TransformedGamma {actuar} | R Documentation |

Density function, distribution function, quantile function, random generation,
raw moments and limited moments for the Transformed Gamma distribution
with parameters `shape1`

, `shape2`

and `scale`

.

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)

`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 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 = (x/s)^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 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 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 > -shape1 * shape2*.

`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.

Distribution also known as the 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(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)

[Package *actuar* version 3.1-4 Index]