Tukey lambda distribution

Tukey lambda distribution
Probability density function
Probability density plots of Tukey lambda distributions
Notation   Tukey(λ)
Parameters   λ ∈ ℝ shape parameter
Support   x[ 1 /λ,  1 /λ ]    if    λ > 0 ,
  x ∈ ℝ           if    λ ≤ 0 .
PDF  
CDF     (general case)
  (special case exact solution)
Mean  
Median   0
Mode   0
Variance  
 
Skewness  
Excess kurtosis  

 

 
Entropy  [1]
CF   [2]

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, λ, and as with other probability distributions, it can be transformed with a location parameter, μ, and a scale parameter, σ. 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.

  1. ^ Vasicek, Oldrich (1976). "A test for normality based on sample entropy". Journal of the Royal Statistical Society. Series B. 38 (1): 54–59.
  2. ^ Shaw, W.T.; McCabe, J. (2009), "Monte Carlo sampling given a characteristic function: Quantile mechanics in momentum space", arXiv:0903.1592 [q-fin.CP]