Pareto {actuar} | R Documentation |

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

and `scale`

.

dpareto(x, shape, scale, log = FALSE) ppareto(q, shape, scale, lower.tail = TRUE, log.p = FALSE) qpareto(p, shape, scale, lower.tail = TRUE, log.p = FALSE) rpareto(n, shape, scale) mpareto(order, shape, scale) levpareto(limit, shape, scale, order = 1)

`x, q` |
vector of quantiles. |

`p` |
vector of probabilities. |

`n` |
number of observations. If |

`shape, scale` |
parameters. Must be strictly positive. |

`log, log.p` |
logical; if |

`lower.tail` |
logical; if |

`order` |
order of the moment. |

`limit` |
limit of the loss variable. |

The Pareto distribution with parameters `shape`

*= a* and `scale`

*= s* has density:

*
f(x) = a s^a / (x + s)^(a + 1)*

for *x > 0*, *a > 0* and *s > 0*.

There are many different definitions of the Pareto distribution in the literature; see Arnold (2015) or Kleiber and Kotz (2003). In the nomenclature of actuar, The “Pareto distribution” does not have a location parameter. The version with a location parameter is the Pareto II.

The *k*th raw moment of the random variable *X* is
*E[X^k]*, *-1 < k < shape*.

The *k*th limited moment at some limit
*d* is *E[min(X, d)^k]*,
*k > -1* and *shape - k* not a
negative integer.

`dpareto`

gives the density,
`ppareto`

gives the distribution function,
`qpareto`

gives the quantile function,
`rpareto`

generates random deviates,
`mpareto`

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

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

Invalid arguments will result in return value `NaN`

, with a warning.

`levpareto`

computes the limited expected value using
`betaint`

.

The version of the Pareto defined for *x > s* is named
Single Parameter Pareto, or Pareto I, in actuar.

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.

`dpareto2`

for an equivalent distribution with location
parameter.

`dpareto1`

for the Single Parameter Pareto distribution.

`"distributions"`

package vignette for details on the
interrelations between the continuous size distributions in
actuar and complete formulas underlying the above functions.

exp(dpareto(2, 3, 4, log = TRUE)) p <- (1:10)/10 ppareto(qpareto(p, 2, 3), 2, 3) ## variance mpareto(2, 4, 1) - mpareto(1, 4, 1)^2 ## case with shape - order > 0 levpareto(10, 3, scale = 1, order = 2) ## case with shape - order < 0 levpareto(10, 1.5, scale = 1, order = 2)

[Package *actuar* version 3.1-4 Index]