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{{Short description|Symmetric probability distribution}}
{{Probability distribution
| name = Tukey lambda distribution
| type = density
| pdf_image = [[File:Several samples of the pdfs of the Tukey lambda distributions.svg|325px|Probability density plots of Tukey lambda distributions]]
| cdf_image =
| notation = {{big| }} Tukey({{mvar|λ}})
| parameters = {{big| }} {{math| ''λ'' ∈ ℝ }} — [[shape parameter]]
| support = {{big| }} {{nobr|{{math| ''x'' ∈ {{big|[}} −{{small|{{sfrac| 1 |''λ''}}}}, {{small|{{sfrac| 1 |''λ''}}}} {{big|]}} }} }} {{nobr|   if    }} {{nobr|{{math| ''λ'' > 0}} ,}} {{big| }} {{nobr|{{math| ''x'' ∈ ℝ }}}} {{nobr|{{big| }} }} &emsp; &emsp;   &emsp; if    {{nobr|{{math| ''λ'' ≤ 0 }}.}}
| pdf = {{big| }} <math> \left(\ Q(\ p\ ; \lambda\ ),\ \frac{ 1 }{\ q(\ p\ ; \lambda\ )\ }\ \right) \quad ~\mathsf{ for\ any }~ \quad p \; : \; 0 \leq\ p\ \leq\ 1 </math>
| cdf = {{big| }} <math> \Bigl(\ Q(p;\lambda),\ p\ \Bigr) ~~\mathsf{ for\ any }~~ p\; : \; 0 \leq\ p\ \leq\ 1 ~</math> &emsp; (general case) {{big| }} <math> \frac{ 1 }{\ e^{-x} + 1\ } \quad ~\mathsf{ if }~ \quad \lambda\ =\ 0 \quad </math> (special case exact solution)
| mean = {{big| }} <math> 0 \quad ~\mathsf{ if }~ \quad \lambda > -1\ </math>
| median = {{big| }} {{math| 0 }}
| mode = {{big| }} {{math| 0 }}
| variance = {{big| }} <math> \frac{ 2 }{\ \lambda^2\ } \left(\ \frac{ 1 }{\ 1 + 2\ \lambda\ } - \frac{\ \Gamma\,\!(\lambda + 1)^2\ }{\ \Gamma\,\!(\ 2\ \lambda + 2\ )\ } \right) \quad ~\mathsf{ if }~ \quad \lambda > -\tfrac{\ 1\ }{ 2 }\ </math> {{big| }} <math>\frac{\ \pi^{2}\ }{ 3 } \qquad \qquad ~\mathsf{ if }~ \quad \lambda\ =\ 0</math>
| skewness = {{big| }} <math> 0 \qquad \qquad ~\mathsf{ if }~ \quad \lambda > -\tfrac{\ 1\ }{ 3 }\ </math>
| kurtosis = {{big| }} <math>~ \frac{\ (2\ \lambda + 1)^2 \cdot g_2^2 \cdot \big(\ 3\ g_2^2 - 4\ g_1\ g_3 + g_4\ \big)\ }{\ (\ 8\ \lambda + 2\ ) \cdot g_4 \cdot \big(\ g_1^2 - g_2\ \big)^2}\ -\ 3 \quad \mathsf{ if }~~ \lambda > 0\ ,</math> {{big| }} <math>\frac{\ 6\ }{ 5 } \qquad \qquad ~\mathsf{ if }~ \quad \lambda\ =\ 0\ ;</math> {{big| }} <math>\mathsf{ where } \quad g_k\ \equiv\ \Gamma\,\!(\ k\,\lambda + 1\ ) \quad \mathsf{ and } \quad \lambda\ > -\tfrac{\ 1\ }{ 4 } ~.</math>
| entropy = {{big| }}<math> h( \lambda ) = \int_0^1 \ln \bigl(\ q(p;\lambda)\ \bigr)\ \operatorname{d}p ~</math><ref>{{cite journal |last=Vasicek |first=Oldrich |year=1976 |title=A test for normality based on sample entropy |journal=[[Journal of the Royal Statistical Society]] |series=Series B |volume=38 |issue=1 |pages=54–59 }}</ref>
| mgf =
| cf = {{big| }} <math>\phi(t;\lambda) = \int_0^1 \exp \bigl( \ i\ t\ Q(p;\lambda)\ \bigr)\ \operatorname{d}p ~</math><ref>{{cite arXiv |last1=Shaw |first1=W.T. |last2=McCabe |first2=J. |year=2009 |title=Monte Carlo sampling given a characteristic function: Quantile mechanics in momentum space |class=q-fin.CP |eprint=0903.1592 |mode=cs2}}</ref>
}}
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 model]]s directly.

The Tukey lambda distribution has a single [[shape parameter]], {{mvar|λ}}, and as with other probability distributions, it can be transformed with a [[location parameter]], {{mvar|μ}}, and a [[scale parameter]], {{mvar|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.

==Quantile function==
For the standard form of the Tukey lambda distribution, the quantile function, <math>~ Q(p) ~,</math> (i.e. the inverse function to the [[cumulative distribution function]]) and the quantile density function, <math>~ q = \frac{\ \operatorname{d}Q\ }{ \operatorname{d}p }\ ,</math> are

:<math>
\ Q\left(\ p\ ; \lambda\ \right) ~=~
\begin{cases}
\tfrac{ 1 }{\ \lambda\ } \left[\ p^\lambda - (1 - p)^\lambda\ \right]\ , &\ \mbox{ if }\ \lambda \ne 0~, \\ {} \\
\ln\left( \frac{ p }{\ 1 - p\ } \right)~, &\ \mbox{ if }\ \lambda = 0 ~.
\end{cases}</math>

:<math> q \left(\ p\ ; \lambda\ \right) ~=~ \frac{\ \operatorname{d}Q\ }{ \operatorname{d}p } ~=~ p^{ \lambda - 1 } + \left(\ 1 - p\ \right)^{ \lambda - 1 } ~.</math>

For most values of the shape parameter, {{mvar|λ}}, the [[probability density function]] (PDF) and [[cumulative 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: {{mvar|λ}} {{small|∈}} {{big|{<nowiki/>}} 2, 1, {{small|{{sfrac| 1 |2}}}}, 0 {{big|}<nowiki/>}} (see [[Continuous uniform distribution|uniform distribution]] {{nobr|[ cases {{math| ''λ'' {{=}} 1 }} }} and {{nobr|{{math| ''λ'' {{=}} 2 }} ]}} and the [[logistic distribution]] {{nobr|[ case {{math| ''λ'' {{=}} 0 }} ].}}

However, for any value of {{mvar|λ}} both the CDF and PDF can be tabulated for any number of cumulative probabilities, {{mvar|p}}, using the quantile function {{mvar|Q}} to calculate the value {{mvar|x}}, for each cumulative probability {{mvar|p}}, with the probability density given by {{sfrac|1|{{mvar|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.

==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 {{nobr|{{math| ''λ'' > −{{small|{{sfrac| 1 |2}} }} }} ,}} and except when {{nobr|{{math| ''λ'' {{=}} 0 }} ,}} is given by the formula
: <math>
\operatorname{Var}[\ X\ ] = \frac{2}{\lambda^2}\bigg(\ \frac{ 1 }{\ 1 + 2\lambda\ } ~ - ~ \frac{\ \Gamma(\lambda+1)^2\ }{\ \Gamma(2\lambda+2)\ }\ \bigg) ~.
</math>

More generally, the {{mvar|n}}-th order moment is finite when {{nobr|{{math| ''λ'' > {{sfrac|−1 |''n''}} }} }} and is expressed (except when {{nobr|{{math| ''λ'' {{=}} 0 }} )}} in terms of the [[beta function]] {{nobr|{{math| ''Β''(''x'',''y'') }} :}}
: <math>
\mu_n \equiv \operatorname{E}[\ X^n\ ] = \frac{1}{\lambda^n} \sum_{k=0}^n\ (-1)^k\ {n \choose k}\ \Beta(\ \lambda\ k + 1\ ,\ (n - k)\ \lambda + 1\ ) ~.
</math>

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 <math>\lambda > -1\ ,</math> the <math>\ r</math>th L-moment, <math>\ \ell_r\ ,</math> is given by<ref name=Karvnn-Nuutnn-2008>{{cite journal | last=Karvanen | first=Juha | last2=Nuutinen |first2=Arto | year=2008 | title=Characterizing the generalized lambda distribution by L-moments| journal=Computational Statistics & Data Analysis | volume=52 | issue=4 | pages=1971–1983 | doi=10.1016/j.csda.2007.06.021 | arxiv=math/0701405 | s2cid=939977 }}</ref>
:<math>
\begin{align}
\ell_{r} &= \frac{\ 1 + (-1)^{r}\ }{ \lambda }\ \sum_{k=0}^{r-1}\ (-1)^{ r - 1 - k }\ \binom{r-1}{k}\ \binom{ r + k -1 }{ k }\ \left(\frac{ 1 }{\ k + 1 + \lambda\ } \right) \\ {} \\
&= \bigl( 1 + (-1)^{r} \bigr) \frac{\ \Gamma( 1 + \lambda )\ \Gamma( r - 1 - \lambda )\ }{\ \Gamma( 1 - \lambda )\ \Gamma( r + 1 + \lambda)\ } ~.
\end{align}
</math>

The first six L-moments can be presented as follows:<ref name=Karvnn-Nuutnn-2008/>
:<math> \ell_{1} = ~~ 0\ ,</math>

:<math> \ell_2 = \frac{ 2 }{\ \lambda\ }\ \left[\ -\frac{ 1 }{\ 1 + \lambda\ } + \frac{ 2 }{\ 2 + \lambda\ }\ \right]\ ,</math>

:<math> \ell_3 = ~~ 0\ ,</math>

:<math> \ell_4 = \frac{2}{\ \lambda\ }\ \left[ - \frac{ 1 }{\ 1 + \lambda\ } + \frac{ 12 }{\ 2 + \lambda\ } - \frac{ 3 0}{\ 3 + \lambda\ } + \frac{ 20 }{\ 4 + \lambda\ }\ \right]\ ,</math>

:<math> \ell_5 = ~~ 0\ ,</math>

:<math> \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] ~. </math>

==Comments==
[[File:Several samples of the pdfs of the Tukey lambda distributions.svg|325px|right|Probability density plots of Tukey lambda distributions]]
The Tukey lambda distribution is actually a family of distributions that can approximate a number of common distributions. For example,
:{| class="wikitable"
|-
| {{nobr| {{math| ''λ'' ≈ −1 }} }}
| approx. [[Cauchy distribution|Cauchy]] {{nobr| {{math| ''C''( 0, ''π'' ) }} }}
|-
| {{nobr| {{math| ''λ'' {{=}} 0 }} }}
| exactly [[logistic distribution|logistic]]
|-
| {{nobr| {{math| ''λ'' ≈ 0.14 }} }}
| approx. [[normal distribution|normal]] {{nobr| {{math| ''N''( 0, 2.142{{sup|±}} ) }} }}
|-
| {{nobr| {{math| ''λ'' {{=}} {{small|{{sfrac| 1 |2}} }} }}
| strictly [[concave function|concave]] (<math>\cap</math>-shaped)
|-
| {{nobr| {{math| ''λ'' {{=}} 1 }} }}
| exactly [[continuous uniform distribution|uniform]] {{nobr| {{math| ''U''( −1, +1 ) }} }}
|-
| {{nobr| {{math| ''λ'' {{=}} 2 }} }}
| exactly [[continuous uniform distribution|uniform]] {{nobr| {{math| ''U''{{big|(}} −{{small|{{sfrac| 1 |2}} }} , +{{small|{{sfrac| 1 |2}} }} {{big|)}} }} }}
|}

The most common use of this distribution is to generate a Tukey lambda [[PPCC plot]] of a [[data set]]. Based on the value for {{mvar| λ }} with best correlation, as shown on the [[PPCC plot]], an appropriate [[statistical model|model]] for the data is suggested. For example, if the best-fit of the curve to the data occurs for a value of {{mvar| λ }} at or near {{math|0.14}}, then empirically the data could be well-modeled with a normal distribution. Values of {{mvar| λ }} less than 0.14 suggests a heavier-tailed distribution.

A milepost at {{nobr| {{math| ''λ'' {{=}} 0 }} }} ([[logistic distribution|logistic]]) would indicate quite fat tails, with the extreme limit at {{nobr| {{math| ''λ'' {{=}} −1 }} ,}} approximating [[Cauchy distribution|Cauchy]] and small sample versions of the [[Student's t distribution|Student's {{mvar|t}}]]. That is, as the best-fit value of {{mvar|λ}} varies from thin tails at {{math|0.14}} towards fat tails {{math|−1}}, a bell-shaped PDF with increasingly heavy tails is suggested. Similarly, an optimal curve-fit value of {{mvar|λ}} greater than {{math|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 {{mvar|λ}} approaching {{math|0}} and those below, all the PDF functions discussed have finite [[support of a function|support]], between   {{sfrac|−1  |{{!}}{{mvar|λ}}{{!}}}}   and   {{sfrac|+1  | {{!}}{{mvar|λ}}{{!}} }} .

Since the Tukey lambda distribution is a [[reflection symmetry|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.<ref>{{cite journal |first1=Brian L. |last1=Joiner |first2=Joan R. |last2=Rosenblatt |year=1971 |title=Some properties of the range in samples from Tukey's symmetric lambda distributions |journal=[[Journal of the American Statistical Association]] |volume=66 |issue=334 |pages=394–399 |doi=10.2307/2283943 |jstor=2283943}}</ref>

==References==
{{reflist|25em}}

==External links==
* {{cite web |title=Tukey-Lambda distribution |publisher=[[National Institute of Standards and Technology|US NIST]] Information Technology Laboratory |department=Gallery of Distributions |series=Engineering Statistics Handbook |at=1.3.6.6.15 |id=EDA 366F |url=http://www.itl.nist.gov/div898/handbook/eda/section3/eda366f.htm}}

{{NIST-PD}}
{{ProbDistributions|continuous-variable}}

[[Category:Continuous distributions]]
[[Category:Probability distributions with non-finite variance]]

Tukey lambda distribution - Revision history

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