R: The Pareto IV Distribution

Pareto4 {actuar} | R Documentation

## The Pareto IV Distribution

### Description

Density function, distribution function, quantile function, random generation, raw moments and limited moments for the Pareto IV distribution with parameters `min`, `shape1`, `shape2` and `scale`.

### Usage

    dpareto4(x, min, shape1, shape2, rate = 1, scale = 1/rate,
             log = FALSE)
    ppareto4(q, min, shape1, shape2, rate = 1, scale = 1/rate,
             lower.tail = TRUE, log.p = FALSE)
    qpareto4(p, min, shape1, shape2, rate = 1, scale = 1/rate,
             lower.tail = TRUE, log.p = FALSE)
    rpareto4(n, min, shape1, shape2, rate = 1, scale = 1/rate)
    mpareto4(order, min, shape1, shape2, rate = 1, scale = 1/rate)
    levpareto4(limit, min, shape1, shape2, 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.
`shape1, shape2, 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 IV (or “type IV”) distribution with parameters `min` = m, `shape1` = a, `shape2` = b and `scale` = s has density:

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

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

The Pareto IV is the distribution of the random variable

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

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

The Pareto IV distribution also has the following direct special cases:

- A Pareto III distribution when `shape1 == 1`;

- A Pareto II distribution when `shape1 == 1`.

The kth raw moment of the random variable X is E[X^k] for nonnegative integer values of k < shape1 * shape2.

The kth limited moment at some limit d is E[min(X, d)^k] for nonnegative integer values of k and shape1 - j/shape2, j = 1, …, k not a negative integer.

### Value

`dpareto4` gives the density, `ppareto4` gives the distribution function, `qpareto4` gives the quantile function, `rpareto4` generates random deviates, `mpareto4` gives the kth raw moment, and `levpareto4` gives the kth moment of the limited loss variable.

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

### Note

`levpareto4` 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

`dburr` for the Burr distribution.

### Examples

    exp(dpareto4(1, min = 10, 2, 3, log = TRUE))
    p <- (1:10)/10
    ppareto4(qpareto4(p, min = 10, 2, 3, 2), min = 10, 2, 3, 2)

    ## variance
    mpareto4(2, min = 10, 2, 3, 1) - mpareto4(1, min = 10, 2, 3, 1) ^ 2

    ## case with shape1 - order/shape2 > 0
    levpareto4(10, min = 10, 2, 3, 1, order = 2)

    ## case with shape1 - order/shape2 < 0
    levpareto4(10, min = 10, 1.5, 0.5, 1, order = 2)

[Package actuar version 3.1-2 Index]