distributional (version 0.6.0)

dist_burr: The Burr distribution

Description

The Burr distribution (Type XII) is a flexible continuous probability distribution often used for modeling income distributions, reliability data, and failure times.

Usage

dist_burr(shape1, shape2, rate = 1, scale = 1/rate)

Arguments

shape1, shape2, scale

parameters. Must be strictly positive.

rate

an alternative way to specify the scale.

Details

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

In the following, let \(X\) be a Burr random variable with parameters shape1 = \(\alpha\), shape2 = \(\gamma\), and rate = \(\lambda\).

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

Mean: \(\frac{\lambda^{-1/\alpha} \gamma B(\gamma - 1/\alpha, 1 + 1/\alpha)}{\gamma}\) (for \(\alpha \gamma > 1\))

Variance: \(\frac{\lambda^{-2/\alpha} \gamma B(\gamma - 2/\alpha, 1 + 2/\alpha)}{\gamma} - \mu^2\) (for \(\alpha \gamma > 2\))

Probability density function (p.d.f):

$$ f(x) = \alpha \gamma \lambda x^{\alpha - 1} (1 + \lambda x^\alpha)^{-\gamma - 1} $$

Cumulative distribution function (c.d.f):

$$ F(x) = 1 - (1 + \lambda x^\alpha)^{-\gamma} $$

Quantile function:

$$ F^{-1}(p) = \lambda^{-1/\alpha} ((1 - p)^{-1/\gamma} - 1)^{1/\alpha} $$

Moment generating function (m.g.f):

Does not exist in closed form.

See Also

actuar::Burr

Examples

dist <- dist_burr(shape1 = c(1,1,1,2,3,0.5), shape2 = c(1,2,3,1,1,2))
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)
}