtrgamma gives the kkth raw moment, and levtrgamma gives the kkth 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 vincent.goulet@act.ulaval.ca 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)

[Package actuar version 3.3-4 Index]