R: The Pareto Distribution

Pareto {actuar} | R Documentation

## The Pareto Distribution

### 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:

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

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

### Value

`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.

### Note

`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.

### Author(s)

Vincent Goulet vincent.goulet@act.ulaval.ca and Mathieu Pigeon

### References

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.

### See Also

`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.

### Examples

    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 3.0-0 Index]