TransformedGamma: 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)

Value

dtrgamma gives the density,

ptrgamma gives the distribution function,

qtrgamma gives the quantile function,

rtrgamma generates random deviates,

mtrgamma gives the $k$th raw moment, and

levtrgamma 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 vincent.goulet@act.ulaval.ca and Mathieu Pigeon

Details

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

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

Run this code

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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