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| In [[mathematics]], the '''multivariate gamma function''', Γ<sub>''p''</sub>(·), is a generalization of the [[gamma function]]. It is useful in [[multivariate statistics]], appearing in the [[probability density function]] of the [[Wishart distribution|Wishart]] and [[inverse Wishart distribution]]s.
| | Glazier Alfonzo from Alma, spends time with hobbies and interests for instance snowshoeing, property developers in singapore and creating a house. Will soon undertake a contiki journey that will consist of taking a trip to the Lumbini.<br><br>Here is my web blog: [http://tukarposisi.com/groups/property-news-in-singapore/ tukarposisi.com] |
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| It has two equivalent definitions. One is
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| :<math>
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| \Gamma_p(a)=
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| \int_{S>0} \exp\left(
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| -{\rm trace}(S)\right)
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| \left|S\right|^{a-(p+1)/2}
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| dS ,
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| </math>
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| where S>0 means S is [[positive-definite matrix|positive-definite]]. The other one, more useful in practice, is
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| :<math>
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| \Gamma_p(a)=
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| \pi^{p(p-1)/4}\prod_{j=1}^p
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| \Gamma\left[ a+(1-j)/2\right].
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| </math>
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| From this, we have the recursive relationships:
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| :<math>
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| \Gamma_p(a) = \pi^{(p-1)/2} \Gamma(a) \Gamma_{p-1}(a-\tfrac{1}{2}) = \pi^{(p-1)/2} \Gamma_{p-1}(a) \Gamma[a+(1-p)/2] .
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| </math> | |
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| Thus
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| * <math>\Gamma_1(a)=\Gamma(a)</math>
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| * <math>\Gamma_2(a)=\pi^{1/2}\Gamma(a)\Gamma(a-1/2)</math>
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| * <math>\Gamma_3(a)=\pi^{3/2}\Gamma(a)\Gamma(a-1/2)\Gamma(a-1)</math>
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| and so on.
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| == Derivatives ==
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| We may define the multivariate [[digamma function]] as
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| :<math>\psi_p(a) = \frac{\partial \log\Gamma_p(a)}{\partial a} = \sum_{i=1}^p \psi(a+(1-i)/2) ,</math> | |
| and the general [[polygamma function]] as
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| :<math>\psi_p^{(n)}(a) = \frac{\partial^n \log\Gamma_p(a)}{\partial a^n} = \sum_{i=1}^p \psi^{(n)}(a+(1-i)/2).</math>
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| === Calculation steps ===
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| * Since
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| ::<math>\Gamma_p(a) = \pi^{p(p-1)/4}\prod_{j=1}^p \Gamma(a+\frac{1-j}{2}),</math>
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| :it follows that
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| ::<math>\frac{\partial \Gamma_p(a)}{\partial a} = \pi^{p(p-1)/4}\sum_{i=1}^p \frac{\partial\Gamma(a+\frac{1-i}{2})}{\partial a}\prod_{j=1, j\neq i}^p\Gamma(a+\frac{1-j}{2}).</math>
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| * By definition of the [[digamma function]], ψ,
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| ::<math>\frac{\partial\Gamma(a+(1-i)/2)}{\partial a} = \psi(a+(i-1)/2)\Gamma(a+(i-1)/2)</math>
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| :it follows that
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| ::<math>\frac{\partial \Gamma_p(a)}{\partial a} = \pi^{p(p-1)/4}\prod_{j=1}^p \Gamma(a+(1-j)/2) \sum_{i=1}^p \psi(a+(1-i)/2) = \Gamma_p(a)\sum_{i=1}^p \psi(a+(1-i)/2).</math>
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| {{single source|date=May 2012}}
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| {{inline|date=May 2012}}
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| ==References==
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| * {{cite journal
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| |title=Distributions of Matrix Variates and Latent Roots Derived from Normal Samples
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| |last=James |first=A.
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| |journal=[[Annals of Mathematical Statistics]]
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| |volume=35 |issue=2 |year=1964 |pages=475–501
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| |doi=10.1214/aoms/1177703550 |mr=181057 | zbl = 0121.36605
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| }}
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| [[Category:Gamma and related functions]]
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Glazier Alfonzo from Alma, spends time with hobbies and interests for instance snowshoeing, property developers in singapore and creating a house. Will soon undertake a contiki journey that will consist of taking a trip to the Lumbini.
Here is my web blog: tukarposisi.com