Index: The Book of Statistical Proofs ▷ Statistical Models ▷ Periodic data ▷ Univariate von Mises ▷ Maximum likelihood estimation

Theorem: Let there be a univariate von Mises data set $y = \left\lbrace y_1, \ldots, y_n \right\rbrace$:

\[\label{eq:vm} y_i \sim \mathrm{vM}(\mu, \kappa), \quad i = 1, \ldots, n \; .\]

Then, the maximum likelihood estimates for circular mean $\mu$ and reciprocal dispersion $\kappa$ are given by

\[\label{eq:vm-mle} \begin{split} \hat{\mu} &= \arctan \left( \frac{\sum_{i=1}^n \sin y_i}{\sum_{i=1}^n \cos y_i} \right) \\ \hat{\kappa} &= A^{-1}\left( \frac{1}{n} \sum_{i=1}^n \cos(y_i-\hat{\mu}) \right) \end{split}\]

where $A^{-1}$ is the inverse function of

\[\label{eq:A} A(\kappa) = \frac{I_1(\kappa)}{I_0(\kappa)}\]

with the zeroth- and first-order modified Bessel function of the first kind:

\[\label{eq:I0-I1} \begin{split} I_0(\kappa) &= \frac{1}{2\pi} \int_0^{2\pi} \exp \left[ \kappa \cos(x) \right] \, \mathrm{d}x \\ \text{and} \quad I_1(\kappa) &= \frac{\mathrm{d}}{\mathrm{d}\kappa} I_0(\kappa) \; . \end{split}\]

Proof: The likelihood function for each observation is given by the probability density function of the von Mises distribution

\[\label{eq:vm-yi} p(y_i|\mu,\kappa) = \mathrm{vM}(y_i; \mu, \kappa) = \frac{1}{\sqrt{2 \pi} I_0(\kappa)} \cdot \exp \left[ \kappa \cos(y_i-\mu) \right]\]

and because observations are independent, the likelihood function for all observations is equal to the product of the individual ones:

\[\label{eq:vm-LF} p(y|\mu,\kappa) = \prod_{i=1}^n p(y_i|\mu, \kappa) = \left( \frac{1}{\sqrt{2 \pi} I_0(\kappa)} \right)^n \cdot \exp \left[ \kappa \sum_{i=1}^n \cos(y_i-\mu) \right] \; .\]

Thus, the log-likelihood function is

\[\label{eq:vm-LL} \mathrm{LL}(\mu,\kappa) = \ln p(y|\mu,\kappa) = - n \ln(2 \pi) - n \ln I_0(\kappa) + \kappa \sum_{i=1}^n \cos(y_i-\mu) \; .\]

We will use the following trigonometric identities:

\[\label{eq:sin-cos} \begin{split} \frac{\sin x}{\cos x} &= \tan x \\ \cos(x-y) &= \cos x \cos y + \sin x \sin y \; . \end{split}\]

The derivative of the log-likelihood function \eqref{eq:vm-LL} with respect to $\mu$ is

\[\label{eq:dLL-dmu} \frac{\mathrm{d}\mathrm{LL}(\mu,\kappa)}{\mathrm{d}\mu} = - \kappa \sum_{i=1}^n \sin(y_i-\mu)\]

and setting this derivative to zero gives the MLE for $\mu$:

\[\label{eq:mu-mle} \begin{split} \frac{\mathrm{d}\mathrm{LL}(\hat{\mu},\kappa)}{\mathrm{d}\mu} &= 0 \\ 0 &= - \kappa \sum_{i=1}^n \sin(y_i-\hat{\mu}) \\ 0 &\overset{\eqref{eq:sin-cos}}{=} - \kappa \sum_{i=1}^n \left[ \cos \hat{\mu} \sin y_i - \cos y_i \sin \hat{\mu} \right] \\ 0 &= - \kappa \cos \hat{\mu} \sum_{i=1}^n \sin y_i + \kappa \sin \hat{\mu} \sum_{i=1}^n \cos y_i \\ \kappa \cos \hat{\mu} \sum_{i=1}^n \sin y_i &= \kappa \sin \hat{\mu} \sum_{i=1}^n \cos y_i \\ \frac{\sin \hat{\mu}}{\cos \hat{\mu}} &= \frac{\sum_{i=1}^n \sin y_i}{\sum_{i=1}^n \cos y_i} \\ \tan \hat{\mu} &\overset{\eqref{eq:sin-cos}}{=} \frac{\sum_{i=1}^n \sin y_i}{\sum_{i=1}^n \cos y_i} \\ \hat{\mu} &= \arctan \left( \frac{\sum_{i=1}^n \sin y_i}{\sum_{i=1}^n \cos y_i} \right) \; . \end{split}\]

The derivative of the log-likelihood function \eqref{eq:vm-LL} at $\hat{\mu}$ with respect to $\kappa$ is

\[\label{eq:dLL-dkappa} \begin{split} \frac{\mathrm{d}\mathrm{LL}(\hat{\mu},\kappa)}{\mathrm{d}\kappa} &= - n \frac{\mathrm{d} \ln I_0(\kappa)}{\mathrm{d}\kappa} + \sum_{i=1}^n \cos(y_i-\hat{\mu}) \\ &\overset{\eqref{eq:I0-I1}}{=} - n \frac{I_1(\kappa)}{I_0(\kappa)} + \sum_{i=1}^n \cos(y_i-\hat{\mu}) \end{split}\]

and setting this derivative to zero gives the MLE for $\kappa$:

\[\label{eq:kappa-mle} \begin{split} \frac{\mathrm{d}\mathrm{LL}(\hat{\mu},\hat{\kappa})}{\mathrm{d}\kappa} &= 0 \\ 0 &= - n \frac{I_1(\hat{\kappa})}{I_0(\hat{\kappa})} + \sum_{i=1}^n \cos(y_i-\hat{\mu}) \\ \frac{I_1(\hat{\kappa})}{I_0(\hat{\kappa})} &= \frac{1}{n} \sum_{i=1}^n \cos(y_i-\hat{\mu}) \\ A(\hat{\kappa}) &\overset{\eqref{eq:A}}{=} \frac{1}{n} \sum_{i=1}^n \cos(y_i-\hat{\mu}) \\ \hat{\kappa} &= A^{-1}\left( \frac{1}{n} \sum_{i=1}^n \cos(y_i-\hat{\mu}) \right) \; . \end{split}\]

Together, \eqref{eq:mu-mle} and \eqref{eq:kappa-mle} constitute the MLE for the univariate von Mises.

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Metadata: ID: P537 | shortcut: vm-mle | author: JoramSoch | date: 2026-04-23, 16:52.