freealg.distributions.Wachter

class freealg.distributions.Wachter(a, b)

Wachter distribution.

Parameters:

afloat

Parameter \(a\) of the distribution. See Notes.

bfloat

Parameter \(b\) of the distribution. See Notes.

Notes

The Wachter distribution has the absolutely-continuous density

\[\mathrm{d} \rho(x) = \frac{(a + b) ) \sqrt{(\lambda_{+} - x) (x - \lambda_{-})}}{2 \pi x (1 - x)} \mathbf{1}_{x \in [\lambda_{-}, \lambda_{+}]} \mathrm{d}{x}\]

where \(a, b\) are the shape parameters of the distributon. The edges of the support are

\[\lambda_{\pm} = \left( \frac{\sqrt{b} \pm \sqrt{a (a+b-1)}} {a+b} \right)^2.\]

References

[1]

Wachter, K. W. (1978). The strong limits of random matrix spectra for sample matrices of independent elements. The Annals of Probability, 6(1), 1-18

Examples

>>> from freealg.distributions import Wachter
>>> wa = Wachter(2, 3)

Methods

density([x, plot, latex, save, eig])	Density of distribution.
hilbert([x, plot, latex, save])	Hilbert transform of the distribution.
stieltjes([x, y, plot, on_disk, latex, save])	Stieltjes transform of distribution.
sample(size[, x_min, x_max, method, seed, ...])	Sample from distribution.
matrix(size[, seed])	Generate matrix with the spectral density of the distribution.