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I was wondering if there is a closed form to calculate the nth percentile of Johnson-S$_U$ distribution with known parameters? I know R could do this easily, but I want a formula instead so I can use it in some analysis I am working on

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If you seek $x_q$, the quantity that is at least as large as a proportion $q$ of the population for a Johnson $S_U$ with parameters $\gamma ,\xi ,\delta,\lambda $ (for the parameterization as at the Wikipedia page), then

$$x_q=\lambda \sinh \left({\frac {\Phi ^{{-1}}(q)-\gamma }{\delta }}\right)+\xi $$

where $\Phi^{-1}$ is the inverse cdf of a standard normal and $\sinh$ is the usual hyperbolic sine function.

This is just the inverse function of the Johnson $S_U$ cdf.

Whether this counts as "closed form" depends on what functions you are willing to admit; for example, by the big table here it wouldn't count, since $\Phi^{-1}$ is a special function (or a simple transformation of one). However as the article notes at the beginning, the set of operations and functions admitted in a closed-form expression may vary with author and context.

Most statisticians would tend to treat it as closed form, however.

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