Index: The Book of Statistical ProofsProbability Distributions ▷ Univariate continuous distributions ▷ Gamma distribution ▷ Special case of Wishart distribution

Theorem: The gamma distribution is a special case of the Wishart distribution where the number of columns of the random matrix is $p = 1$.

Proof: Let $X$ be a $p \times p$ positive-definite symmetric matrix, such that $X$ follows a Wishart distribution:

\[\label{eq:wish} Y \sim \mathcal{W}(V, n) \; .\]

Then, $Y$ is described by the probability density function

\[\label{eq:wish-pdf} \begin{split} p(Y) = \frac{1}{\Gamma_p \left( \frac{n}{2} \right)} \cdot \frac{1}{\sqrt{2^{n p} |V|^n}} \cdot |X|^{(n-p-1)/2} \cdot \exp\left[ -\frac{1}{2} \mathrm{tr}\left( V^{-1} X \right) \right] \end{split}\]

where $\lvert A \rvert$ is a matrix determinant, $A^{-1}$ is a matrix inverse and $\Gamma_p(x)$ is the multivariate gamma function of order $p$. If $p = 1$, then $\Gamma_p(x) = \Gamma(x)$ is the ordinary gamma function, $x = X$ and $v = V$ are real numbers. Thus, the probability density function of $x$ can be developed as

\[\label{eq:gam-pdf-s1} \begin{split} p(x) &= \frac{1}{\Gamma\left( \frac{n}{2} \right)} \cdot \frac{1}{\sqrt{2^n \, v^n}} \cdot x^{(n-2)/2} \cdot \exp\left[ -\frac{1}{2} \mathrm{tr}\left( v^{-1} x \right) \right] \\ &= \frac{(2v)^{-n/2}}{\Gamma\left( \frac{n}{2} \right)} \cdot x^{n/2-1} \cdot \exp\left[ -\frac{1}{2v} x \right] \\ \end{split}\]

Finally, substituting $a = \frac{n}{2}$ and $b = \frac{1}{2v}$, we get

\[\label{eq:gam-pdf-s2} p(x) = \frac{b^a}{\Gamma(a)} \, x^{a-1} \, \exp[-b x]\]

which is the probability density function of the gamma distribution.


Metadata: ID: P328 | shortcut: gam-wish | author: JoramSoch | date: 2022-07-14, 07:45.