Tukey lambda distribution

Tukey lambda distribution

Probability density function
Notation	Tukey(λ)
Parameters	λ ∈ R — shape parameter
Support	x ∈ [−1/λ, 1/λ] for λ > 0, x ∈ R for λ ≤ 0
PDF	${\displaystyle (Q(p;\lambda )\,,Q'(p;\lambda )^{-1}),\,0\leq \,p\,\leq \,1}$
CDF	${\displaystyle (e^{-x}+1)^{-1},\,\,\lambda \,=\,0}$
Mean	${\displaystyle 0,\,\,\lambda >-1}$
Median	0
Mode	0
Variance	${\displaystyle {\frac {2}{\lambda ^{2}}}{\bigg (}{\frac {1}{1+2\lambda }}-{\frac {\Gamma (\lambda +1)^{2}}{\Gamma (2\lambda +2)}}{\bigg )},\,\,\lambda >-1/2}$ ${\displaystyle {\frac {\pi ^{2}}{3}},\,\,\lambda \,=\,0}$
Skewness	${\displaystyle 0,\,\,\lambda >-1/3}$
Ex. kurtosis	${\displaystyle {\frac {(2\lambda +1)^{2}}{2(4\lambda +1)}}{\frac {g_{2}^{2}{\big (}3g_{2}^{2}-4g_{1}g_{3}+g_{4}{\big )}}{g_{4}{\big (}g_{1}^{2}-g_{2}{\big )}^{2}}}-3,}$ ${\displaystyle 1.2,\,\,\lambda \,=\,0,}$ where gk = Γ(kλ+1) and λ > -1/4.
Entropy	${\displaystyle h(\lambda )=\int _{0}^{1}\log(Q'(p;\lambda ))\,dp}$
CF	${\displaystyle \phi (t;\lambda )=\int _{0}^{1}\exp(\,it\,Q(p;\lambda ))\,dp}$

Formalized by John Tukey, the Tukey lambda distribution is a continuous 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. As with other probability distributions, the Tukey lambda distribution 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.

For the standard form of the Tukey lambda distribution, the quantile function, Q(p), (i.e. the inverse of the cumulative distribution function) and the quantile density function (i.e. the derivative of the quantile function) are

The probability density function (pdf) and cumulative distribution function (cdf) are both computed numerically, as the Tukey lambda distribution does not have a simple, closed form for any values of the parameters except λ = 0 (see logistic distribution). However, the pdf can be expressed in parametric form, for all values of λ, in terms of the quantile function and the reciprocal of the quantile density function.

The Tukey lambda distribution is symmetric around zero, therefore the expected value of this distribution is equal to zero. The variance exists for λ > −½ and is given by the formula (except when λ = 0)

More generally, the n-th order moment is finite when λ > −1/n and is expressed in terms of the beta function Β(x,y) (except when λ = 0) :

Note that due to symmetry of the density function, all moments of odd orders are equal to zero.

Differently from the central moments, L-moments can be expressed in a closed form. The L-moment of order r>1 is given by