Three-parameter Burr Type XII distribution with scale
\(s\), first shape \(\zeta\) (shape1), and
second shape \(\theta\) (shape2, the tail index). The
distribution is supported on \(x > 0\) and is widely used for
modelling heavy-tailed positive variables. L-moments are computed via
closed-form expressions involving Beta-function PWMs; a two-step
seeding procedure initialises the shapes from a pre-computed lookup
table (Burr_InitValues) and derives the scale analytically.
Fitting uses L-BFGS-B minimisation of the normalised L-moment error.
The probability density function is: $$f(x) = \zeta s^{-\zeta} x^{\zeta-1} \left(\zeta\theta\left(\frac{x}{s}\right)^\zeta + 1\right)^{-1/(\zeta\theta) - 1}, \quad x > 0$$ where:
\(s\) — scale parameter (
scale)\(\zeta\) — first shape parameter (
shape1), controls lower-tail behaviour\(\theta\) — second shape parameter (
shape2), tail index; the mean exists only when \(\theta < 1\)
Usage
dburr(x, scale, shape1, shape2, PW = 1)
pburr(q, scale, shape1, shape2, PW = 1)
qburr(p, scale, shape1, shape2, PW = 1)
rburr(n, scale, shape1, shape2, PW = 1)