represents Tukey's lambda distribution with shape parameter λ.

represents Tukey's lambda distribution with location parameter μ and scale parameter σ.

represents the generalized Tukey's lambda distribution with location parameter μ, scale parameters σ1 and σ2, and shape parameters λ1 and λ2.

# Details • The quantile function for value in a Tukey lambda distribution is equal to for and , and when .
• The quantile function for value in a generalized Tukey lambda distribution is equal to for .
• TukeyLambdaDistribution allows λ, λ1, λ2, and μ to be any real numbers, and σ, σ1, and σ2 any positive real numbers.
• TukeyLambdaDistribution allows μ, σ, σ1, and σ2 to be any quantities of the same unit dimensions, and λ, λ1, and λ2 to be dimensionless quantities. »
• TukeyLambdaDistribution can be used with such functions as Mean, CDF, and RandomVariate.

# Background & Context

• TukeyLambdaDistribution[{λ1,λ2},μ,{σ1,σ2}] represents a continuous statistical distribution parametrized by the real numbers λ1, λ2, and μ (two "shape parameters" and a "location parameter", respectively) and by positive real numbers σ1, and σ2 (called "scale parameters"), which together determine the overall behavior of its probability density function (PDF). Depending on the values of λ1, λ2, μ, σ1, and σ2, the PDF of a Tukey lambda distribution may have any of a number of shapes, including unimodal with a single "peak" (i.e. a global maximum), linear, and monotone decreasing. In addition, the tails of the PDF (which may or may not exist, depending on the value of the parameters) may be "fat" (i.e. the PDF decreases non-exponentially for large values ) or "thin" (i.e. the PDF decreases exponentially for large ), depending on the values of λ1, λ2, μ, σ1, and σ2. (When defined, this behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.) The five-parameter form TukeyLambdaDistribution[{λ1,λ2},μ,{σ1,σ2}] is usually referred to as the generalized Tukey lambda distribution, while the three- and one-parameter forms TukeyLambdaDistribution[λ,μ,σ] and (equivalent to TukeyLambdaDistribution[{λ,λ},μ,{σ,σ}] and TukeyLambdaDistribution[{λ,λ},0,{1,1}], respectively) are often referred to as "the" (Tukey) lambda distribution and the standard (Tukey) lambda distribution, respectively.
• The one-parameter Tukey lambda distribution was conceived by American mathematician John Tukey in the 1940s as a tool to model the means, variances, and covariances of small samples of normally distributed (NormalDistribution) random variates. The lambda distribution is unique among probability distributions in the sense that its PDF has no closed-form expression (except when in the one-parameter form) and must instead be described in terms of its quantile function (Quantile). Even so, the generalized lambda distribution is able to assume a wide range of shapes and is therefore an extremely versatile distribution that can be used to approximate a wide array of distributions, including the NormalDistribution, UniformDistribution, BetaDistribution, StudentTDistribution, ExponentialDistribution, and various so-called extreme-value distributions (e.g. WeibullDistribution, ExtremeValueDistribution, GumbelDistribution, and FrechetDistribution).
• RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a Tukey lambda distribution. Distributed[x,TukeyLambdaDistribution[{λ1,λ2},μ,{σ1,σ2}]], written more concisely as xTukeyLambdaDistribution[{λ1,λ2},μ,{σ1,σ2}], can be used to assert that a random variable x is distributed according to a Tukey lambda distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
• The probability density and cumulative distribution functions for Tukey lambda distributions may be given using PDF[TukeyLambdaDistribution[{λ1,λ2},μ,{σ1,σ2}],x] and CDF[TukeyLambdaDistribution[{λ1,λ2},μ,{σ1,σ2}],x]. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively.
• DistributionFitTest can be used to test if a given dataset is consistent with a Tukey lambda distribution, EstimatedDistribution to estimate a Tukey lambda parametric distribution from given data, and FindDistributionParameters to fit data to a Tukey lambda distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic Tukey lambda distribution, and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic Tukey lambda distribution.
• TransformedDistribution can be used to represent a transformed Tukey lambda distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higher-dimensional distributions that contain a Tukey lambda distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving Tukey lambda distributions.
• TukeyLambdaDistribution is related to a number of other distributions. TukeyLambdaDistribution generalizes a number of distributions including LogisticDistribution and UniformDistribution, in the sense that the PDFs of , , and are precisely the same as those of LogisticDistribution[0,1], UniformDistribution[{-1,1}], and UniformDistribution[{-1/2,1/2}], respectively. For certain values of λ, shares qualitative properties with CauchyDistribution and NormalDistribution and hence is also closely related to LogNormalDistribution, BetaDistribution, and PearsonDistribution.

# Examples

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## Basic Examples(15)

Quantile function for symmetric case:

Quantile function with location and scale parameters:

Quantile function for generalized Tukey's lambda distribution:

Probability density function does not have closed form but can be found numerically:

Cumulative distribution function does not have closed form but can be found numerically:

Mean and variance have closed forms:

Median:

Probability density function with location and scale:

Cumulative distribution function with location and scale:

Mean and variance with location and scale:

Median with location and scale:

Probability density function of generalized Tukey lambda distribution:

Cumulative distribution function of generalized Tukey lambda distribution:

Mean and variance of generalized Tukey lambda distribution:

Median of generalized Tukey lambda distribution:

## Scope(7)

Generate a sample of pseudorandom numbers from a Tukey's lambda distribution:

Compare its histogram to the PDF:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Compare the density histogram of the sample with the PDF of the estimated distribution:

Skewness is zero for symmetric case:

With location and scale:

Kurtosis:

Location and scale does not change the kurtosis:

Kurtosis attains its minimum:

Different moments with closed forms as functions of parameters:

Moment for symmetric case:

Hazard function for symmetric case:

With location and scale:

Hazard function of generalized Tukey lambda distribution:

Consistent use of Quantity in parameters yields QuantityDistribution:

Find the average distance:

## Generalizations & Extensions(1)

Useful shortcuts evaluate to their full-form counterparts:

## Applications(2)

Symmetric TukeyLambdaDistribution is often used to approximate other symmetric distributions. The distribution with approximates NormalDistribution:

Tukey lambda distribution with approximately corresponds to CauchyDistribution:

Probability plot correlation coefficient plot is often used to determine best fit of symmetric TukeyLambdaDistribution to the data:

Visually confirm that is a good fit for normal samples:

Visually find best fit for standard samples from LogisticDistribution:

## Properties & Relations(5)

Shifting and scaling by positive factor:

Relationships to other distributions: For , it is exactly UniformDistribution:

For , it is exactly LogisticDistribution:

Tukey lambda with location and scale simplifies to one parameter case:

## Neat Examples(1)

PDFs for different λ values with CDF contours: