Pareto2 {actuar} | R Documentation |

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

, `shape`

and `scale`

.

dpareto2(x, min, shape, rate = 1, scale = 1/rate, log = FALSE) ppareto2(q, min, shape, rate = 1, scale = 1/rate, lower.tail = TRUE, log.p = FALSE) qpareto2(p, min, shape, rate = 1, scale = 1/rate, lower.tail = TRUE, log.p = FALSE) rpareto2(n, min, shape, rate = 1, scale = 1/rate) mpareto2(order, min, shape, rate = 1, scale = 1/rate) levpareto2(limit, min, shape, 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. |

`shape, 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 II (or “type II”) distribution with parameters
`min`

*= m*,
`shape`

*= a* and
`scale`

*= s* has density:

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

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

The Pareto II is the distribution of the random variable

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

where *X* has a beta distribution with parameters *1* and
*a*. It derives from the Feller-Pareto
distribution with *shape2 = shape3 = 1*.
Setting *min = 0* yields the familiar
Pareto distribution.

The Pareto I (or Single parameter Pareto)
distribution is a special case of the Pareto II with ```
min ==
scale
```

.

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

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

`dpareto2`

gives the density,
`ppareto2`

gives the distribution function,
`qpareto2`

gives the quantile function,
`rpareto2`

generates random deviates,
`mpareto2`

gives the *k*th raw moment, and
`levpareto2`

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

Invalid arguments will result in return value `NaN`

, with a warning.

`levpareto2`

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.

`dpareto`

for the Pareto distribution without a location
parameter.

exp(dpareto2(1, min = 10, 3, 4, log = TRUE)) p <- (1:10)/10 ppareto2(qpareto2(p, min = 10, 2, 3), min = 10, 2, 3) ## variance mpareto2(2, min = 10, 4, 1) - mpareto2(1, min = 10, 4, 1)^2 ## case with shape - order > 0 levpareto2(10, min = 10, 3, scale = 1, order = 2) ## case with shape - order < 0 levpareto2(10, min = 10, 1.5, scale = 1, order = 2)

[Package *actuar* version 3.1-4 Index]