Three-parameter Dagum (Burr Type III) distribution with
scale \(s\), first shape \(\alpha\) (shape1), and
second shape \(\theta\) (shape2, the tail index). The
distribution is supported on \(x > 0\) and, unlike the Burr XII,
has a unimodal density with mode at the origin for certain parameter
ranges. L-moments are computed via closed-form Beta-function PWMs;
fitting uses L-BFGS-B minimisation seeded from the
Dagum_InitValues lookup table with scale-free L-ratio matching
and analytic scale derivation.
The probability density function is: $$f(x) = \frac{1}{s} \left(\frac{x}{s}\right)^{-1/\theta - 1} \left(1 + \frac{\theta}{\alpha}\left(\frac{x}{s}\right)^{-1/\theta}\right)^{-\alpha-1}, \quad x > 0$$ where:
\(s\) — scale parameter (
scale)\(\alpha\) — first shape parameter (
shape1), controls upper-tail heaviness\(\theta\) — second shape parameter (
shape2), tail index; the mean exists only when \(\theta < 1\)
Usage
pdagum(q, scale, shape1, shape2, PW = 1)
ddagum(x, scale, shape1, shape2, PW = 1)
qdagum(p, scale, shape1, shape2, PW = 1)
rdagum(n, scale, shape1, shape2, PW = 1)