Pareto4 {actuar} | R Documentation |

Density function, distribution function, quantile function, random generation,
raw moments and limited moments for the Pareto IV distribution with
parameters `min`

, `shape1`

, `shape2`

and `scale`

.

dpareto4(x, min, shape1, shape2, rate = 1, scale = 1/rate, log = FALSE) ppareto4(q, min, shape1, shape2, rate = 1, scale = 1/rate, lower.tail = TRUE, log.p = FALSE) qpareto4(p, min, shape1, shape2, rate = 1, scale = 1/rate, lower.tail = TRUE, log.p = FALSE) rpareto4(n, min, shape1, shape2, rate = 1, scale = 1/rate) mpareto4(order, min, shape1, shape2, rate = 1, scale = 1/rate) levpareto4(limit, min, shape1, shape2, rate = 1, scale = 1/rate, order = 1)

`x, q` |
vector of quantiles. |

`p` |
vector of probabilities. |

`n` |
number of observations. If |

`min` |
lower bound of the support of the distribution. |

`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 Pareto IV (or “type IV”) distribution with parameters
`min`

*= m*,
`shape1`

*= a*,
`shape2`

*= b* and
`scale`

*= s* has density:

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

for *x > m*, *-Inf < m < Inf*,
*a > 0*, *b > 0*
and *s > 0*.

The Pareto IV is the distribution of the random variable

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

where *X* has a beta distribution with parameters *1* and
*a*. It derives from the Feller-Pareto
distribution with *shape3 = 1*. Setting *min
= 0* yields the Burr distribution.

The Pareto IV distribution also has the following direct special cases:

A Pareto III distribution when

`shape1 == 1`

;A Pareto II distribution when

`shape1 == 1`

.

The *k*th raw moment of the random variable *X* is
*E[X^k]* for nonnegative integer values of *k < shape1 * shape2*.

The *k*th limited moment at some limit *d* is *E[min(X, d)^k]* for nonnegative integer values of *k*
and *shape1 - j/shape2*, *j = 1, …, k*
not a negative integer.

`dpareto4`

gives the density,
`ppareto4`

gives the distribution function,
`qpareto4`

gives the quantile function,
`rpareto4`

generates random deviates,
`mpareto4`

gives the *k*th raw moment, and
`levpareto4`

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

Invalid arguments will result in return value `NaN`

, with a warning.

`levpareto4`

computes the limited expected value using
`betaint`

.

For Pareto distributions, we use the classification of Arnold (2015) with the parametrization of Klugman et al. (2012).

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

Arnold, B.C. (2015), *Pareto Distributions*, Second Edition, CRC
Press.

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.

`dburr`

for the Burr distribution.

exp(dpareto4(1, min = 10, 2, 3, log = TRUE)) p <- (1:10)/10 ppareto4(qpareto4(p, min = 10, 2, 3, 2), min = 10, 2, 3, 2) ## variance mpareto4(2, min = 10, 2, 3, 1) - mpareto4(1, min = 10, 2, 3, 1) ^ 2 ## case with shape1 - order/shape2 > 0 levpareto4(10, min = 10, 2, 3, 1, order = 2) ## case with shape1 - order/shape2 < 0 levpareto4(10, min = 10, 1.5, 0.5, 1, order = 2)

[Package *actuar* version 3.1-4 Index]