Index: The Book of Statistical Proofs ▷ Probability Distributions ▷ Univariate continuous distributions ▷ t-distribution ▷ Special case of multivariate t-distribution

Theorem: The t-distribution is a special case of the multivariate t-distribution with number of variables $n = 1$, i.e. random vector $x \in \mathbb{R}$, mean $\mu = 0$ and covariance matrix $\Sigma = 1$.

Proof: The probability density function of the multivariate t-distribution is

\[\label{eq:mvt-pdf} t(x; \mu, \Sigma, \nu) = \sqrt{\frac{1}{(\nu \pi)^{n} |\Sigma|}} \, \frac{\Gamma([\nu+n]/2)}{\Gamma(\nu/2)} \, \left[ 1 + \frac{1}{\nu} (x-\mu)^\mathrm{T} \Sigma^{-1} (x-\mu) \right]^{-(\nu+n)/2} \; .\]

Setting $n = 1$, such that $x \in \mathbb{R}$, as well as $\mu = 0$ and $\Sigma = 1$, we obtain

\[\label{eq:t-pdf} \begin{split} t(x; 0, 1, \nu) &= \sqrt{\frac{1}{(\nu \pi)^{1} |1|}} \, \frac{\Gamma([\nu+1]/2)}{\Gamma(\nu/2)} \, \left[ 1 + \frac{1}{\nu} (x-0)^\mathrm{T} 1^{-1} (x-0) \right]^{-(\nu+1)/2} \\ &= \sqrt{\frac{1}{\nu \pi}} \, \frac{\Gamma([\nu+1]/2)}{\Gamma(\nu/2)} \, \left[ 1 + \frac{x^2}{\nu} \right]^{-(\nu+1)/2} \\ &= \frac{1}{\sqrt{\nu \pi}} \cdot \frac{\Gamma\left( \frac{\nu+1}{2} \right)}{\Gamma\left( \frac{\nu}{2} \right)} \cdot \left[ 1 + \frac{x^2}{\nu} \right]^{-\frac{\nu+1}{2}} \; . \end{split}\]

which is equivalent to the probability density function of the t-distribution.

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Metadata: ID: P332 | shortcut: t-mvt | author: JoramSoch | date: 2022-08-25, 12:38.