The Tukey lambda distribution, also called the symmetric lambda distribution, is a family of symmetric distributions with truncated tails. Typically, it aids in identification of an appropriate distribution and is not directly used in statistical modeling. This is because it lacks a general form of a probability density function (PDF) or cumulative distribution function (CDF). However, there are some useful special cases, including :
- λ = -1 : approximately Cauchy;
- λ = 0 : exactly logistic;
- λ = 0.14 : approximately normal;
- λ = 0.5 : U-shaped;
- λ= 1 : exactly uniform.
Another common use for the Tukey lambda distribution is to generate PPCC plots, where the software processes inputted data to suggest models via a technique such as the Tukey-Lambda PPCC plot (Probability Plot Correlation Coefficient Plot).
Properties of the Tukey Lambda distribution
The Tukey lambda distribution is defined numerically with three parameters:
- λ, = shape parameter
- μ= location parameter
- σ = scale parameter.
Most probability distributions have a formula for the probability density function (PDF) and cumulative distribution function (CDF) that fits all shapes of the distribution, but this isn’t the case for the Tukey lambda distribution; In general, neither its PDF not its CDF is known, but the CDF’s inverse — the quantile function — is known [2, 3]. Thus, you’ll usually find the distribution described in terms of quantiles:
This function is not always analytically invertible and only allows for the following values of λ :−1, 0, 1/4, 1/3, 1/2, 1, 3/2, 2, 3, 4=−1, 0, 1/4, 1/3, 1/2, 1, 3/2, 2, 3, 4. You must use numerical inversion to get a CDF for other λ values .
The generalized lambda distribution is often defined in terms of its percentile function:
- λ1, the location parameter,
- λ2, the scale parameter,
- λ3, skewness,
- λ4, kurtosis.
If you know the percentile function, you can generate a PDF for specific values of λ.
History of the Tukey lambda distribution
John Tukey was the first to propose Tukey’s lambda distribution in 1960 , based on work from Hastings et. al in 1947 . In the early 1970s, John Ramberg and Bruce Schmeiser  generalized the distribution for Monte Carlo simulations. In the late 70’s, Ramberg and colleagues developed the curve-fitting properties of the distribution . Once the curve is fit, you can then model the residuals. Curve fitting algorithms you can use include gradient descent, Gauss-Newton and the Levenberg-Marquardt algorithm.
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 Ramberg, J. and Schmeiser, B. (1972) An Approximate Method for Generating Symmetric Random Variables. Communications of the ACM, 15, 987-990. https://doi.org/10.1145/355606.361888
 Ramberg, J., et al. (1979) A Probability Distribution and Its Uses in Fitting Data. Technometrics, 21, 201-214.
 Sarabia, J.M. (1997) A Hierarchy of Lorenz Curves Based on the Generalized Tukey’s Lambda Distribution. Econometric Reviews, 16, 305-320.
 Girone, G. eta l. Mean Difference and Mean Deviation of Tukey Lambda Distribution. Applied Mathematics > Vol.11 No.8, August 2020. 10.4236/am.2020.118051
 Tukey, J. (1960). The Practical Relationship Between the Common Transformations of Percentages of Counts and Amounts, Technical Report 36. Statistical Techniques Research Group, Princeton University.
 Hastings, C. et. al (1947). Low moments for small samples: a comparative study of statistics. Annals of Mathematical Statistics, 18, 413-426.
 Ramberg, J & Schmeiser, B. (1972). An approximate method for generating symmetric random variables. Commun. ACM, 15:987-990.
 Ramberg, J. et. al. (1979). A probability distribution and its uses in fitting data. Technometrics, 21(2):201-214, May 1979.