The Transformed Gamma Distribution
Density function, distribution function, quantile function, random generation,
raw moments and limited moments for the Transformed Gamma distribution
with parameters shape1, shape2 and scale.
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)x, q |
vector of quantiles. |
p |
vector of probabilities. |
n |
number of observations. If |
shape1, shape2, scale |
parameters. Must be strictly positive. |
rate |
an alternative way to specify the scale. |
log, log.p |
logical; if |
lower.tail |
logical; if |
order |
order of the moment. |
limit |
limit of the loss variable. |
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:
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 > -shape1 * shape2.
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.
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.
Vincent Goulet vincent.goulet@act.ulaval.ca and Mathieu Pigeon
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.
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)
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