TransformedBeta {actuar} | R Documentation |

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

, `shape2`

, `shape3`

and
`scale`

.

dtrbeta(x, shape1, shape2, shape3, rate = 1, scale = 1/rate, log = FALSE) ptrbeta(q, shape1, shape2, shape3, rate = 1, scale = 1/rate, lower.tail = TRUE, log.p = FALSE) qtrbeta(p, shape1, shape2, shape3, rate = 1, scale = 1/rate, lower.tail = TRUE, log.p = FALSE) rtrbeta(n, shape1, shape2, shape3, rate = 1, scale = 1/rate) mtrbeta(order, shape1, shape2, shape3, rate = 1, scale = 1/rate) levtrbeta(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 transformed beta distribution with parameters `shape1`

*= a*, `shape2`

*= b*, `shape3`

*= c* and `scale`

*= s*, has
density:

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

for *x > 0*, *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 transformed beta is the distribution of the random variable

*
s (X/(1 - X))^(1/b),*

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

The transformed beta distribution defines a family of distributions with the following special cases:

A Burr distribution when

`shape3 == 1`

;A loglogistic distribution when

`shape1 == shape3 == 1`

;A paralogistic distribution when

`shape3 == 1`

and`shape2 == shape1`

;A generalized Pareto distribution when

`shape2 == 1`

;A Pareto distribution when

`shape2 == shape3 == 1`

;An inverse Burr distribution when

`shape1 == 1`

;An inverse Pareto distribution when

`shape2 == shape1 == 1`

;An inverse paralogistic distribution when

`shape1 == 1`

and`shape3 == shape2`

.

The *k*th raw moment of the random variable *X* is
*E[X^k]*, *-shape3 * shape2 <
k < shape1 * shape2*.

The *k*th limited moment at some limit *d* is *E[min(X, d)^k]*, *k > -shape3 * shape2*
and *shape1 - k/shape2* not a negative integer.

`dtrbeta`

gives the density,
`ptrbeta`

gives the distribution function,
`qtrbeta`

gives the quantile function,
`rtrbeta`

generates random deviates,
`mtrbeta`

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

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

Invalid arguments will result in return value `NaN`

, with a warning.

`levtrbeta`

computes the limited expected value using
`betaint`

.

Distribution also known as the Generalized Beta of the Second Kind and Pearson Type VI. 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.

`dfpareto`

for an equivalent distribution with a location
parameter.

exp(dtrbeta(2, 2, 3, 4, 5, log = TRUE)) p <- (1:10)/10 ptrbeta(qtrbeta(p, 2, 3, 4, 5), 2, 3, 4, 5) qpearson6(0.3, 2, 3, 4, 5, lower.tail = FALSE) ## variance mtrbeta(2, 2, 3, 4, 5) - mtrbeta(1, 2, 3, 4, 5)^2 ## case with shape1 - order/shape2 > 0 levtrbeta(10, 2, 3, 4, scale = 1, order = 2) ## case with shape1 - order/shape2 < 0 levtrbeta(10, 1/3, 0.75, 4, scale = 0.5, order = 2)

[Package *actuar* version 3.1-4 Index]