R: The Pareto III Distribution

Pareto3 {actuar} | R Documentation

## The Pareto III Distribution

### Description

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

### Usage

    dpareto3(x, min, shape, rate = 1, scale = 1/rate,
             log = FALSE)
    ppareto3(q, min, shape, rate = 1, scale = 1/rate,
             lower.tail = TRUE, log.p = FALSE)
    qpareto3(p, min, shape, rate = 1, scale = 1/rate,
             lower.tail = TRUE, log.p = FALSE)
    rpareto3(n, min, shape, rate = 1, scale = 1/rate)
    mpareto3(order, min, shape, rate = 1, scale = 1/rate)
    levpareto3(limit, min, shape, rate = 1, scale = 1/rate,
               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.
`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 `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 III (or “type III”) distribution with parameters `min` = m, `shape` = b and `scale` = s has density:

f(x) = (b ((x - m)/s)^(b - 1))/(s [1 + ((x - m)/s)^b]^2)

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

The Pareto III is the distribution of the random variable

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

where X has a uniform distribution on (0, 1). It derives from the Feller-Pareto distribution with shape1 = shape3 = 1. Setting min = 0 yields the loglogistic distribution.

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

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

### Value

`dpareto3` gives the density, `ppareto3` gives the distribution function, `qpareto3` gives the quantile function, `rpareto3` generates random deviates, `mpareto3` gives the kth raw moment, and `levpareto3` gives the kth moment of the limited loss variable.

Invalid arguments will result in return value `NaN`, with a warning.

### Note

`levpareto3` 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.

### Author(s)

Vincent Goulet vincent.goulet@act.ulaval.ca

### References

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.

### See Also

`dllogis` for the loglogistic distribution.

### Examples

    exp(dpareto3(1, min = 10, 3, 4, log = TRUE))
    p <- (1:10)/10
    ppareto3(qpareto3(p, min = 10, 2, 3), min = 10, 2, 3)

    ## mean
    mpareto3(1, min = 10, 2, 3)

    ## case with 1 - order/shape > 0
    levpareto3(20, min = 10, 2, 3, order = 1)

    ## case with 1 - order/shape < 0
    levpareto3(20, min = 10, 2/3, 3, order = 1)

[Package actuar version 3.0-0 Index]