The Inverse Transformed Gamma Distribution
Density function, distribution function, quantile function, random generation,
raw moments, and limited moments for the Inverse Transformed Gamma
distribution with parameters shape1, shape2 and
scale.
dinvtrgamma(x, shape1, shape2, rate = 1, scale = 1/rate,
log = FALSE)
pinvtrgamma(q, shape1, shape2, rate = 1, scale = 1/rate,
lower.tail = TRUE, log.p = FALSE)
qinvtrgamma(p, shape1, shape2, rate = 1, scale = 1/rate,
lower.tail = TRUE, log.p = FALSE)
rinvtrgamma(n, shape1, shape2, rate = 1, scale = 1/rate)
minvtrgamma(order, shape1, shape2, rate = 1, scale = 1/rate)
levinvtrgamma(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 inverse 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 = (s/x)^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 inverse 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 inverse transformed gamma distribution defines a family of distributions with the following special cases:
An Inverse Gamma distribution when
shape2 == 1;
An Inverse Weibull distribution when
shape1 == 1;
An Inverse Exponential distribution when
shape1 == shape2 == 1;
The kth raw moment of the random variable X is E[X^k], k < shape1 * shape2, and the kth limited moment at some limit d is E[min(X, d)^k] for all k.
dinvtrgamma gives the density,
pinvtrgamma gives the distribution function,
qinvtrgamma gives the quantile function,
rinvtrgamma generates random deviates,
minvtrgamma gives the kth raw moment, and
levinvtrgamma gives the kth moment of the limited loss
variable.
Invalid arguments will result in return value NaN, with a warning.
levinvtrgamma computes the limited expected value using
gammainc from package expint.
Distribution also known as the Inverse 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(dinvtrgamma(2, 3, 4, 5, log = TRUE)) p <- (1:10)/10 pinvtrgamma(qinvtrgamma(p, 2, 3, 4), 2, 3, 4) minvtrgamma(2, 3, 4, 5) levinvtrgamma(200, 3, 4, 5, order = 2)
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