Tukey Lambda Distribution - Quantile Function

Quantile Function

For the standard form of the Tukey lambda distribution, the quantile function, Q(p), (i.e. the inverse of the cumulative distribution function) and the quantile density function (i.e. the derivative of the quantile function) are

$Q\left(p;\lambda\right) = \begin{cases} \frac{ 1 }{ \lambda } \left, & \mbox{if } \lambda \ne 0 \\ \log(p) - \log(1-p), & \mbox{if } \lambda = 0, \end{cases}$

The probability density function (pdf) and cumulative distribution function (cdf) are both computed numerically, as the Tukey lambda distribution does not have a simple, closed form for any values of the parameters except λ = 0 (see logistic distribution). However, the pdf can be expressed in parametric form, for all values of λ, in terms of the quantile function and the reciprocal of the quantile density function.

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