Tukey lambda distribution
Probability density function
Notation	Tukey(λ)
Parameters	λ ∈ ℝ —shape parameter
Support	x ∈[ −⁠ 1 /λ⁠,⁠ 1 /λ⁠]   if   λ > 0 ,  x ∈ ℝ          if   λ ≤ 0.
PDF	${\displaystyle \left(Q(p;\lambda ),\frac{1}{q(p;\lambda )}\right)\mathsf{f}\mathsf{o}\mathsf{r}\mathsf{a}\mathsf{n}\mathsf{y}p:0\le p\le 1}$
CDF	${\displaystyle (Q(p;\lambda ),p)\mathsf{f}\mathsf{o}\mathsf{r}\mathsf{a}\mathsf{n}\mathsf{y}p:0\le p\le 1}$   (general case)  ${\displaystyle \frac{1}{e^{-x}+1}\mathsf{i}\mathsf{f}\lambda =0}$ (special case exact solution)
Mean	${\displaystyle 0\mathsf{i}\mathsf{f}\lambda >-1}$
Median	0
Mode	0
Variance	${\displaystyle \frac{2}{{\lambda }^2}\left(\frac{1}{1+2\lambda }-\frac{\mathrm{\Gamma }(\lambda +1{)}^2}{\mathrm{\Gamma }(2\lambda +2)}\right)\mathsf{i}\mathsf{f}\lambda >-\frac{1}{2}}$  ${\displaystyle \frac{{\pi }^2}{3}\mathsf{i}\mathsf{f}\lambda =0}$
Skewness	${\displaystyle 0\mathsf{i}\mathsf{f}\lambda >-\frac{1}{3}}$
Excess kurtosis	${\displaystyle \frac{(2\lambda +1{)}^2\cdot g_2^2\cdot (3g_2^2-4g_1{g}_3+g_4)}{(8\lambda +2)\cdot g_4\cdot (g_1^2-g_2{)}^2}-3\mathsf{i}\mathsf{f}\lambda >0,}$  ${\displaystyle \frac{6}{5}\mathsf{i}\mathsf{f}\lambda =0;}$  ${\displaystyle \mathsf{w}\mathsf{h}\mathsf{e}\mathsf{r}\mathsf{e}{g}_k\equiv \mathrm{\Gamma }(k\lambda +1)\mathsf{a}\mathsf{n}\mathsf{d}\lambda >-\frac{1}{4}.}$
Entropy	${\displaystyle h(\lambda )={\int }_0^1\ln (q(p;\lambda ))\mathrm{d}p}$[1]
CF	${\displaystyle \varphi (t;\lambda )={\int }_0^1\exp (itQ(p;\lambda ))\mathrm{d}p}$[2]

Formalized byJohn Tukey, theTukey lambda distribution is a continuous, symmetric probability distribution defined in terms of itsquantile function. It is typically used to identify an appropriate distribution (see the comments below) and not used instatistical models directly.

The Tukey lambda distribution has a singleshape parameter,λ, and as with other probability distributions, it can be transformed with alocation parameter,μ, and ascale 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,${\displaystyle Q(p),}$ (i.e. the inverse function to thecumulative distribution function) and the quantile density function,${\displaystyle q=\frac{\mathrm{d}Q}{\mathrm{d}p},}$ are

${\displaystyle q\left(p;\lambda \right)=\frac{\mathrm{d}Q}{\mathrm{d}p}=p^{\lambda -1}+{\left(1-p\right)}^{\lambda -1}.}$

For most values of the shape parameter,λ, theprobability density function (PDF) andcumulative distribution function (CDF) must be computed numerically. The Tukey lambda distribution has a simple, closed form for the CDF and / or PDF only for a few exceptional values of the shape parameter, for example:λ∈{ 2, 1,⁠ 1 /2⁠, 0} (seeuniform distribution[ casesλ = 1 andλ = 2 ] and thelogistic distribution[ caseλ = 0 ].

However, for any value ofλ both the CDF and PDF can be tabulated for any number of cumulative probabilities,p, using the quantile functionQ to calculate the valuex, for each cumulative probabilityp, with the probability density given by⁠1/q⁠, the reciprocal of the quantile density function. As is the usual case with statistical distributions, the Tukey lambda distribution can readily be used by looking up values in a prepared table.

The Tukey lambda distribution is symmetric around zero, therefore the expected value of this distribution, if it exists, is equal to zero. The variance exists forλ > −⁠ 1 /2⁠ , and except whenλ = 0 , is given by the formula

${\displaystyle \mathrm{Var}[X]=\frac{2}{{\lambda }^2}(\frac{1}{1+2\lambda }-\frac{\mathrm{\Gamma }(\lambda +1{)}^2}{\mathrm{\Gamma }(2\lambda +2)}).}$

More generally, then-th order moment is finite whenλ >⁠−1 /n⁠ and is expressed (except whenλ = 0 ) in terms of thebeta functionΒ(x,y) :

${\displaystyle {\mu }_n\equiv \mathrm{E}[X^n]=\frac{1}{{\lambda }^n}\sum _{k=0}^n(-1{)}^k(\genfrac{}{}{0ex}{}{n}{k})\mathrm{B}(\lambda k+1,(n-k)\lambda +1).}$

Due to symmetry of the density function, all moments of odd orders, if they exist, are equal to zero.

Differently from the central moments,L-moments can be expressed in a closed form. For${\displaystyle \lambda >-1,}$ the${\displaystyle r}$th L-moment,${\displaystyle {\ell }_r,}$ is given by^[3]

The first six L-moments can be presented as follows:^[3]

${\displaystyle {\ell }_1=0,}$

${\displaystyle {\ell }_2=\frac{2}{\lambda }\left[-\frac{1}{1+\lambda }+\frac{2}{2+\lambda }\right],}$

${\displaystyle {\ell }_3=0,}$

${\displaystyle {\ell }_4=\frac{2}{\lambda }\left[-\frac{1}{1+\lambda }+\frac{12}{2+\lambda }-\frac{30}{3+\lambda }+\frac{20}{4+\lambda }\right],}$

${\displaystyle {\ell }_5=0,}$

${\displaystyle {\ell }_6=\frac{2}{\lambda }\left[-\frac{1}{1+\lambda }+\frac{30}{2+\lambda }-\frac{210}{3+\lambda }+\frac{560}{4+\lambda }-\frac{630}{5+\lambda }+\frac{252}{6+\lambda }\right].}$

The Tukey lambda distribution is actually a family of distributions that can approximate a number of common distributions. For example,

λ ≈ −1	approx.CauchyC( 0,π )
λ = 0	exactlylogistic
λ ≈ 0.14	approx.normalN( 0, 2.142± )
λ =⁠ 1 /2⁠	strictlyconcave (${\displaystyle \cap }$-shaped)
λ = 1	exactlyuniformU( −1, +1 )
λ = 2	exactlyuniformU( −⁠ 1 /2⁠ , +⁠ 1 /2⁠)

The most common use of this distribution is to generate a Tukey lambdaPPCC plot of adata set. Based on the value for λ with best correlation, as shown on thePPCC plot, an appropriatemodel for the data is suggested. For example, if the best-fit of the curve to the data occurs for a value of λ at or near0.14, then empirically the data could be well-modeled with a normal distribution. Values of λ less than 0.14 suggests a heavier-tailed distribution.

A milepost at λ = 0 (logistic) would indicate quite fat tails, with the extreme limit at λ = −1 , approximatingCauchy and small sample versions of theStudent'st. That is, as the best-fit value ofλ varies from thin tails at0.14 towards fat tails−1, a bell-shaped PDF with increasingly heavy tails is suggested. Similarly, an optimal curve-fit value ofλ greater than0.14 suggests a distribution withexceptionally thin tails (based on the point of view that the normal distribution itself is thin-tailed to begin with; theexponential distribution is often chosen as the exemplar of tails intermediate between fat and thin).

Except for values ofλ approaching0 and those below, all the PDF functions discussed have finitesupport, between  ⁠−1  /|λ|⁠   and  ⁠+1  / |λ|⁠ .

Since the Tukey lambda distribution is asymmetric distribution, the use of the Tukey lambda PPCC plot to determine a reasonable distribution to model the data only applies to symmetric distributions. Ahistogram of the data should provide evidence as to whether the data can be reasonably modeled with a symmetric distribution.^[4]

• "Tukey-Lambda distribution". Gallery of Distributions. Engineering Statistics Handbook.US NIST Information Technology Laboratory. 1.3.6.6.15. EDA 366F.

 This article incorporatespublic domain material from the National Institute of Standards and Technology

• Continuous distributions
• Probability distributions with non-finite variance

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