R: The Single-parameter Pareto Distribution

SingleParameterPareto {actuar} | R Documentation

## The Single-parameter Pareto Distribution

### Description

Density function, distribution function, quantile function, random generation, raw moments, and limited moments for the Single-parameter Pareto distribution with parameter `shape`.

### Usage

    dpareto1(x, shape, min, log = FALSE)
    ppareto1(q, shape, min, lower.tail = TRUE, log.p = FALSE)
    qpareto1(p, shape, min, lower.tail = TRUE, log.p = FALSE)
    rpareto1(n, shape, min)
    mpareto1(order, shape, min)
    levpareto1(limit, shape, min, 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` | parameter. Must be strictly positive.
`min` | lower bound of the support of the distribution.
`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 single-parameter Pareto, or Pareto I, distribution with parameter `shape` = a has density:

f(x) = a b^a/x^(a + 1)

for x > b, a > 0 and b > 0.

Although there appears to be two parameters, only `shape` is a true parameter. The value of `min` = b must be set in advance.

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

### Value

`dpareto1` gives the density, `ppareto1` gives the distribution function, `qpareto1` gives the quantile function, `rpareto1` generates random deviates, `mpareto1` gives the kth raw moment, and `levpareto1` gives the kth moment of the limited loss variable.

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

### Note

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 and Mathieu Pigeon

### References

Arnold, B.C. (2015), Pareto Distributions, Second Edition, CRC Press.

Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2012), Loss Models, From Data to Decisions, Fourth Edition, Wiley.

### See Also

`dpareto` for the two-parameter Pareto distribution.

### Examples

    exp(dpareto1(5, 3, 4, log = TRUE))
    p <- (1:10)/10
    ppareto1(qpareto1(p, 2, 3), 2, 3)
    mpareto1(2, 3, 4) - mpareto(1, 3, 4) ^ 2
    levpareto(10, 3, 4, order = 2)

[Package actuar version 3.0-0 Index]