Index: The Book of Statistical Proofs ▷ Probability Distributions ▷ Univariate continuous distributions ▷ von Mises distribution ▷ Full width at half maximum

Theorem: Let $X$ be a random variable following a von Mises distribution:

\[\label{eq:vm} X \sim \mathrm{vM}(\mu, \kappa) \; .\]

Then, the full width at half maximum (FWHM) of $X$ is

\[\label{eq:vm-fwhm} \mathrm{FWHM}(X) = 2 \left[ \pi - \arccos\left(1 - \frac{\ln 2}{\kappa} \right) \right] \; .\]

Moreover, the FWHM of $X$ is not defined, if

\[\label{eq:vm-fwhm-nd} \kappa < \frac{1}{2} \ln 2 \approx 0.347 \; .\]

Proof:

1) The probability density function of the von Mises distribution is

\[\label{eq:vm-pdf} f_X(x) = \frac{1}{2 \pi I_0(\kappa)} \cdot \exp \left[ \kappa \cos(x-\mu) \right]\]

and the mode of the von Mises distribution is

\[\label{eq:vm-mode} \mathrm{mode}(X) = \mu \; ,\]

such that

\[\label{eq:vm-pdf-max} f_\mathrm{max} = f_X(\mathrm{mode}(X)) \overset{\eqref{eq:vm-mode}}{=} f_X(\mu) \overset{\eqref{eq:vm-pdf}}{=} \frac{\exp[\kappa]}{2 \pi I_0(\kappa)} \; .\]

The FWHM bounds satisfy the equation

\[\label{eq:x-FHWM} f_X(x_\mathrm{FWHM}) = \frac{1}{2} f_\mathrm{max} \overset{\eqref{eq:vm-pdf-max}}{=} \frac{\exp[\kappa]}{4 \pi I_0(\kappa)} \; .\]

Using \eqref{eq:vm-pdf}, we can develop this equation as follows:

\[\label{eq:x-FHWM-s1} \begin{split} \frac{1}{2 \pi I_0(\kappa)} \cdot \exp \left[ \kappa \cos(x-\mu) \right] &= \frac{\exp[\kappa]}{4 \pi I_0(\kappa)} \\ \exp \left[ \kappa \cos(x-\mu) \right] &= \frac{1}{2} \exp[\kappa] \\ \kappa \cos(x-\mu) &= \kappa + \ln \frac{1}{2} \\ \cos(x-\mu) &= 1 - \frac{\ln 2}{\kappa} \\ x_{\mathrm{FHWM}} &= \arccos\left(1 - \frac{\ln 2}{\kappa} \right) + \mu \; . \end{split}\]

Considering the nature of the arccosine function, this implies the following two solutions for $x_\mathrm{FWHM}$

\[\label{eq:x-FHWM-s2} \begin{split} x_1 &= \arccos\left(1 - \frac{\ln 2}{\kappa} \right) + \mu \\ x_2 &= \left( 2\pi - \arccos\left(1 - \frac{\ln 2}{\kappa} \right) \right) + \mu \; , \end{split}\]

such that the full width at half maximum of $X$ is

\[\label{eq:vm-fwhm-qed} \begin{split} \mathrm{FWHM}(X) &= \Delta x = x_2 - x_1 \\ &\overset{\eqref{eq:x-FHWM-s2}}{=} \left[ \left( 2\pi - \arccos\left(1 - \frac{\ln 2}{\kappa} \right) \right) + \mu \right] - \left[ \arccos\left(1 - \frac{\ln 2}{\kappa} \right) + \mu \right] \\ &= 2 \left[ \pi - \arccos\left(1 - \frac{\ln 2}{\kappa} \right) \right] \; . \end{split}\]

2) Since $\arccos(x)$ is only defined for $-1 \leq x \leq +1$ and since $\ln 2 \approx 0.693$ and $\kappa$ are positive, the following must hold for $\mathrm{FWHM}(X)$ to be defined:

\[\label{eq:vm-fwhm-nd-qed} \begin{split} 1 - \frac{\ln 2}{\kappa} &\geq -1 \\ - \frac{\ln 2}{\kappa} &\geq -2 \\ \ln 2 &\leq 2\kappa \\ \kappa &\geq \frac{1}{2} \ln 2 \; . \end{split}\]

In the case that $\kappa < \frac{1}{2} \ln 2$, the probability density function is so flat that $f_\mathrm{min} = f_X(\mu-\pi)$ exceeds $\frac{1}{2} f_\mathrm{max} = \frac{1}{2} f_X(\mu)$ which means that condition \eqref{eq:x-FHWM} cannot be satisfied, such that the FWHM of $X$ is not defined.

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Metadata: ID: P535 | shortcut: vm-fwhm | author: JoramSoch | date: 2026-04-21, 16:15.