# How do you find shape parameters of a Johnson distribution from skew and kurtosis?

I'm trying to generate random numbers from a distribution with given moments, in my field the Johnson family of distributions is typically used but I can't find anywhere how to set it's shape parameters to give specific population moments?

If anyone knows about any code that does it, that would be ideal but I would be happy with a book recommendation.

Thanks

• I'm not sure if I understood your question correctly, but the Wikipedia article describes a method of how to generate random numbers from a Johnson distribution. If this doesn't fit your needs, could you clarify exactly what it is you want to do? Dec 5 '18 at 10:46
• Thanks for the reply, the wikipedia article uses parameters for the distribution (gamma, delta, etc.) I want to know how these relate to the skew and kurtosis of the distribution.
– Mike
Dec 5 '18 at 10:53
• The following should do what you seek: stats.stackexchange.com/questions/58719/… See also Chapter 5 of Rose and Smith, Mathematical Statistics with Mathematica (section 5.3: Johnson transformations) - a free download is at: mathstatica.com/book/bookcontents.html Dec 5 '18 at 11:47

There is this paper by Shenton and Bowman (1975) titled Johnson's $$S_U$$ and the Skewness and Kurtosis Statistics. In it, the authors provide the third and fourth moments of Johnson's $$S_U$$ distribution.

Following the second section therein...

Define the density as follows $$p(y) = \frac{\delta}{\sqrt{2\pi}} \frac{\exp(-\frac{1}{2} z^2)}{\sqrt{1+y^2}} \, ,$$ with $$z = \gamma + \delta \ln \left(y + \sqrt{1+y^2}\right)$$.

Further, define $$\ln \omega = 1 /\delta^2$$ and $$\Omega = \gamma / \delta$$, then the third and fourth moments are

1. Skewness. $$\mu_3 = -(\omega -1)^2 \sqrt{\omega}\left[\omega(\omega + 2)\sinh(3\Omega)\, + \,3\sinh(\Omega) \right] /4 \,.$$
2. Kurtosis. $$\mu_4 = (\omega -1)^2 \left[d_4 \cosh(4\Omega)\, + \, d_2 \cosh(2\Omega) + d_0 \right],$$ with $$d_4 = \omega^2(\omega^4 + 2\omega^3 + 3\omega^2 - 3)/8 ,\, d_2 = \frac{1}{2}\omega^2(\omega + 2) ,\, d_0 = \frac{3}{8}(2\omega + 1)$$.

With regard to generating a random sample, these R packages may be useful:

• Thanks very much that's exactly what I was looking for!
– Mike
Dec 5 '18 at 12:04
• Take care: (i) the above only applies if your data happens to be characterised by the JohnsonSU system ... and that is just one of the 3 distributions that make up the Johnson family. Before applying the above, you first need to check that your data is of the SU 'type'. (ii) Even then, the above is describing the moments in terms of the Johnson parameters -- the OP's question is asking the opposite: namely, given the moments, how to define the parameters of the distribution. Dec 5 '18 at 13:36