If mu
, sigma
, nu
and tau
are not specified they assume the default values of 0, 1, 0 and 0.5, respectively.
The Johnson SU distribution with parameters mu =
\(\mu\), sigma =
\(\sigma\), nu =
\(\nu\) and tau =
\(\tau\) has density:
$$ \frac{1}{c\sigma\tau}\frac{1}{\sqrt{z^2+1}}\frac{1}{\sqrt{2\pi}}e^{-r^2/2} $$
where \(r = -\nu + (1/\tau)sinh^-1(z)\),
\(z = (x - (\mu + c*\sigma (\sqrt(\omega)) sinh(w)))/(c*\sigma)\),
\(c = ((w-1)(w cosh(2\omega)+1)/2)^-1/2\),
\(w = e^(\tau^2)\) and \(\omega = -\nu\tau\).