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Tukey lambda distribution
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<h2 id="quantile-function">Quantile function</h2> <p>For the standard form of the Tukey lambda distribution, the quantile function, Q ( p ) , {\displaystyle ~Q(p)~,} (i.e. the inverse function to the <a href="/facts/Cumulative_distribution_function/WaKU8tp4">cumulative distribution function</a>) and the quantile density function, q = d Q d p , {\displaystyle ~q={\frac {\ \operatorname {d} Q\ }{\operatorname {d} p}}\ ,} are </p> Q ( p ; λ ) = { 1 λ [ p λ − ( 1 − p ) λ ] , if λ ≠ 0 , ln ( p 1 − p ) , if λ = 0 . {\displaystyle \ Q\left(\ p\ ;\lambda \ \right)~=~{\begin{cases}{\tfrac {1}{\ \lambda \ }}\left[\ p^{\lambda }-(1-p)^{\lambda }\ \right]\ ,&\ {\mbox{ if }}\ \lambda \neq 0~,\\{}\\\ln \left({\frac {p}{\ 1-p\ }}\right)~,&\ {\mbox{ if }}\ \lambda =0~.\end{cases}}} q ( p ; λ ) = d Q d p = p λ − 1 + ( 1 − p ) λ − 1 . {\displaystyle q\left(\ p\ ;\lambda \ \right)~=~{\frac {\ \operatorname {d} Q\ }{\operatorname {d} p}}~=~p^{\lambda -1}+\left(\ 1-p\ \right)^{\lambda -1}~.} <p>For most values of the shape parameter, λ, the <a href="/facts/Probability_density_function/zvfybna4">probability density function</a> (PDF) and <a href="/facts/Cumulative_distribution_function/WaKU8tp4">cumulative distribution function</a> (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 } (see <a href="/facts/Continuous_uniform_distribution/XbnlVljT">uniform distribution</a> [ cases <i>λ</i> = 1 and <i>λ</i> = 2 ] and the <a href="/facts/Logistic_distribution/QlQNt0uC">logistic distribution</a> [ case <i>λ</i> = 0 ]. </p><p>However, for any value of λ both the CDF and PDF can be tabulated for any number of cumulative probabilities, p, using the quantile function Q to calculate the value x, for each cumulative probability p, with the probability density given by 1/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. </p> <h2 id="moments">Moments</h2> <p>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 <i>λ</i> > − 1 /2 , and except when <i>λ</i> = 0 , is given by the formula </p> Var [ X ] = 2 λ 2 ( 1 1 + 2 λ − Γ ( λ + 1 ) 2 Γ ( 2 λ + 2 ) ) . {\displaystyle \operatorname {Var} [\ X\ ]={\frac {2}{\lambda ^{2}}}{\bigg (}\ {\frac {1}{\ 1+2\lambda \ }}~-~{\frac {\ \Gamma (\lambda +1)^{2}\ }{\ \Gamma (2\lambda +2)\ }}\ {\bigg )}~.} <p>More generally, the n-th order moment is finite when <i>λ</i> > −1 /<i>n</i> and is expressed (except when <i>λ</i> = 0 ) in terms of the <a href="/facts/Beta_function/Dcm1JAxb">beta function</a> <i>Β</i>(<i>x</i>,<i>y</i>) : </p> μ n ≡ E [ X n ] = 1 λ n ∑ k = 0 n ( − 1 ) k ( n k ) B ( λ k + 1 , ( n − k ) λ + 1 ) . {\displaystyle \mu _{n}\equiv \operatorname {E} [\ X^{n}\ ]={\frac {1}{\lambda ^{n}}}\sum _{k=0}^{n}\ (-1)^{k}\ {n \choose k}\ \mathrm {B} (\ \lambda \ k+1\ ,\ (n-k)\ \lambda +1\ )~.} <p>Due to symmetry of the density function, all moments of odd orders, if they exist, are equal to zero. </p> <h2 id="l-moments">L-moments</h2> <p>Differently from the central moments, <a href="/facts/L-moments/xJuzASyy">L-moments</a> can be expressed in a closed form. For λ > − 1 , {\displaystyle \lambda >-1\ ,} the r {\displaystyle \ r} th L-moment, ℓ r , {\displaystyle \ \ell _{r}\ ,} is given by<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> </p> ℓ r = 1 + ( − 1 ) r λ ∑ k = 0 r − 1 ( − 1 ) r − 1 − k ( r − 1 k ) ( r + k − 1 k ) ( 1 k + 1 + λ ) = ( 1 + ( − 1 ) r ) Γ ( 1 + λ ) Γ ( r − 1 − λ ) Γ ( 1 − λ ) Γ ( r + 1 + λ ) . {\displaystyle {\begin{aligned}\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{aligned}}} <p>The first six L-moments can be presented as follows:<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p> ℓ 1 = 0 , {\displaystyle \ell _{1}=~~0\ ,} ℓ 2 = 2 λ [ − 1 1 + λ + 2 2 + λ ] , {\displaystyle \ell _{2}={\frac {2}{\ \lambda \ }}\ \left[\ -{\frac {1}{\ 1+\lambda \ }}+{\frac {2}{\ 2+\lambda \ }}\ \right]\ ,} ℓ 3 = 0 , {\displaystyle \ell _{3}=~~0\ ,} ℓ 4 = 2 λ [ − 1 1 + λ + 12 2 + λ − 30 3 + λ + 20 4 + λ ] , {\displaystyle \ell _{4}={\frac {2}{\ \lambda \ }}\ \left[-{\frac {1}{\ 1+\lambda \ }}+{\frac {12}{\ 2+\lambda \ }}-{\frac {30}{\ 3+\lambda \ }}+{\frac {20}{\ 4+\lambda \ }}\ \right]\ ,} ℓ 5 = 0 , {\displaystyle \ell _{5}=~~0\ ,} ℓ 6 = 2 λ [ − 1 1 + λ + 30 2 + λ − 210 3 + λ + 560 4 + λ − 630 5 + λ + 252 6 + λ ] . {\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]~.} <h2 id="comments">Comments</h2> <p>The Tukey lambda distribution is actually a family of distributions that can approximate a number of common distributions. For example, </p> <table><tbody><tr><td> <i>λ</i> ≈ −1 </td><td>approx. <a href="/facts/Cauchy_distribution/3HS6D3Xk">Cauchy</a> <i>C</i>( 0, <i>π</i> ) </td></tr><tr><td> <i>λ</i> = 0 </td><td>exactly <a href="/facts/Logistic_distribution/QlQNt0uC">logistic</a></td></tr><tr><td> <i>λ</i> ≈ 0.14 </td><td>approx. <a href="/facts/Normal_distribution/UapjjPyQ">normal</a> <i>N</i>( 0, 2.142± ) </td></tr><tr><td> <i>λ</i> = 1 /2 </td><td>strictly <a href="/facts/Concave_function/HxVEBkce">concave</a> ( ∩ {\displaystyle \cap } -shaped)</td></tr><tr><td> <i>λ</i> = 1 </td><td>exactly <a href="/facts/Continuous_uniform_distribution/XbnlVljT">uniform</a> <i>U</i>( −1, +1 ) </td></tr><tr><td> <i>λ</i> = 2 </td><td>exactly <a href="/facts/Continuous_uniform_distribution/XbnlVljT">uniform</a> <i>U</i>( − 1 /2 , + 1 /2 ) </td></tr></tbody></table> <p>The most common use of this distribution is to generate a Tukey lambda <a href="/facts/PPCC_plot/CmumPPza">PPCC plot</a> of a <a href="/facts/Data_set/eDvegUTK">data set</a>. Based on the value for λ with best correlation, as shown on the <a href="/facts/PPCC_plot/CmumPPza">PPCC plot</a>, an appropriate <a href="/facts/Statistical_model/jKkT5ftm">model</a> 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. </p><p>A milepost at <i>λ</i> = 0 (<a href="/facts/Logistic_distribution/QlQNt0uC">logistic</a>) would indicate quite fat tails, with the extreme limit at <i>λ</i> = −1 , approximating <a href="/facts/Cauchy_distribution/3HS6D3Xk">Cauchy</a> and small sample versions of the <a href="/facts/Student%2527s_t_distribution/DeT1SDqH">Student's t</a>. 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 <i>exceptionally</i> thin tails (based on the point of view that the normal distribution itself is thin-tailed to begin with; the <a href="/facts/Exponential_distribution/yY3EdV0u">exponential distribution</a> is often chosen as the exemplar of tails intermediate between fat and thin). </p><p>Except for values of λ approaching 0 and those below, all the PDF functions discussed have finite <a href="/facts/Support_of_a_function/BTuMGbsb">support</a>, between −1 /|λ| and +1 / |λ| . </p><p>Since the Tukey lambda distribution is a <a href="/facts/Reflection_symmetry/eSeR8nbD">symmetric</a> distribution, the use of the Tukey lambda PPCC plot to determine a reasonable distribution to model the data only applies to symmetric distributions. A <a href="/facts/Histogram/oMmvqGEA">histogram</a> of the data should provide evidence as to whether the data can be reasonably modeled with a symmetric distribution.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> <h2 id="external-links">External links</h2> <ul><li><a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda366f.htm">"Tukey-Lambda distribution"</a>. Gallery of Distributions. Engineering Statistics Handbook. <a href="/facts/National_Institute_of_Standards_and_Technology/gr9Fnh85">US NIST</a> Information Technology Laboratory. 1.3.6.6.15. EDA 366F.</li></ul> <p> This article incorporates <a href="/facts/Copyright_status_of_works_by_the_federal_government_of_the_United_States/Q1jvr96g">public domain material</a> from the <a href="https://www.nist.gov">National Institute of Standards and Technology</a> </p> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Karvanen, Juha; Nuutinen, Arto (2008). "Characterizing the generalized lambda distribution by L-moments". Computational Statistics & Data Analysis. 52 (4): 1971–1983. arXiv:math/0701405. doi:10.1016/j.csda.2007.06.021. S2CID 939977. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Karvanen, Juha; Nuutinen, Arto (2008). "Characterizing the generalized lambda distribution by L-moments". Computational Statistics & Data Analysis. 52 (4): 1971–1983. arXiv:math/0701405. doi:10.1016/j.csda.2007.06.021. S2CID 939977. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Joiner, Brian L.; Rosenblatt, Joan R. (1971). "Some properties of the range in samples from Tukey's symmetric lambda distributions". Journal of the American Statistical Association. 66 (334): 394–399. doi:10.2307/2283943. JSTOR 2283943. <a href="/wiki/Journal_of_the_American_Statistical_Association" target="_blank">/wiki/Journal_of_the_American_Statistical_Association</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> </ol>