1.
Exploratory Data Analysis
1.3. EDA Techniques 1.3.6. Probability Distributions 1.3.6.6. Gallery of Distributions


Probability Density Function 
The TukeyLambda density function does not have a simple, closed form.
It is computed numerically.
The TukeyLambda distribution has the shape parameter λ. As with other probability distributions, the TukeyLambda distribution can be transformed with a location parameter, μ, and a scale parameter, σ. Since the general form of probability functions can be expressed in terms of the standard distribution, all subsequent formulas in this section are given for the standard form of the function. The following is the plot of the TukeyLambda probability density function for four values of λ.


Cumulative Distribution Function 
The TukeyLambda distribution does not have a simple, closed form.
It is computed numerically.
The following is the plot of the TukeyLambda cumulative distribution function with the same values of λ as the pdf plots above.


Percent Point Function 
The formula for the percent point
function of the standard form of the TukeyLambda distribution is
\( G(p;\lambda) = \frac{p^{\lambda}  (1  p)^{\lambda}} {\lambda} \) The following is the plot of the TukeyLambda percent point function with the same values of λ as the pdf plots above.


Other Probability Functions  The TukeyLambda distribution is typically used to identify an appropriate distribution (see the comments below) and not used in statistical models directly. For this reason, we omit the formulas, and plots for the hazard, cumulative hazard, survival, and inverse survival functions. We also omit the common statistics and parameter estimation sections.  
Comments 
The TukeyLambda distribution is actually a family of distributions
that can approximate a number of common distributions. For example,
The most common use of this distribution is to generate a TukeyLambda PPCC plot of a data set. Based on the ppcc plot, an appropriate model for the data is suggested. For example, if the maximum correlation occurs for a value of λ at or near 0.14, then the data can be modeled with a normal distribution. Values of λ less than this imply a heavytailed distribution (with 1 approximating a Cauchy). That is, as the optimal value of λ goes from 0.14 to 1, increasingly heavy tails are implied. Similarly, as the optimal value of λ becomes greater than 0.14, shorter tails are implied. As the TukeyLambda distribution is a symmetric distribution, the use of the TukeyLambda PPCC plot to determine a reasonable distribution to model the data only applies to symmetric distributions. A histogram of the data should provide evidence as to whether the data can be reasonably modeled with a symmetric distribution. 

Software  Most general purpose statistical software programs do not support the probability functions for the TukeyLambda distribution. 