# Pareto {actuar}

### Description

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

and `scale`

.

### Usage

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)

### Arguments

- x, q
- vector of quantiles.
- p
- vector of probabilities.
- n
- number of observations. If
`length(n) > 1`

, the length is taken to be the number required. - shape, scale
- parameters. Must be strictly positive.
- log, log.p
- logical; if
`TRUE`

, probabilities/densities p are returned as log(p). - lower.tail
- logical; if
`TRUE`

(default), probabilities are P[X <= x], otherwise, P[X > x]. - order
- order of the moment.
- limit
- limit of the loss variable.

### Details

The Pareto distribution with parameters `shape`

= a and `scale`

= s has density: for x > 0, a > 0 and s > 0.

The kth raw moment of the random variable X is E[X^k] and the kth limited moment at some limit d is E[min(X, d)^k].

### Values

`dpareto`

gives the density, `ppareto`

gives the distribution function, `qpareto`

gives the quantile function, `rpareto`

generates random deviates, `mpareto`

gives the kth raw moment, and `levpareto`

gives the kth moment of the limited loss variable.

Invalid arguments will result in return value `NaN`

, with a warning.

### References

Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2008), *Loss Models, From Data to Decisions, Third Edition*, Wiley.

### Note

Distribution also known as the Pareto Type II or Lomax distribution.

### Examples

Documentation reproduced from package actuar, version 1.2-1. License: GPL (>= 2)