actuar (version 3.3-4)

Pareto: 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)

Value

dpareto gives the density,

ppareto gives the distribution function,

qpareto gives the quantile function,

rpareto generates random deviates,

mpareto gives the \(k\)th raw moment, and

levpareto gives the \(k\)th moment of the limited loss variable.

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

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 \(P[X \le x]\), otherwise, \(P[X > x]\).

order

order of the moment.

limit

limit of the loss variable.

Author

Vincent Goulet and Mathieu Pigeon

Details

The Pareto distribution with parameters shape \(= \alpha\) and scale \(= \theta\) has density: $$f(x) = \frac{\alpha \theta^\alpha}{(x + \theta)^{\alpha + 1}}$$ for \(x > 0\), \(\alpha > 0\) and \(\theta\).

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 \(k\)th raw moment of the random variable \(X\) is \(E[X^k]\), \(-1 < k < \alpha\).

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

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)