en>Vaiveahtoish at 17:35, 9 December 2013

2013-12-09T17:35:35Z

← Older revision		Revision as of 19:35, 9 December 2013
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	~~Hi there~~. ~~Let me start by introducing~~ the ~~writer~~, ~~her title~~ is ~~Sophia~~. ~~Invoicing~~ is ~~what I do~~. ~~To perform lacross~~ is ~~something I truly appreciate performing~~. ~~My wife~~ and ~~I reside~~ in ~~Mississippi~~ but ~~now I~~'~~m considering other options~~.<br><br>~~Feel free~~ to ~~visit my web blog accurate psychic readings~~ ([http://www.~~skullrocker~~.~~com~~/~~blogs~~/~~post~~/~~10991~~ www.~~skullrocker~~.com])		{{Probability distribution\|
			name =Multivariate t\|
			type =density\|
			pdf_image =\|
			cdf_image =\|
			notation =<math>t_\nu(\boldsymbol\mu,\boldsymbol\Sigma)</math>\|
			parameters =<math>\boldsymbol\mu = [\mu_1, \dots, \mu_p]^T</math> [[location parameter\|location]] ([[real number\|real]] <math>p\times 1</math> [[random vector\|vector]])<br/><math>\boldsymbol\Sigma</math> [[scale matrix]] ([[positive-definite matrix\|positive-definite]] real <math>p\times p</math> [[matrix (mathematics)\|matrix]]) <br/> <math>\nu</math> is the [[Degrees of freedom (statistics)\|degrees of freedom]] \|
			support =<math>\mathbf{x} \in\mathbb{R}^p\!</math>\|
			pdf =<math>
			\frac{\Gamma\left[(\nu+p)/2\right]}{\Gamma(\nu/2)\nu^{p/2}\pi^{p/2}\left\|{\boldsymbol\Sigma}\right\|^{1/2}\left[1+\frac{1}{\nu}({\mathbf x}-{\boldsymbol\mu})^{\rm T}{\boldsymbol\Sigma}^{-1}({\mathbf x}-{\boldsymbol\mu})\right]^{(\nu+p)/2}}</math>\|
			cdf =No analytic expression\|
			mean =<math>\boldsymbol\mu</math> if <math>\nu > 1</math>; else undefined\|
			median =<math>\boldsymbol\mu</math>\|
			mode =<math>\boldsymbol\mu</math>\|
			variance =<math>\frac{\nu}{\nu-2} \boldsymbol\Sigma</math> if <math>\nu > 2</math>; else undefined\|
			skewness =0\|
			kurtosis =\|
			entropy =\|
			mgf =\|
			char =\|
			}}

			In [[statistics]], the '''multivariate t-distribution''' (or '''multivariate Student distribution''') is a [[multivariate probability distribution]]. It is a generalization to [[random vector]]s of the [[Student's t-distribution]], which is a distribution applicable to univariate [[random variable]]s. While the case of a [[random matrix]] could be treated within this structure, the [[matrix t-distribution]] is distinct and makes particular use of the matrix structure.

			==Definition==
			One common method of construction of a multivariate t distribution, for the case of <math>p</math> dimensions, is based on the observation that if <math>\mathbf y</math> and <math>u</math> are independent and distributed as <math>{\mathcal N}({\mathbf 0},{\boldsymbol\Sigma})</math> and <math>\chi^2_\nu</math> (i.e. [[multivariate normal distribution\|multivariate normal]] and [[chi-squared distribution]]s) respectively, the covariance <math>\mathbf{\Sigma}\,</math> is a ''p'' × ''p'' matrix, and <math>{\mathbf y}\sqrt{\nu/u}={\mathbf x}-{\boldsymbol\mu}</math>, then <math>{\mathbf x}</math> has the density

			:<math>
			\frac{\Gamma\left[(\nu+p)/2\right]}{\Gamma(\nu/2)\nu^{p/2}\pi^{p/2}\left\|{\boldsymbol\Sigma}\right\|^{1/2}\left[1+\frac{1}{\nu}({\mathbf x}-{\boldsymbol\mu})^T{\boldsymbol\Sigma}^{-1}({\mathbf x}-{\boldsymbol\mu})\right]^{(\nu+p)/2}}</math>

			and is said to be distributed as a multivariate t-distribution with parameters <math>{\boldsymbol\Sigma},{\boldsymbol\mu},\nu</math>.

			In the special case <math>\nu=1</math> , the distribution is a [[Cauchy distribution#Multivariate Cauchy distribution\|multivariate Cauchy distribution]].

			==Derivation==

			There are in fact many candidates for the multivariate generalization of [[Student's t-distribution]]. An extensive survey of the field has been given by Kotz and Nadarajah (2004). The essential issue is to define a probability density function of several variables that is the appropriate generalization of the formula for the univariate case. In one dimension (<math>p=1</math>), with <math>t=x-\mu</math> and <math>\Sigma=1</math>, we have the [[probability density function]]
			:<math>f(t) = \frac{\Gamma[(\nu+1)/2]}{\sqrt{\nu\pi\,}\,\Gamma[\nu/2]} (1+t^2/\nu)^{-(\nu+1)/2}</math>
			and one approach is to write down a corresponding function of several variables. This is the basic idea of [[elliptical distribution]] theory, where one writes down a corresponding function of <math>p</math> variables <math>t_i</math> that replaces <math>t^2</math> by a quadratic function of all the <math>t_i</math>. It is clear that this only makes sense when all the marginal distributions have the same [[Degrees of freedom (statistics)\|degrees of freedom]] <math>\nu</math>. With <math> \mathbf{A} = \boldsymbol\Sigma^{-1}</math>, one has a simple choice of multivariate density function

			:<math>f(\mathbf t) = \frac{\Gamma((\nu+p)/2)\left\|\mathbf{A}\right\|^{1/2}}{\sqrt{\nu^p\pi^p\,}\,\Gamma(\nu/2)} (1+\sum_{i,j=1}^{p,p} A_{ij} t_i t_j/\nu)^{-(\nu+p)/2}</math>

			which is the standard but not the only choice.

			An important special case is the standard '''bivariate t-distribution'''{{anchor\|bivariate}}, ''p'' = 2:

			:<math>f(t_1,t_2) = \frac{\left\|\mathbf{A}\right\|^{1/2}}{2\pi} (1+\sum_{i,j=1}^{2,2} A_{ij} t_i t_j/\nu)^{-(\nu+2)/2}</math>

			Note that <math>\frac{\Gamma \left(\frac{\nu +2}{2}\right)}{\pi \ \nu \Gamma \left(\frac{\nu }{2}\right)}= \frac {1} {2\pi}</math>.

			Now, if <math>\mathbf{A}</math> is the identity matrix, the density is

			:<math>f(t_1,t_2) = \frac{1}{2\pi} (1+(t_1^2 + t_2^2)/\nu)^{-(\nu+2)/2}.</math>



			The difficulty with the standard representation is revealed by this formula, which does not factorize into the product of the marginal one-dimensional distributions. When <math> \Sigma</math> is diagonal the standard representation can be shown to have zero [[Pearson product-moment correlation coefficient\|correlation]] but the [[marginal distribution]]s do not agree with [[statistical independence]]. There are differing views on this issue, which is under discussion in the research literature as of early 2007.{{Citation needed\|date=December 2011}}

			==Further theory==
			Many such distributions may be constructed by considering the quotients of normal random variables with the square root of a sample from a chi-squared distribution. These are surveyed in the references and links below.

			==Copulas based on the multivariate t ==
			The use of such distributions is enjoying renewed interest due to applications in [[mathematical finance]], especially through the use of the Student t [[copula (statistics)\|copula]].

			==Related concepts==

			In univariate statistics, the [[Student's t-test]] makes use of [[Student's t-distribution]]. [[Hotelling's T-squared distribution]] is a distribution that arises in multivariate statistics. The [[matrix t-distribution]] is a distribution for random variables arranged in a matrix structure.

			{{inline\|date=May 2012}}
			==References==
			{{refbegin}}
			* {{cite book \|title= Multivariate ''t'' Distributions and Their Applications \|last= Kotz \|first= Samuel \|authorlink= \|coauthors= Nadarajah, Saralees \|year= 2004 \|publisher= Cambridge University Press \|location= \|isbn= 0521826543 \|page= \|pages= \|url= }}
			* {{cite book \|title= Copula methods in finance \|last= Cherubini \|first= Umberto \|authorlink= \|coauthors= Luciano, Elisa; Vecchiato, Walter \|year= 2004 \|publisher= John Wiley & Sons \|location= \|isbn= 0470863447 \|page= \|pages= \|url= }}
			{{refend}}

			==External links==
			*[http://www.mth.kcl.ac.uk/~shaww/web_page/papers/MultiStudentc.pdf Copula Methods vs Canonical Multivariate Distributions: the multivariate Student T distribution with general degrees of freedom]
			*[http://www.statlect.com/mcdstu1.htm Multivariate Student's t distribution]

			{{ProbDistributions\|multivariate}}

			[[Category:Continuous distributions]]
			[[Category:Multivariate continuous distributions]]
			[[Category:Probability distributions]]

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Hi there. Let me start by introducing the writer, her title is Sophia. Invoicing is what I do. To perform lacross is something I truly appreciate performing. My wife and I reside in Mississippi but now I'm considering other options.<br><br>Feel free to visit my web blog accurate psychic readings ([http://www.skullrocker.com/blogs/post/10991 www.skullrocker.com])

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en>Vaiveahtoish at 17:35, 9 December 2013

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