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 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 Transformed Gamma distribution with parameters shape1 = a, shape2 = b and scale = s has density:

f(x) = b u^a exp(-u) / (x Gamma(a)), u = (x/s)^b

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

The Transformed Gamma is the distribution of the random variable s X^(1/b), where X has a Gamma distribution with shape parameter a and scale parameter 1 or, equivalently, of the random variable Y^(1/b) with Y a Gamma distribution with shape parameter a and scale parameter s^b.

The Transformed Gamma probability distribution defines a family of distributions with the following special cases:

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].

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.

Author(s)

Vincent Goulet vincent.goulet@act.ulaval.ca and Mathieu Pigeon

References

Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2004), Loss Models, From Data to Decisions, Second 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 1.0-2 Index]