GeneralizedBeta {actuar} | R Documentation |

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

, `shape2`

, `shape3`

and
`scale`

.

dgenbeta(x, shape1, shape2, shape3, rate = 1, scale = 1/rate, log = FALSE) pgenbeta(q, shape1, shape2, shape3, rate = 1, scale = 1/rate, lower.tail = TRUE, log.p = FALSE) qgenbeta(p, shape1, shape2, shape3, rate = 1, scale = 1/rate, lower.tail = TRUE, log.p = FALSE) rgenbeta(n, shape1, shape2, shape3, rate = 1, scale = 1/rate) mgenbeta(order, shape1, shape2, shape3, rate = 1, scale = 1/rate) levgenbeta(limit, shape1, shape2, shape3, rate = 1, scale = 1/rate, order = 1)

`x, q` |
vector of quantiles. |

`p` |
vector of probabilities. |

`n` |
number of observations. If |

`shape1, shape2, shape3, 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 generalized beta distribution with parameters `shape1`

*= a*, `shape2`

*= b*, `shape3`

*= c* and `scale`

*= s*, has
density:

*
f(x) = Gamma(a + b)/(Gamma(a) * Gamma(b))
(c (x/s)^(ac) [1 - (x/s)^c]^(b - 1))/x*

for *0 < x < s*, *a > 0*,
*b > 0*, *c > 0* and *s >
0*. (Here *Gamma(a)* is the function implemented
by **R**'s `gamma()`

and defined in its help.)

The generalized beta is the distribution of the random variable

*s X^(1/c),*

where *X* has a beta distribution with parameters *a*
and *b*.

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
> -shape1 * shape3*.

`dgenbeta`

gives the density,
`pgenbeta`

gives the distribution function,
`qgenbeta`

gives the quantile function,
`rgenbeta`

generates random deviates,
`mgenbeta`

gives the *k*th raw moment, and
`levgenbeta`

gives the *k*th moment of the limited loss
variable.

Invalid arguments will result in return value `NaN`

, with a warning.

This is *not* the generalized three-parameter beta distribution
defined on page 251 of Johnson et al, 1995.

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

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995)
*Continuous Univariate Distributions, Volume 2*, Wiley.

Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2012),
*Loss Models, From Data to Decisions, Fourth Edition*, Wiley.

exp(dgenbeta(2, 2, 3, 4, 0.2, log = TRUE)) p <- (1:10)/10 pgenbeta(qgenbeta(p, 2, 3, 4, 0.2), 2, 3, 4, 0.2) mgenbeta(2, 1, 2, 3, 0.25) - mgenbeta(1, 1, 2, 3, 0.25) ^ 2 levgenbeta(10, 1, 2, 3, 0.25, order = 2)

[Package *actuar* version 3.1-4 Index]