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| {{Probability distribution
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| | type = density
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| | pdf_image =
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| | cdf_image =
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| | notation = Tukey(''λ'')
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| | parameters = ''λ'' ∈ '''R''' — [[shape parameter]]
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| | support = ''x'' ∈ [−1/''λ'', 1/''λ''] for ''λ'' > 0,<br/>''x'' ∈ '''R''' for ''λ'' ≤ 0
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| | pdf = <math>(Q(p;\lambda)\,,Q'(p;\lambda)^{-1}),\, 0\leq\,p\,\leq\,1</math>
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| | cdf = <math>(e^{-x}+1)^{-1},\,\,\lambda\,=\,0</math>
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| | mean = <math>0,\,\,\lambda > -1</math>
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| | median = 0
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| | mode = 0
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| | variance = <math>\frac{2}{\lambda^2}\bigg(\frac{1}{1+2\lambda}-\frac{\Gamma(\lambda+1)^2}{\Gamma(2\lambda+2)}\bigg),\,\,\lambda > -1/2</math><br/><math>\frac{ \pi^{2} }{ 3 },\,\,\lambda\,=\,0</math>
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| | skewness = <math>0,\,\,\lambda > -1/3</math>
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| | kurtosis = <math>\frac{(2\lambda+1)^2}{2(4\lambda+1)} \frac{ g_2^2\big(3g_2^2-4g_1g_3+g_4\big)}{g_4\big(g_1^2-g_2\big)^2} - 3,</math><br/><math> 1.2,\,\,\lambda\,=\,0,</math> where ''g''<sub>''k''</sub> = Γ(''kλ''+1) and ''λ'' > -1/4.
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| | entropy = <math>h(\lambda) = \int_0^1 \log (Q'(p;\lambda))\,dp</math><ref>{{Citation |last1=Vasicek |first1=Oldrich |year=1976 |title=A Test for Normality Based on Sample Entropy |journal=Journal of the Royal Statistical Society, Series B |volume=38 |issue=1 |pages=54–59 |postscript=. }}</ref>
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| | mgf =
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| | cf = <math>\phi(t;\lambda) = \int_0^1 \exp (\,i t\,Q(p;\lambda))\,dp</math><ref>{{Citation |last1=Shaw |first1=W. T. |last2=McCabe |first2=J. |year=2009 |title=Monte Carlo sampling given a Characteristic Function: Quantile Mechanics in Momentum Space |journal=Eprint-arXiv:0903,1592 }}</ref>
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| }}
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| 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 model]]s directly.
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| 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.
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| ==Quantile function==
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| 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
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| :<math>
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| Q\left(p;\lambda\right) =
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| \begin{cases}
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| \frac{ 1 }{ \lambda } \left[p^\lambda - (1 - p)^\lambda\right], & \mbox{if } \lambda \ne 0 \\
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| \log(\frac{p}{1-p}), & \mbox{if } \lambda = 0,
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| \end{cases}</math>
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| :<math>Q'\left(p;\lambda\right) = p^{(\lambda-1)} + \left(1-p\right)^{(\lambda-1)}.</math>
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| 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.
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| ==Moments==
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| The Tukey lambda distribution is symmetric around zero, therefore the expected value of this distribution is equal to zero. The variance exists for {{nowrap|''λ'' > −½}} and is given by the formula (except when ''λ'' = 0)
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| : <math>
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| \operatorname{Var}[X] = \frac{2}{\lambda^2}\bigg(\frac{1}{1+2\lambda} - \frac{\Gamma(\lambda+1)^2}{\Gamma(2\lambda+2)}\bigg).
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| </math>
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| More generally, the ''n''-th order moment is finite when {{nowrap|''λ'' > −1/''n''}} and is expressed in terms of the [[beta function]] ''Β''(''x'',''y'') (except when ''λ'' = 0) :
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| : <math>
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| \mu_n = \operatorname{E}[X^n] = \frac{1}{\lambda^n} \sum_{k=0}^n (-1)^k {n \choose k}\, \Beta(\lambda k+1,\, \lambda(n-k)+1 ).
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| </math>
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| Note that due to symmetry of the density function, all moments of odd orders are equal to zero.
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| ==Comments==
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| The Tukey lambda distribution is actually a family of distributions that can approximate a number of common distributions. For example,
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| {| class="wikitable"
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| |-
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| | ''λ'' = −1
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| | approx. [[Cauchy distribution|Cauchy]] ''C''(0,''π'')
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| |-
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| | ''λ'' = 0
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| | exactly [[logistic distribution|logistic]]
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| |-
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| | ''λ'' = 0.14
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| | approx. [[normal distribution|normal]] ''N''(0, 2.142)
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| |-
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| | ''λ'' = 0.5
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| | strictly [[concave function|concave]] (<math>\cap</math>-shaped)
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| |-
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| | ''λ'' = 1
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| | exactly [[continuous uniform distribution|uniform]] ''U''(−1, 1)
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| |-
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| | ''λ'' = 2
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| | exactly [[continuous uniform distribution|uniform]] ''U''(−½, ½)
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| |}
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| The most common use of this distribution is to generate a Tukey lambda [[PPCC plot]] of a [[data set]]. Based on the PPCC plot, an appropriate [[statistical model|model]] for the data is suggested. For example, if the maximum [[correlation]] occurs for a value of ''λ'' at or near 0.14, then the data can be modeled with a normal distribution. Values of ''λ'' less than this imply a heavy-tailed distribution (with −1 approximating a Cauchy). That is, as the optimal value of lambda goes from 0.14 to −1, increasingly heavy tails are implied. Similarly, as the optimal value of ''λ'' becomes greater than 0.14, shorter tails are implied.
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| 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>{{Citation| title=Some Properties of the Range in Samples from Tukey's Symmetric Lambda Distributions| first1=Brian L. |last1=Joiner|first2=Joan R. |last2=Rosenblatt| journal=Journal of the American Statistical Association| volume=66 |issue=334 |year=1971| pages=394–399| doi=10.2307/2283943| jstor=2283943}}</ref>
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| ==References==
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| {{Reflist}}
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| ==External links==
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| *[http://www.itl.nist.gov/div898/handbook/eda/section3/eda366f.htm Tukey-Lambda Distribution]
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| {{NIST-PD}}
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| {{ProbDistributions|continuous-variable}}
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| [[Category:Continuous distributions]]
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| [[Category:Probability distributions with non-finite variance]]
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| [[Category:Probability distributions]]
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Not much to write about me at all.
Hurrey Im here and a part of this community.
I really wish I am useful in some way here.
Feel free to surf to my homepage; backup plugin