Tukey lambda distribution

Moments

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 for λ > −⁠ 1 /2⁠ , and except when λ = 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, the n-th order moment is finite when λ > ⁠−1 /n⁠ and is expressed (except when λ = 0 ) in terms of the beta function Β(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.

L-moments

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

${\displaystyle \begin{array}{cc}{\ell }_r& =\frac{1+(-1{)}^r}{\lambda }\sum _{k=0}^{r-1}(-1{)}^{r-1-k}(\genfrac{}{}{0ex}{}{r-1}{k})(\genfrac{}{}{0ex}{}{r+k-1}{k})\left(\frac{1}{k+1+\lambda }\right)\\ \\ & =(1+(-1{)}^r)\frac{\mathrm{\Gamma }(1+\lambda )\mathrm{\Gamma }(r-1-\lambda )}{\mathrm{\Gamma }(1-\lambda )\mathrm{\Gamma }(r+1+\lambda )}.\end{array}}$

The first six L-moments can be presented as follows:

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

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

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

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

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

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

Comments

The Tukey lambda 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 Tukey lambda PPCC plot of a data set. Based on the value for λ with best correlation, as shown on the PPCC plot, an appropriate model 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 near 0.14, 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 , approximating Cauchy and small sample versions of the Student's t. That is, as the best-fit value of λ varies from thin tails at 0.14 towards fat tails −1, a bell-shaped PDF with increasingly heavy tails is suggested. Similarly, an optimal curve-fit value of λ greater than 0.14 suggests a distribution with exceptionally thin tails (based on the point of view that the normal distribution itself is thin-tailed to begin with; the exponential distribution is often chosen as the exemplar of tails intermediate between fat and thin).

Except for values of λ approaching 0 and those below, all the PDF functions discussed have finite support, between ⁠−1 /|λ|⁠ and ⁠+1 / |λ| ⁠ .

Since the Tukey lambda distribution is a symmetric distribution, the use of the Tukey lambda 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.

Generalization

The generalized lambda distribution (GLD) generalizes the lambda distribution by splitting the occurrences of λ in the quantile into two shape parameters (accounts for asymmetry), a scale parameter, and adds a location parameter. It is defined by the quantile function

${\displaystyle Q_{\text{RS74}}={\lambda }_1+\frac{1}{{\lambda }_2}\left[p^{{\lambda }_3}-(1-p{)}^{{\lambda }_4}\right].}$

The inverse function is the cumulative distribution function. This distribution also has (rather complex) expressions for μ, σ, and the moments.

The above parameterization is the original form proposed by Ramberg and Schmeiser; some choices of λ_3,4 yield invalid quantile functions (a quantile function should not be negative). A more commonly used form is FMKL (1988) which is valid as long as λ2 > 0;. There is also a variant of FMKL that adds skewness. The other alternative parameterization is GPD of van Staden and Loots (2009), which has simpler expressions for L-moments, a simple rule for parameter validity, and a skew parameter δ.

${\displaystyle \begin{array}{ccc}{Q}_{\text{FMKL}}=& {\lambda }_1+\frac{1}{{\lambda }_2}\left[\frac{p^{{\lambda }_3}-1}{{\lambda }_3}-\frac{{(1-p)}^{{\lambda }_4}-1}{{\lambda }_4}\right]& \text{ if }{\lambda }_2>0,\\ Q_{\text{GPD}}=& \alpha +\frac{\beta }{\lambda }\left[(1-\delta )(p^{\lambda }-1)-\delta ({(1-p)}^{\lambda }-1)\right]& \text{ if }\beta >0\text{ and }-1\le \delta \le 1.\end{array}}$

The GLD was developed by Ramberg and Schmeiser in 1974 for generating random numbers for Monte Carlo simulation. Other researchers found it useful for fitting various measurements in the natural sciences, finance, and economy. Its use spread to the field of statistical quality control and procedures are available for using it in the calculation of process capability indices.