Formalized by John Tukey, the Tukey lambda distribution is a continuous, symmetric probability distribution defined in terms of its quantile function. It is typically used to identify an appropriate distribution (see the comments below) and not used in statistical models directly.
The Tukey lambda distribution has a single shape parameter, , and as with other probability distributions, it can be transformed with a location parameter, , and a scale parameter, . Since the general form of probability distribution can be expressed in terms of the standard distribution, the subsequent formulas are given for the standard form of the function.
For the standard form of the Tukey lambda distribution, the quantile function, (i.e. the inverse function to the cumulative distribution function) and the quantile density function, are
For most values of the shape parameter, , the probability density function (PDF) and cumulative distribution function (CDF) must be computed numerically. The Tukey lambda distribution has a simple, closed form for the CDF and / or PDF only for a few exceptional values of the shape parameter, for example: 2, 1, , 0 (see uniform distribution and and the logistic distribution
However, for any value of both the CDF and PDF can be tabulated for any number of cumulative probabilities, , using the quantile function to calculate the value , for each cumulative probability , with the probability density given by , the reciprocal of the quantile density function. As is the usual case with statistical distributions, the Tukey lambda distribution can readily be used by looking up values in a prepared table.
The Tukey lambda distribution is symmetric around zero, therefore the expected value of this distribution, if it exists, is equal to zero. The variance exists for and except when is given by the formula
More generally, the -th order moment is finite when and is expressed (except when in terms of the beta function
Due to symmetry of the density function, all moments of odd orders, if they exist, are equal to zero.
Differently from the central moments, L-moments can be expressed in a closed form. For the th L-moment, is given by
The first six L-moments can be presented as follows:
The Tukey lambda distribution is actually a family of distributions that can approximate a number of common distributions. For example,