actuar (version 3.3-4)

InverseTransformedGamma: The Inverse Transformed Gamma Distribution

Description

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.

Usage

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)

Value

dinvtrgamma gives the density,

pinvtrgamma gives the distribution function,

qinvtrgamma gives the quantile function,

rinvtrgamma generates random deviates,

minvtrgamma gives the \(k\)th raw moment, and

levinvtrgamma gives the \(k\)th moment of the limited loss variable.

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

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.

Author

Vincent Goulet and Mathieu Pigeon

Details

The inverse transformed gamma distribution with parameters shape1 \(= \alpha\), shape2 \(= \tau\) and scale \(= \theta\), has density: $$f(x) = \frac{\tau u^\alpha e^{-u}}{x \Gamma(\alpha)}, % \quad u = (\theta/x)^\tau$$ 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 inverse 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 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 \(k\)th raw moment of the random variable \(X\) is \(E[X^k]\), \(k < \alpha\tau\), and the \(k\)th limited moment at some limit \(d\) is \(E[\min(X, d)^k]\) for all \(k\).

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(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)