InverseTransformedGamma {actuar} | R Documentation |

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 *k*th raw moment of the random variable *X* is
*E[X^k]*, *k < shape1 * shape2*, and
the *k*th 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 *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.

`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)

[Package *actuar* version 3.1-4 Index]