distributional (version 0.6.0)

dist_pareto: The Pareto Distribution

Description

The Pareto distribution is a power-law probability distribution commonly used in actuarial science to model loss severity and in economics to model income distributions and firm sizes.

Usage

dist_pareto(shape, scale)

Arguments

shape, scale

parameters. Must be strictly positive.

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_pareto.html

In the following, let \(X\) be a Pareto random variable with parameters shape = \(\alpha\) and scale = \(\theta\).

Support: \((0, \infty)\)

Mean: \(\frac{\theta}{\alpha - 1}\) for \(\alpha > 1\), undefined otherwise

Variance: \(\frac{\alpha\theta^2}{(\alpha - 1)^2(\alpha - 2)}\) for \(\alpha > 2\), undefined otherwise

Probability density function (p.d.f):

$$ f(x) = \frac{\alpha\theta^\alpha}{(x + \theta)^{\alpha + 1}} $$

for \(x > 0\), \(\alpha > 0\) and \(\theta > 0\).

Cumulative distribution function (c.d.f):

$$ F(x) = 1 - \left(\frac{\theta}{x + \theta}\right)^\alpha $$

for \(x > 0\).

Moment generating function (m.g.f):

Does not exist in closed form, but the \(k\)th raw moment \(E[X^k]\) exists for \(-1 < k < \alpha\).

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

actuar::Pareto

Examples

dist <- dist_pareto(shape = c(10, 3, 2, 1), scale = rep(1, 4))
dist

if (FALSE) { # requireNamespace("actuar", quietly = TRUE)
mean(dist)
variance(dist)
support(dist)
generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)
}