actuar (version 0.9-3)

InverseTransformedGamma: The Inverse Transformed Gamma Distribution

Description

Density, 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)

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.

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.

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:

• AnInverse Gammadistribution whenshape2 == 1;
• AnInverse Weibulldistribution whenshape1 == 1;
• AnInverse Exponentialdistribution whenshape1 == shape2 == 1;

The $k$th raw moment of the random variable $X$ is $E[X^k]$ and the $k$ limited moment at some limit $d$ is $E[\min(X, d)]$.

References

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